Research ArticleThe Exact Controllability of the Molecular Graph Networks

Liang Wei 12 Faxu Li 3 Haixing Zhao 3 Bo Deng2 and Zhonglin Ye 3

1School of Computer Science Shaanxi Normal University Xirsquoan 710062 China2School of Mathematics and Statistics Qinghai Normal University Xining 810008 China3College of Computer Qinghai Normal University Xining 810008 China

Correspondence should be addressed to Faxu Li lifaxuqhnueducn

Received 30 July 2019 Accepted 18 October 2019 Published 30 November 2019

Guest Editor Jia-Bao Liu

Copyright copy 2019 Liang Wei et al -is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper we mainly study the exact controllability of several types of extended-path molecular graph networks Based on theconstruction of extended-path molecular graph networks with C3 C4 or K4 using the determinant operation of the matrix andthe recursive method the exact characteristic polynomial of these networks is deduced According to the definition of minimumdriver node number ND the exact controllability of these networks is obtained Moreover we give the minimum driver node setsof partial small networks

1 Introduction

In recent years the research on complex network is one ofthe hottest research fields and controllability of complexnetwork is a challenging problem in modern network sci-ence such as molecular graph networks computer net-works social networks biological networks andtransportation networks [1ndash8] -e study of networkstructure has always been a basic object in complex networkresearch -e majority of core issues of research in manyfields are based on the structure of the networks such asconnectivity robustness and controllability of networks

In chemical graph theory the vertices represent eachatom and the edges represent the bonds between them in themolecule and the correspondingmolecular graph representsdifferent chemical structures when they represent differentthings Figure 1(a) shows the representation of two smallmolecular structures As the molecular scale increases themolecular graph structure gradually becomes a complexsystem an example as shown in Figure 1(b) Generallyresearchers investigate the related problems of complexsystems with large molecular structure by using methods ofstudying complex networks In addition molecular graphscan be used to express some topological properties ofmolecules -e topological properties of molecules with the

same connected structure should be identical -e quantitydescribing the topological properties of molecules is topo-logical invariants of molecular graphs such as number ofnodes number of edges shortest distance and spectrum Inorder to quantify the topological properties of moleculesresearchers proposed a lot of indexes such as Wiener indexRandic index Hosoya index MerrifieldndashSimmons indexand molecular information index

With the development of network science people beginto pay much attention to study how to control some nodes inthe network to achieve the desirable goals namely makingthe network controllable -e structural controllability ofmolecular graphs is studied as a molecular topological indexLin firstly gives the concept of structural controllability inthe literature [9] and presents the structural controllabilitytheorem In the literature [10] the structural controllabilitytheorem is applied to the directed and unweighted complexnetworks and the maximum matching theory and theminimum input theorem of structural controllability areproposed

However when the edges of the networks are un-directed or weighted the previous theory is not suitable forthis type of networks -en researchers present anothertheory framework which is the exact controllabilityanalysis of the complex network Yuan et al [11] proves

HindawiJournal of ChemistryVolume 2019 Article ID 6073905 7 pageshttpsdoiorg10115520196073905

that the number of driver nodes of a network is equal to themaximum geometric multiplicity and gives a method tosolve the minimum set of driver nodes In [12] Wu et alconstructs a star-type network and gives its characteristicpolynomial which directly expresses the exact controlla-bility of this kind of complex networks Furthermore re-search on controllability of multilayer networks and fractalnetworks which is common in the chemical system hassome results [13 14]

In chemistry hexagonal chain and polyomino chain aretwo kinds of classical linear systems In order to observedifferent linear systems we consider the controllability ofcomplex networks with linear dynamics A network with Nnodes is described by the following set of ordinary differ-ential equations [15]

_x Ax + Bu (1)

where x (x1 x2 x3 xN)T is the status of nodesA isin RNtimesN denotes the coupling matrix of the system inwhich aij represents the weight of a link (i j) andu (u1 u2 u3 uM)T is the status ofM controllers and Bis the control matrix with the size of N times M In classiccontrol theory the system (1) can be controlled from anyinitial state to any final state with finite time if and only if

rank B AB A2B A

Nminus 1B1113960 11139611113872 1113873 N (2)

Generally solving the controllability problems is toconstruct a suitable control matrix B which consists of aminimum set of driver nodes for satisfying the Kalman rankcondition [16] And there are many possible control matricesB for system (1) which satisfies the controllable condition-e main task of exact controllability is to find a set of Bcorresponding to the minimum number ND of controllerswhich can control the whole network -en it is importantto measure the exact controllability of the system (1) For

undirected graphs the system controllability can be given bythe following terms

ND maxi

δ λi( 11138571113864 1113865 (3)

where δ(λi) is the algebraic multiplicity of eigenvalue ofmatrix A It is well known that the theory of graph spectra isrelated to the chemistry through the HMO (Huckel Mo-lecular Orbital) theory [1] At an early stage it was supposedthat HMO theory could be reduced to the study of graphspectra of the adjacency matrix of molecular graphs

In subsequent sections we mainly discuss the undirectedgraph with link unweight Firstly we give some constructionmethods of the extended-path molecular graph networks Byusing the controllability theory in Sections 3 and 4 we thenanalyze the exact controllability of these networks andcharacterize all minimum sets of driver nodes of partialsmall networks by simulation

2 Construction of the Extended-Path Networks

In real world we can find a class of regular networks [17ndash19]which is constructed like a path network and each com-ponent looks like a star circle or clique In order to carry outour research we must construct some networks with suchcharacteristics

Firstly we give the construction method of the H-ex-tended-path networks (H-EPN) which is composed ofseveral subgraphs pasted together by one common node Asshown in Figure 2(b) H-EPN is composed of subgraph Hthe number ofH isN and it satisfies that every two subgraphare connected by one common node In other words eachedge of the path (Figure 2(a)) can be extended by a subgraphH which can be connected to each other at a common nodeWe likewise construct several types of H-EPNs as shown inFigures 3(a)ndash3(d)

H

H

HH

H

H

H

H

H

H

H H

CCC

CCC

CC O

(a)

HO

HO

HO

HO

HO

HO

HO

HO

HO

HO HO

HO

OH OH

OH

OH

OOOO

OO

O

O OO

O

O

OOO O O

OO

O

O

OH

OH

OH

OHOH

OH

OHOHOH

(b)

Figure 1 Transformation from molecular structure to molecular graph network Representation of (a) two small molecular structures and(b) a large molecular structure

2 Journal of Chemistry

3 Main Results

In this section we will study the exact controllability ofseveral extended-path networks which are extended by C3C4 and K4

-e adjacent matrix of C3-EPN can be formalized asfollows

A

0 1 1

1 0 1

1 1 0 1 1

1 0 1

1 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4)

-erefore we can obtain the characteristic polynomial ofthe coupling matrix A of C3-EPN with 2N + 1 nodes Forconvenience we use B2N+1 to represent the determinant ofmatrix λI minus A as follows

|λI minus A| B2N+1

λ minus 1 minus 1

minus 1 λ minus 1

minus 1 minus 1 λ minus 1 minus 1

minus 1 λ minus 1

minus 1 minus 1 λ

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

λ2 minus 11113872 1113873B2Nminus 1 minus 2(λ + 1)C2Nminus 2

(5)

where

C2Nminus 2

λ minus 1

minus 1 λ minus 1 minus 1

minus 1 λ minus 1

minus 1 minus 1 λ

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(6)

-us it follows that

B2N+1 λ2 minus 11113872 1113873B2Nminus 1 minus 2(λ + 1)C2Nminus 2 (7)

C2Nminus 2 λB2Nminus 3 + λC2Nminus 4 (8)

According to the equations (7) and (8) we have

B2N+1 minus λ2 + λ minus 11113872 1113873B2Nminus 1 + λ3 + 2λ2 + λ1113872 1113873B2Nminus 3 0

(9)

For convenience of calculation take SN B2N+1 -en

SNminus 1 B2Nminus 1

SNminus 2 B2Nminus 3(10)

So we haveSN minus λ2 + λ minus 11113872 1113873SNminus 1 + λ3 + 2λ2 + λ1113872 1113873SNminus 2 0 (11)

According to the recursive method in the theory of thedeterminant the characteristic equation of recursive re-lations [11] is given by

m2

minus λ2 + λ minus 11113872 1113873m + λ3 + 2λ2 + λ1113872 1113873 0 (12)

It is easy to get

m1 λ2 + λ minus 11113872 1113873 + p

2

m2 λ2 + λ minus 11113872 1113873 minus p

2

(13)

where p λ4 minus 2λ3 minus 9λ2 minus 6λ + 1

1113968

-us we give the form of solution of SN in the followingexpression

SN k1mN1 + k2m

N2 (14)

For S0 B1 λ and S1 B3 λ3 minus 3λ minus 2 we have k1 +

k2 λ and k1m1 + k2m2 λ3 minus 3λ minus 2

So we get

(a)

H H

(b)

Figure 2 Construction of extended-path networks (a) Path networks (b) Extended-path networks

(a) (b)

(c) (d)

Figure 3 H-EPN extended by C3 C4 andK4 (a) C3-EPN (b) C4-EPN1 (c) C4-EPN2 (d) K4-EPN

Journal of Chemistry 3

k1 λ3 minus 3λ minus 2 minus λm2

m1 minus m21p

λ3 minus 3λ minus 2 minus λm21113872 1113873

k2 λ3 minus 3λ minus 2 minus λm1

m1 minus m21p

λ3 minus 3λ minus 2 minus λm11113872 1113873

(15)

After an arrangement of equations (13)ndash(15) we have

SN k1mN1 + k2m

N2

12p

(λ + 1)21113876(λ minus 2) m

N1 minus m

N21113872 1113873

minus λ2 mNminus 11 minus m

Nminus 121113872 11138731113877

(16)According to equations (5) and (10) the characteristic

polynomial of matrix A is as follows|λI minus A| B2N+1 SN

12p

(λ + 1)2

(λ minus 2) mN1 minus m

N21113872 1113873 minus λ2 m

Nminus 11 minus m

Nminus 121113872 11138731113960 1113961

(17)Let g(λ) (λ minus 2)(mN

1 minus mN2 ) minus λ2(mNminus 1

1 minus mNminus 12 ) then

it is easy to get that g(minus 1)ne 0 g(1)ne 0 and g(0)ne 0 So themultiplicity of eigenvalue minus 1 is 2 Now we discuss themultiplicity of the remaining eigenvalues We all know fromequations (7) and (9) that if there is an eigenvalue λ0 makingB2N+1 0 and suppose that C2Nminus 2 0 then it must beB2Nminus 1 0 and B2Nminus 3 0 (note that 1 and 0 are not roots ofthe B2N+1 0) which is a contradiction with λ0 being theroot of all B2i+1(λ0) 0 (i 1 2 N)

Now we show that there is a principal minor determinantof order 2N for B2N+1 such that the value is not 0 as follows

Aprime

λ minus 1

minus 1 λ

λ minus 1

minus 1 λ minus 1 minus 1

minus 1 λ minus 1

minus 1 minus 1 λ

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

λ minus 1

minus 1 λ

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868C2Nminus 2

(18)which is constructed by deleting the 3th row and the 3th volume

of B2N+1 For C2Nminus 2 ne 0 and λ minus 1minus 1 λ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868ne 0 when λne minus 1 we

have Aprime ne 0 By B2N+1 0 and Aprime ne 0 in the case of λne minus 1 wesee that the rank of the corresponding matrix of B2N+1 is 2N-en for all the eigenvalues λne minus 1 its multiplicity is 1 So weknow that all roots of B2N+1 0 are distinct except minus 1

According to the above analysis it can be known that themaximum multiplicity of the solution of the equation (17) is2-erefore we get the minimum number of driver nodes inthe C3-EPN as follows

ND 2 (19)

Using the same way we investigate the exact control-lability of the C4-EPN1 -e adjacent matrix A of theC4-EPN1 can be formalized as follows

A

0 1 0 1

1 0 1 0

0 1 0 1

1 0 1 0 1 0 1

1 0 1 0

0 1 0 1

1 0 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

We obtain the characteristic polynomial of the matrix Aas follows

|λI minus A|

λ minus 1 0 minus 1

minus 1 λ minus 1 0

0 minus 1 λ minus 1

minus 1 0 minus 1 λ minus 1 0 minus 1

minus 1 λ minus 1 0

0 minus 1 λ minus 1

minus 1 0 minus 1 λ

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(21)

Let B3N+1 |λI minus A| and taking a series of determinantoperations one gets the form as follows

B3N+1 |λI minus A| λ2C3Nminus 1 minus 2λB3Nminus 2 (22)

C3Nminus 1 λB3Nminus 2 minus 2D3Nminus 3 (23)

D3Nminus 3 λ2 minus 11113872 1113873B3Nminus 5 minus λD3Nminus 6 (24)

where C3Nminus 1 and D3Nminus 3 are determinants with the size of(3N minus 1) times (3N minus 1) and (3N minus 3) times (3N minus 3) respectivelyWe can formalise them as follows

C3Nminus 1

λ minus 1

minus 2 λ minus 1 0 minus 1

minus 1 λ minus 1 0

0 minus 1 λ minus 1

minus 1 0 minus 1 λ

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

D3Nminus 3

λ minus 1 0

minus 1 λ minus 1

0 minus 1 λ minus 1 0 minus 1

minus 1 λ minus 1 0

0 minus 1 λ minus 1

minus 1 0 minus 1 λ

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(25)

-rough the equations (22)ndash(24) we get the followingform of recurrence relationship

4 Journal of Chemistry

B3N+1 λ3 minus 3λ1113872 1113873B3Nminus 2 minus λ4B3Nminus 5 (26)

Now let us define SN B3N+1 SNminus 1 B3Nminus 2 andSNminus 2 B3Nminus 5 then the equation (26) can be formalized as

SN λ3 minus 3λ1113872 1113873SNminus 1 minus λ4SNminus 2 (27)

According to the above definitions and equation (27) wehave S0 B1 λ and S1 B4 λ2(λ2 minus 4) so

S2 λ3 minus 3λ1113872 1113873S1 minus λ4S0 λ3 λ4 minus 8λ2 + 121113872 1113873 λ3f2(λ)

(28)

where fi(x) represents the i-th polynomial about x forwhich the lowest order is 0 Fox example f0(λ) 1f1(λ) λ2 minus 4 and f2(λ) λ4 minus 8λ2 + 12 Obviously wecan get remaining items formalized as

S3 λ3 minus 3λ1113872 1113873S2 minus λ4S1 λ4f3(λ)

S4 λ3 minus 3λ1113872 1113873S3 minus λ4S2 λ5f4(λ)

SN λ3 minus 3λ1113872 1113873SNminus 1 minus λ4SNminus 2 λN+1fN(λ)

(29)

Using a simple induction and computation we can easilyget (29) -at is to say the lowest orders of polynomials S0S1 S2 and SN are 1 2 3 and N + 1 respectively andthe multiplicity of the eigenvalue 0 of SN is N + 1 Since aconnected graph with diameter D has at least D + 1 distincteigenvalues noticing that the network C4-EPN1 withN cellshas 3N + 1 nodes and the diameter is N + 2 we have thatC4-EPN1 has at least N + 3 distinct eigenvalues Howeverthe eigenvalue 0 has a multiplicity of N + 1 and theremaining 2N eigenvalues have at least N + 2 distinctnamely the maximummultiplicity of these eigenvalues shallnot exceed N minus 1 -erefore the maximum multiplicity ofC4-EPN1rsquos eigenvalues is N + 1 (the eigenvalue is 0) -usthe minimum number of driver nodes of the system cor-responding to the graph C4-EPN1 is as follows

ND N + 1 (30)

By the analytical method of the preceding part we canobtain the recursive relations of characteristic polynomial ofC4-EPN2 and K4-EPN (see (31) and (32)) as follows

SN λ3 minus 4λ1113872 1113873SNminus 1 minus λ4SNminus 2 λN+1fN(λ) (31)

SN (λ + 1) λ2 minus λ minus 41113872 1113873SNminus 1 minus (λ + 1)4SNminus 2

(λ + 1)N+2

fN(λ + 1)(32)

Using a method similar to that of the C3-EPN andC4-EPN1 it is easy to get that the minimum number ofdriver nodes of C4-EPN2 and K4-EPN shown in Table 1

4 Characterization of Driver Nodes

-e topology of the complex network determines thecontrollability and the minimum driver nodes canquantitatively reflect the controllability of the network

structures -erefore the minimum driver nodes are thekey factors in the structural controllability analysis Infact evaluating the exact controllability of complexnetworks is to obtain the minimum driver nodes ofnetworks

Using the method based on the PBH rank criterion[11 20] we can characterize the minimum driver node set indetail -e condition of the full rank of the matrixλI minus A B1113858 1113859 is

(1) Matrix λI minus A avoids zero vector(2) Matrix λI minus A avoids repetitive vector(3) Matrix λI minus A avoids correlation vector

Because of the symmetry of graph structures the min-imum driver nodes set is not unique and its number ismainly determined by three factors such as zero row vectorrepetitive row vector and correlation row vector of [λI minus A]

Let zero vector have x rows repetitive row vector have ygroups in which each group has r1 r2 ry rows and thecorrelation row vectors have z groups in which each grouphas c1 c2 cz rows then the number of the minimumdriver nodes ND and the number of groups obtained NC areshown in Table 2

Based on the above conditions we design a correspondingalgorithm to obtain the minimum driver nodes set -roughsimulation analysis of Matlab platform we compute thecontrollability and describe all sets ofminimal driver nodes ofC3-EPN C4-EPN and K4-EPN(N 4 5) respectively theyare shown in Tables 3 4 and 5 H-EPNs of N 4 are shownin Figure 4

-e simulation results show that in order to fullycontrol these extended-path networks the driver nodesselected are usually nodes with small degree rather thanthe maximum degree nodes -e point is consistent withthe conclusion in the literature [11] And we can obtainfrom Table 3 that the C3-EPN has a fixed number ofminimum driver node set which can be fully controlled byminimal cost Namely the C3-EPN has higher control-lability In addition although the structures of theC4-EPN1 and C4-EPN2 are different we find that thecontrollability of the two networks is exactly the same indifferent network scales

Table 1 ND of extended-path networks

Networks Number of cells n ND

C3-EPN N 2N + 1 2C4-EPN1 N 3N + 1 N + 1C4-EPN2 N 3N + 1 N + 1K4-EPN N 3N + 1 N + 2

Table 2 ND and NC of three kinds of situations

Case ND NC

Zero row vector x 1Repetitive row vector 1113936

y

i1(ri minus 1) 1113937y

i1(Cri minus 1ri

)

Correlation row vector z 1113937zj1(C1

cj)

Journal of Chemistry 5

5 Conclusion

To summarize we mainly study the controllability of severaltypes of the extended-path molecular graph networks -estructural characteristics of these networks are linearly con-nected by a cycle or complete graph For these special network

structures we derive the general formula of the characteristicpolynomial of the networkrsquos adjacent matrix Furthermore theexact controllability of these kinds of networks is analyzed andsome results are obtained In addition in order to directly reflectthe concrete minimum driver node set of each network wedetermine the distribution of the nodes by simulation

Table 3 Controllability of C3-EPN C4-EPN and K4-EPN (N 4 5)

Networks Number of cell (N) Number of nodes (n) Minimum driver node (ND) Controllability (NDn)

C3-EPN 4 9 2 02222C4-EPN1 4 13 5 03846C4-EPN2 4 13 5 03846K4-EPN 4 13 6 04615C3-EPN 5 11 2 01818C4-EPN1 5 16 6 03750C4-EPN2 5 16 6 03750K4-EPN 5 16 7 04375

Table 4 Minimum driver nodes set of C3-EPN C4-EPN and K4-EPN (N 4)

Networks Minimum driver node setC3-EPN (n1 n2)| n1 isin 1 2 n2 isin 8 9

C4-EPN1

(1 n1 n2 n3 n4)|

n1 isin 2 3 n2 isin 5 6 n3 isin 8 9 n4 isin 11 12 or(n1 n2 n3 n4 13)| n1 isin 2 3

n2 isin 5 6 n3 isin 8 9 n4 isin 11 12

C4-EPN2 ( n1 10 11 12 13)| n1 isin 1 3 or (1 2 3 4 n1)| n1

isin 11 13

K4-EPN

(n1 n2 n3 n4 n5 n6)| n1 isin 1 2 3 n2 isin

1 2 3 n1ne n2 n3 isin 5 6 n4 isin 8 9 n5isin 11 12 13 n6 isin 11 12 13 n5ne n6

Table 5 Minimum driver node set of C3-EPN C4-EPN and K4-EPN (N 5)

Networks Minimum driver node setC3-EPN (n1 n2)| n1 isin 1 2 n2 isin 10 11

C4-EPN1 (1 n1 n2 n3 n4 n5)| n1 isin 2 3 n2 isin 5 6 n3 isin

8 9 n4 isin 11 12 n5 isin 14 15 n3 isin 8 9 n4 isin 11 12 n5 isin 14 15

C4-EPN2 (n1 12 13 14 15 16)| n1 isin 1 3 or (1 2 3 4 5 n1)| n2 isin 14 16

K4-EPN(n1 n2 n3 n4 n5 n6 n7)| n1 isin 1 2 3 n2 isin 1 2 3 n1ne n2 n3 isin 5 6 n4 isin 8 9

n5 isin 11 12 n6 isin 14 15 16 n7 isin 14 15 16 n6ne n7

1

2

3

4

5 9

6

7

8

(a)

1

2 3

4

5 96

7

8

10

11 12

13

(b)

1

2

3

4

5

96

8

7 10

11

12

13

(c)

1

2

34

5

967

8

10

11

12

13

(d)

Figure 4 C3-EPN C4-EPN and K4-EPN (N 4)

6 Journal of Chemistry

Data Availability

All data models and codes generated or used during thestudy are available from the corresponding author uponrequest

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

-is work was supported by the Nature Science Foundationfrom Qinghai Province (No 2018-ZJ-718) the Tibetan In-formation Processing and Machine Translation Key Labo-ratory of Qinghai Province and the Key Laboratory ofTibetan Information Processing Ministry of Education

References

[1] A Graovac I Gutman and N Trinajstic Topological Ap-proach to the Chemistry of Conjugated Molecules Springer-Verlag Berlin Germany 1977

[2] A Lombardi and M Hornquist ldquoControllability analysis ofnetworksrdquo Physical Review E vol 75 no 5 Article ID 0561102007

[3] X F Wang and G R Chen ldquoPinning control of scale-freedynamical networksrdquo Physica A Statistical Mechanics and itsApplications vol 310 no 3-4 pp 521ndash531 2002

[4] B Liu T Chu L Wang and G Xie ldquoControllability of aleader-follower dynamic network with switching topologyrdquoIEEE Transactions on Automatic Control vol 53 no 4pp 1009ndash1013 2008

[5] A Rahmani M Ji M Mesbahi and M Egerstedt ldquoCon-trollability of multi-agent systems from a graph-theoreticperspectiverdquo SIAM Journal on Control and Optimizationvol 48 no 1 pp 162ndash186 2009

[6] G-R Chen ldquoProblems and challenges in control theory undercomplex dynamical network environmentsrdquoActa AutomaticaSinica vol 39 no 4 pp 312ndash321 2013

[7] G Chen ldquoPinning control and synchronization on complexdynamical networksrdquo International Journal of Control Au-tomation and Systems vol 12 no 2 pp 221ndash230 2014

[8] L Hou S Y Lao Y D Xiao et al ldquoRecent progressioncontrollability of complex networkrdquo Acta Physica Sinicavol 64 no 18 pp 188901_1ndash188901_11 2015 (in Chinese)

[9] C T Lin ldquoStructural controllabilityrdquo IEEE Trans AutomaticControl vol 19 no 3 pp 201ndash208 1974

[10] Y-Y Liu J-J Slotine and A-L Barabasi ldquoControllability ofcomplex networksrdquo Nature vol 473 no 7346 pp 167ndash1732011

[11] Z Z Yuan C Zhao Z R Di et al ldquoExact controllability ofcomplex networksrdquo Nature Communications vol 4 no 1p 2447 2013

[12] L P Wu J K Zhou Z Z Yuan et al ldquoExact controllability ofa star-type networkrdquo in Proceedings of the 27th ChineseControl and Decision Conference (2015 CCDC) QingdaoChina May 2015 pp 5187-5191 in Chinese

[13] Z Yuan C Zhao W-X Wang Z Di and Y-C Lai ldquoExactcontrollability of multiplex networksrdquoNew Journal of Physicsvol 16 no 10 Article ID 103036 2014

[14] J Li Z Yuan Y Fan W-X Wang and Z Di ldquoControllabilityof fractal networks an analytical approachrdquo EPL (EurophysicsLetters) vol 105 no 5 Article ID 58001 2014

[15] J J Slotine and W Li Applied Nonlinear Control Prentice-Hall Upper Saddle River NJ USA 1991

[16] R E Kalman ldquoMathematical description of linear dynamicalsystemsrdquo Journal of the Society for Industrial and AppliedMathematics Series A Control vol 1 no 2 pp 152ndash192 1963

[17] A Shafieinejad F Hendessi and F Fekri ldquoStar-structurenetwork coding for multiple unicast sessions in wireless meshnetworksrdquo Wireless Personal Communications vol 72 no 4pp 2185ndash2214 2013

[18] A Jain A Vyas and H Yadav ldquoAnalysis of ethernet load onvarious nodes of star network using opnetrdquo InternationalJournal of Computer Science and Applications (TIJCSA) vol 3no 1 2014

[19] X J Li and J M Xu ldquoFault-tolerance of (n k) star networkrdquoAppliedMathematics and Computation vol 248 pp 525ndash5302014

[20] X K Chen Y Lu and L Y Chen ldquoNetwork structuralcontrollability analysis based on PBHrdquo Journal of PLA Uni-versity of Science and Technology (Natural Science Edition)

Journal of Chemistry 7

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Submit your manuscripts atwwwhindawicom

3 Main Results

In this section we will study the exact controllability ofseveral extended-path networks which are extended by C3C4 and K4

-e adjacent matrix of C3-EPN can be formalized asfollows

A

0 1 1

1 0 1

1 1 0 1 1

1 0 1

1 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(4)

-erefore we can obtain the characteristic polynomial ofthe coupling matrix A of C3-EPN with 2N + 1 nodes Forconvenience we use B2N+1 to represent the determinant ofmatrix λI minus A as follows

|λI minus A| B2N+1

λ minus 1 minus 1

minus 1 λ minus 1

minus 1 minus 1 λ minus 1 minus 1

minus 1 λ minus 1

minus 1 minus 1 λ

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

λ2 minus 11113872 1113873B2Nminus 1 minus 2(λ + 1)C2Nminus 2

(5)

where

C2Nminus 2

λ minus 1

minus 1 λ minus 1 minus 1

minus 1 λ minus 1

minus 1 minus 1 λ

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(6)

-us it follows that

B2N+1 λ2 minus 11113872 1113873B2Nminus 1 minus 2(λ + 1)C2Nminus 2 (7)

C2Nminus 2 λB2Nminus 3 + λC2Nminus 4 (8)

According to the equations (7) and (8) we have

B2N+1 minus λ2 + λ minus 11113872 1113873B2Nminus 1 + λ3 + 2λ2 + λ1113872 1113873B2Nminus 3 0

(9)

For convenience of calculation take SN B2N+1 -en

SNminus 1 B2Nminus 1

SNminus 2 B2Nminus 3(10)

So we haveSN minus λ2 + λ minus 11113872 1113873SNminus 1 + λ3 + 2λ2 + λ1113872 1113873SNminus 2 0 (11)

According to the recursive method in the theory of thedeterminant the characteristic equation of recursive re-lations [11] is given by

m2

minus λ2 + λ minus 11113872 1113873m + λ3 + 2λ2 + λ1113872 1113873 0 (12)

It is easy to get

m1 λ2 + λ minus 11113872 1113873 + p

2

m2 λ2 + λ minus 11113872 1113873 minus p

2

(13)

where p λ4 minus 2λ3 minus 9λ2 minus 6λ + 1

1113968

-us we give the form of solution of SN in the followingexpression

SN k1mN1 + k2m

N2 (14)

For S0 B1 λ and S1 B3 λ3 minus 3λ minus 2 we have k1 +

k2 λ and k1m1 + k2m2 λ3 minus 3λ minus 2

So we get

(a)

H H

(b)

Figure 2 Construction of extended-path networks (a) Path networks (b) Extended-path networks

(a) (b)

(c) (d)

Figure 3 H-EPN extended by C3 C4 andK4 (a) C3-EPN (b) C4-EPN1 (c) C4-EPN2 (d) K4-EPN

Journal of Chemistry 3

k1 λ3 minus 3λ minus 2 minus λm2

m1 minus m21p

λ3 minus 3λ minus 2 minus λm21113872 1113873

k2 λ3 minus 3λ minus 2 minus λm1

m1 minus m21p

λ3 minus 3λ minus 2 minus λm11113872 1113873

(15)

After an arrangement of equations (13)ndash(15) we have

SN k1mN1 + k2m

N2

12p

(λ + 1)21113876(λ minus 2) m

N1 minus m

N21113872 1113873

minus λ2 mNminus 11 minus m

Nminus 121113872 11138731113877

(16)According to equations (5) and (10) the characteristic

polynomial of matrix A is as follows|λI minus A| B2N+1 SN

12p

(λ + 1)2

(λ minus 2) mN1 minus m

N21113872 1113873 minus λ2 m

Nminus 11 minus m

Nminus 121113872 11138731113960 1113961

(17)Let g(λ) (λ minus 2)(mN

1 minus mN2 ) minus λ2(mNminus 1

1 minus mNminus 12 ) then

it is easy to get that g(minus 1)ne 0 g(1)ne 0 and g(0)ne 0 So themultiplicity of eigenvalue minus 1 is 2 Now we discuss themultiplicity of the remaining eigenvalues We all know fromequations (7) and (9) that if there is an eigenvalue λ0 makingB2N+1 0 and suppose that C2Nminus 2 0 then it must beB2Nminus 1 0 and B2Nminus 3 0 (note that 1 and 0 are not roots ofthe B2N+1 0) which is a contradiction with λ0 being theroot of all B2i+1(λ0) 0 (i 1 2 N)

Now we show that there is a principal minor determinantof order 2N for B2N+1 such that the value is not 0 as follows

Aprime

λ minus 1

minus 1 λ

λ minus 1

minus 1 λ minus 1 minus 1

minus 1 λ minus 1

minus 1 minus 1 λ

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

λ minus 1

minus 1 λ

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868C2Nminus 2

(18)which is constructed by deleting the 3th row and the 3th volume

of B2N+1 For C2Nminus 2 ne 0 and λ minus 1minus 1 λ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868ne 0 when λne minus 1 we

have Aprime ne 0 By B2N+1 0 and Aprime ne 0 in the case of λne minus 1 wesee that the rank of the corresponding matrix of B2N+1 is 2N-en for all the eigenvalues λne minus 1 its multiplicity is 1 So weknow that all roots of B2N+1 0 are distinct except minus 1

According to the above analysis it can be known that themaximum multiplicity of the solution of the equation (17) is2-erefore we get the minimum number of driver nodes inthe C3-EPN as follows

ND 2 (19)

Using the same way we investigate the exact control-lability of the C4-EPN1 -e adjacent matrix A of theC4-EPN1 can be formalized as follows

A

0 1 0 1

1 0 1 0

0 1 0 1

1 0 1 0 1 0 1

1 0 1 0

0 1 0 1

1 0 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

We obtain the characteristic polynomial of the matrix Aas follows

|λI minus A|

λ minus 1 0 minus 1

minus 1 λ minus 1 0

0 minus 1 λ minus 1

minus 1 0 minus 1 λ minus 1 0 minus 1

minus 1 λ minus 1 0

0 minus 1 λ minus 1

minus 1 0 minus 1 λ

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(21)

Let B3N+1 |λI minus A| and taking a series of determinantoperations one gets the form as follows

B3N+1 |λI minus A| λ2C3Nminus 1 minus 2λB3Nminus 2 (22)

C3Nminus 1 λB3Nminus 2 minus 2D3Nminus 3 (23)

D3Nminus 3 λ2 minus 11113872 1113873B3Nminus 5 minus λD3Nminus 6 (24)

where C3Nminus 1 and D3Nminus 3 are determinants with the size of(3N minus 1) times (3N minus 1) and (3N minus 3) times (3N minus 3) respectivelyWe can formalise them as follows

C3Nminus 1

λ minus 1

minus 2 λ minus 1 0 minus 1

minus 1 λ minus 1 0

0 minus 1 λ minus 1

minus 1 0 minus 1 λ

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

D3Nminus 3

λ minus 1 0

minus 1 λ minus 1

0 minus 1 λ minus 1 0 minus 1

minus 1 λ minus 1 0

0 minus 1 λ minus 1

minus 1 0 minus 1 λ

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(25)

-rough the equations (22)ndash(24) we get the followingform of recurrence relationship

4 Journal of Chemistry

B3N+1 λ3 minus 3λ1113872 1113873B3Nminus 2 minus λ4B3Nminus 5 (26)

Now let us define SN B3N+1 SNminus 1 B3Nminus 2 andSNminus 2 B3Nminus 5 then the equation (26) can be formalized as

SN λ3 minus 3λ1113872 1113873SNminus 1 minus λ4SNminus 2 (27)

According to the above definitions and equation (27) wehave S0 B1 λ and S1 B4 λ2(λ2 minus 4) so

S2 λ3 minus 3λ1113872 1113873S1 minus λ4S0 λ3 λ4 minus 8λ2 + 121113872 1113873 λ3f2(λ)

(28)

where fi(x) represents the i-th polynomial about x forwhich the lowest order is 0 Fox example f0(λ) 1f1(λ) λ2 minus 4 and f2(λ) λ4 minus 8λ2 + 12 Obviously wecan get remaining items formalized as

S3 λ3 minus 3λ1113872 1113873S2 minus λ4S1 λ4f3(λ)

S4 λ3 minus 3λ1113872 1113873S3 minus λ4S2 λ5f4(λ)

SN λ3 minus 3λ1113872 1113873SNminus 1 minus λ4SNminus 2 λN+1fN(λ)

(29)

Using a simple induction and computation we can easilyget (29) -at is to say the lowest orders of polynomials S0S1 S2 and SN are 1 2 3 and N + 1 respectively andthe multiplicity of the eigenvalue 0 of SN is N + 1 Since aconnected graph with diameter D has at least D + 1 distincteigenvalues noticing that the network C4-EPN1 withN cellshas 3N + 1 nodes and the diameter is N + 2 we have thatC4-EPN1 has at least N + 3 distinct eigenvalues Howeverthe eigenvalue 0 has a multiplicity of N + 1 and theremaining 2N eigenvalues have at least N + 2 distinctnamely the maximummultiplicity of these eigenvalues shallnot exceed N minus 1 -erefore the maximum multiplicity ofC4-EPN1rsquos eigenvalues is N + 1 (the eigenvalue is 0) -usthe minimum number of driver nodes of the system cor-responding to the graph C4-EPN1 is as follows

ND N + 1 (30)

By the analytical method of the preceding part we canobtain the recursive relations of characteristic polynomial ofC4-EPN2 and K4-EPN (see (31) and (32)) as follows

SN λ3 minus 4λ1113872 1113873SNminus 1 minus λ4SNminus 2 λN+1fN(λ) (31)

SN (λ + 1) λ2 minus λ minus 41113872 1113873SNminus 1 minus (λ + 1)4SNminus 2

(λ + 1)N+2

fN(λ + 1)(32)

Using a method similar to that of the C3-EPN andC4-EPN1 it is easy to get that the minimum number ofdriver nodes of C4-EPN2 and K4-EPN shown in Table 1

4 Characterization of Driver Nodes

-e topology of the complex network determines thecontrollability and the minimum driver nodes canquantitatively reflect the controllability of the network

structures -erefore the minimum driver nodes are thekey factors in the structural controllability analysis Infact evaluating the exact controllability of complexnetworks is to obtain the minimum driver nodes ofnetworks

Using the method based on the PBH rank criterion[11 20] we can characterize the minimum driver node set indetail -e condition of the full rank of the matrixλI minus A B1113858 1113859 is

(1) Matrix λI minus A avoids zero vector(2) Matrix λI minus A avoids repetitive vector(3) Matrix λI minus A avoids correlation vector

Because of the symmetry of graph structures the min-imum driver nodes set is not unique and its number ismainly determined by three factors such as zero row vectorrepetitive row vector and correlation row vector of [λI minus A]

Let zero vector have x rows repetitive row vector have ygroups in which each group has r1 r2 ry rows and thecorrelation row vectors have z groups in which each grouphas c1 c2 cz rows then the number of the minimumdriver nodes ND and the number of groups obtained NC areshown in Table 2

Based on the above conditions we design a correspondingalgorithm to obtain the minimum driver nodes set -roughsimulation analysis of Matlab platform we compute thecontrollability and describe all sets ofminimal driver nodes ofC3-EPN C4-EPN and K4-EPN(N 4 5) respectively theyare shown in Tables 3 4 and 5 H-EPNs of N 4 are shownin Figure 4

-e simulation results show that in order to fullycontrol these extended-path networks the driver nodesselected are usually nodes with small degree rather thanthe maximum degree nodes -e point is consistent withthe conclusion in the literature [11] And we can obtainfrom Table 3 that the C3-EPN has a fixed number ofminimum driver node set which can be fully controlled byminimal cost Namely the C3-EPN has higher control-lability In addition although the structures of theC4-EPN1 and C4-EPN2 are different we find that thecontrollability of the two networks is exactly the same indifferent network scales

Table 1 ND of extended-path networks

Networks Number of cells n ND

C3-EPN N 2N + 1 2C4-EPN1 N 3N + 1 N + 1C4-EPN2 N 3N + 1 N + 1K4-EPN N 3N + 1 N + 2

Table 2 ND and NC of three kinds of situations

Case ND NC

Zero row vector x 1Repetitive row vector 1113936

y

i1(ri minus 1) 1113937y

i1(Cri minus 1ri

)

Correlation row vector z 1113937zj1(C1

cj)

Journal of Chemistry 5

5 Conclusion

To summarize we mainly study the controllability of severaltypes of the extended-path molecular graph networks -estructural characteristics of these networks are linearly con-nected by a cycle or complete graph For these special network

structures we derive the general formula of the characteristicpolynomial of the networkrsquos adjacent matrix Furthermore theexact controllability of these kinds of networks is analyzed andsome results are obtained In addition in order to directly reflectthe concrete minimum driver node set of each network wedetermine the distribution of the nodes by simulation

Table 3 Controllability of C3-EPN C4-EPN and K4-EPN (N 4 5)

Networks Number of cell (N) Number of nodes (n) Minimum driver node (ND) Controllability (NDn)

C3-EPN 4 9 2 02222C4-EPN1 4 13 5 03846C4-EPN2 4 13 5 03846K4-EPN 4 13 6 04615C3-EPN 5 11 2 01818C4-EPN1 5 16 6 03750C4-EPN2 5 16 6 03750K4-EPN 5 16 7 04375

Table 4 Minimum driver nodes set of C3-EPN C4-EPN and K4-EPN (N 4)

Networks Minimum driver node setC3-EPN (n1 n2)| n1 isin 1 2 n2 isin 8 9

C4-EPN1

(1 n1 n2 n3 n4)|

n1 isin 2 3 n2 isin 5 6 n3 isin 8 9 n4 isin 11 12 or(n1 n2 n3 n4 13)| n1 isin 2 3

n2 isin 5 6 n3 isin 8 9 n4 isin 11 12

C4-EPN2 ( n1 10 11 12 13)| n1 isin 1 3 or (1 2 3 4 n1)| n1

isin 11 13

K4-EPN

(n1 n2 n3 n4 n5 n6)| n1 isin 1 2 3 n2 isin

1 2 3 n1ne n2 n3 isin 5 6 n4 isin 8 9 n5isin 11 12 13 n6 isin 11 12 13 n5ne n6

Table 5 Minimum driver node set of C3-EPN C4-EPN and K4-EPN (N 5)

Networks Minimum driver node setC3-EPN (n1 n2)| n1 isin 1 2 n2 isin 10 11

C4-EPN1 (1 n1 n2 n3 n4 n5)| n1 isin 2 3 n2 isin 5 6 n3 isin

8 9 n4 isin 11 12 n5 isin 14 15 n3 isin 8 9 n4 isin 11 12 n5 isin 14 15

C4-EPN2 (n1 12 13 14 15 16)| n1 isin 1 3 or (1 2 3 4 5 n1)| n2 isin 14 16

K4-EPN(n1 n2 n3 n4 n5 n6 n7)| n1 isin 1 2 3 n2 isin 1 2 3 n1ne n2 n3 isin 5 6 n4 isin 8 9

n5 isin 11 12 n6 isin 14 15 16 n7 isin 14 15 16 n6ne n7

1

2

3

4

5 9

6

7

8

(a)

1

2 3

4

5 96

7

8

10

11 12

13

(b)

1

2

3

4

5

96

8

7 10

11

12

13

(c)

1

2

34

5

967

8

10

11

12

13

(d)

Figure 4 C3-EPN C4-EPN and K4-EPN (N 4)

6 Journal of Chemistry

Data Availability

All data models and codes generated or used during thestudy are available from the corresponding author uponrequest

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

-is work was supported by the Nature Science Foundationfrom Qinghai Province (No 2018-ZJ-718) the Tibetan In-formation Processing and Machine Translation Key Labo-ratory of Qinghai Province and the Key Laboratory ofTibetan Information Processing Ministry of Education

References

[1] A Graovac I Gutman and N Trinajstic Topological Ap-proach to the Chemistry of Conjugated Molecules Springer-Verlag Berlin Germany 1977

[2] A Lombardi and M Hornquist ldquoControllability analysis ofnetworksrdquo Physical Review E vol 75 no 5 Article ID 0561102007

[3] X F Wang and G R Chen ldquoPinning control of scale-freedynamical networksrdquo Physica A Statistical Mechanics and itsApplications vol 310 no 3-4 pp 521ndash531 2002

[4] B Liu T Chu L Wang and G Xie ldquoControllability of aleader-follower dynamic network with switching topologyrdquoIEEE Transactions on Automatic Control vol 53 no 4pp 1009ndash1013 2008

[5] A Rahmani M Ji M Mesbahi and M Egerstedt ldquoCon-trollability of multi-agent systems from a graph-theoreticperspectiverdquo SIAM Journal on Control and Optimizationvol 48 no 1 pp 162ndash186 2009

[6] G-R Chen ldquoProblems and challenges in control theory undercomplex dynamical network environmentsrdquoActa AutomaticaSinica vol 39 no 4 pp 312ndash321 2013

[7] G Chen ldquoPinning control and synchronization on complexdynamical networksrdquo International Journal of Control Au-tomation and Systems vol 12 no 2 pp 221ndash230 2014

[8] L Hou S Y Lao Y D Xiao et al ldquoRecent progressioncontrollability of complex networkrdquo Acta Physica Sinicavol 64 no 18 pp 188901_1ndash188901_11 2015 (in Chinese)

[9] C T Lin ldquoStructural controllabilityrdquo IEEE Trans AutomaticControl vol 19 no 3 pp 201ndash208 1974

[10] Y-Y Liu J-J Slotine and A-L Barabasi ldquoControllability ofcomplex networksrdquo Nature vol 473 no 7346 pp 167ndash1732011

[11] Z Z Yuan C Zhao Z R Di et al ldquoExact controllability ofcomplex networksrdquo Nature Communications vol 4 no 1p 2447 2013

[12] L P Wu J K Zhou Z Z Yuan et al ldquoExact controllability ofa star-type networkrdquo in Proceedings of the 27th ChineseControl and Decision Conference (2015 CCDC) QingdaoChina May 2015 pp 5187-5191 in Chinese

[13] Z Yuan C Zhao W-X Wang Z Di and Y-C Lai ldquoExactcontrollability of multiplex networksrdquoNew Journal of Physicsvol 16 no 10 Article ID 103036 2014

[14] J Li Z Yuan Y Fan W-X Wang and Z Di ldquoControllabilityof fractal networks an analytical approachrdquo EPL (EurophysicsLetters) vol 105 no 5 Article ID 58001 2014

[15] J J Slotine and W Li Applied Nonlinear Control Prentice-Hall Upper Saddle River NJ USA 1991

[16] R E Kalman ldquoMathematical description of linear dynamicalsystemsrdquo Journal of the Society for Industrial and AppliedMathematics Series A Control vol 1 no 2 pp 152ndash192 1963

[17] A Shafieinejad F Hendessi and F Fekri ldquoStar-structurenetwork coding for multiple unicast sessions in wireless meshnetworksrdquo Wireless Personal Communications vol 72 no 4pp 2185ndash2214 2013

[18] A Jain A Vyas and H Yadav ldquoAnalysis of ethernet load onvarious nodes of star network using opnetrdquo InternationalJournal of Computer Science and Applications (TIJCSA) vol 3no 1 2014

[19] X J Li and J M Xu ldquoFault-tolerance of (n k) star networkrdquoAppliedMathematics and Computation vol 248 pp 525ndash5302014

[20] X K Chen Y Lu and L Y Chen ldquoNetwork structuralcontrollability analysis based on PBHrdquo Journal of PLA Uni-versity of Science and Technology (Natural Science Edition)

Journal of Chemistry 7

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2018

Bioinorganic Chemistry and ApplicationsHindawiwwwhindawicom Volume 2018

SpectroscopyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Medicinal ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

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Biochemistry Research International

Hindawiwwwhindawicom Volume 2018

Enzyme Research

Hindawiwwwhindawicom Volume 2018

Journal of

SpectroscopyAnalytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

MaterialsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

BioMed Research International Electrochemistry

International Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

k1 λ3 minus 3λ minus 2 minus λm2

m1 minus m21p

λ3 minus 3λ minus 2 minus λm21113872 1113873

k2 λ3 minus 3λ minus 2 minus λm1

m1 minus m21p

λ3 minus 3λ minus 2 minus λm11113872 1113873

(15)

After an arrangement of equations (13)ndash(15) we have

SN k1mN1 + k2m

N2

12p

(λ + 1)21113876(λ minus 2) m

N1 minus m

N21113872 1113873

minus λ2 mNminus 11 minus m

Nminus 121113872 11138731113877

(16)According to equations (5) and (10) the characteristic

polynomial of matrix A is as follows|λI minus A| B2N+1 SN

12p

(λ + 1)2

(λ minus 2) mN1 minus m

N21113872 1113873 minus λ2 m

Nminus 11 minus m

Nminus 121113872 11138731113960 1113961

(17)Let g(λ) (λ minus 2)(mN

1 minus mN2 ) minus λ2(mNminus 1

1 minus mNminus 12 ) then

it is easy to get that g(minus 1)ne 0 g(1)ne 0 and g(0)ne 0 So themultiplicity of eigenvalue minus 1 is 2 Now we discuss themultiplicity of the remaining eigenvalues We all know fromequations (7) and (9) that if there is an eigenvalue λ0 makingB2N+1 0 and suppose that C2Nminus 2 0 then it must beB2Nminus 1 0 and B2Nminus 3 0 (note that 1 and 0 are not roots ofthe B2N+1 0) which is a contradiction with λ0 being theroot of all B2i+1(λ0) 0 (i 1 2 N)

Now we show that there is a principal minor determinantof order 2N for B2N+1 such that the value is not 0 as follows

Aprime

λ minus 1

minus 1 λ

λ minus 1

minus 1 λ minus 1 minus 1

minus 1 λ minus 1

minus 1 minus 1 λ

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

λ minus 1

minus 1 λ

111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868C2Nminus 2

(18)which is constructed by deleting the 3th row and the 3th volume

of B2N+1 For C2Nminus 2 ne 0 and λ minus 1minus 1 λ

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868ne 0 when λne minus 1 we

have Aprime ne 0 By B2N+1 0 and Aprime ne 0 in the case of λne minus 1 wesee that the rank of the corresponding matrix of B2N+1 is 2N-en for all the eigenvalues λne minus 1 its multiplicity is 1 So weknow that all roots of B2N+1 0 are distinct except minus 1

According to the above analysis it can be known that themaximum multiplicity of the solution of the equation (17) is2-erefore we get the minimum number of driver nodes inthe C3-EPN as follows

ND 2 (19)

Using the same way we investigate the exact control-lability of the C4-EPN1 -e adjacent matrix A of theC4-EPN1 can be formalized as follows

A

0 1 0 1

1 0 1 0

0 1 0 1

1 0 1 0 1 0 1

1 0 1 0

0 1 0 1

1 0 1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

We obtain the characteristic polynomial of the matrix Aas follows

|λI minus A|

λ minus 1 0 minus 1

minus 1 λ minus 1 0

0 minus 1 λ minus 1

minus 1 0 minus 1 λ minus 1 0 minus 1

minus 1 λ minus 1 0

0 minus 1 λ minus 1

minus 1 0 minus 1 λ

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(21)

Let B3N+1 |λI minus A| and taking a series of determinantoperations one gets the form as follows

B3N+1 |λI minus A| λ2C3Nminus 1 minus 2λB3Nminus 2 (22)

C3Nminus 1 λB3Nminus 2 minus 2D3Nminus 3 (23)

D3Nminus 3 λ2 minus 11113872 1113873B3Nminus 5 minus λD3Nminus 6 (24)

where C3Nminus 1 and D3Nminus 3 are determinants with the size of(3N minus 1) times (3N minus 1) and (3N minus 3) times (3N minus 3) respectivelyWe can formalise them as follows

C3Nminus 1

λ minus 1

minus 2 λ minus 1 0 minus 1

minus 1 λ minus 1 0

0 minus 1 λ minus 1

minus 1 0 minus 1 λ

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

D3Nminus 3

λ minus 1 0

minus 1 λ minus 1

0 minus 1 λ minus 1 0 minus 1

minus 1 λ minus 1 0

0 minus 1 λ minus 1

minus 1 0 minus 1 λ

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868111386811138681113868

(25)

-rough the equations (22)ndash(24) we get the followingform of recurrence relationship

4 Journal of Chemistry

B3N+1 λ3 minus 3λ1113872 1113873B3Nminus 2 minus λ4B3Nminus 5 (26)

Now let us define SN B3N+1 SNminus 1 B3Nminus 2 andSNminus 2 B3Nminus 5 then the equation (26) can be formalized as

SN λ3 minus 3λ1113872 1113873SNminus 1 minus λ4SNminus 2 (27)

According to the above definitions and equation (27) wehave S0 B1 λ and S1 B4 λ2(λ2 minus 4) so

S2 λ3 minus 3λ1113872 1113873S1 minus λ4S0 λ3 λ4 minus 8λ2 + 121113872 1113873 λ3f2(λ)

(28)

where fi(x) represents the i-th polynomial about x forwhich the lowest order is 0 Fox example f0(λ) 1f1(λ) λ2 minus 4 and f2(λ) λ4 minus 8λ2 + 12 Obviously wecan get remaining items formalized as

S3 λ3 minus 3λ1113872 1113873S2 minus λ4S1 λ4f3(λ)

S4 λ3 minus 3λ1113872 1113873S3 minus λ4S2 λ5f4(λ)

SN λ3 minus 3λ1113872 1113873SNminus 1 minus λ4SNminus 2 λN+1fN(λ)

(29)

Using a simple induction and computation we can easilyget (29) -at is to say the lowest orders of polynomials S0S1 S2 and SN are 1 2 3 and N + 1 respectively andthe multiplicity of the eigenvalue 0 of SN is N + 1 Since aconnected graph with diameter D has at least D + 1 distincteigenvalues noticing that the network C4-EPN1 withN cellshas 3N + 1 nodes and the diameter is N + 2 we have thatC4-EPN1 has at least N + 3 distinct eigenvalues Howeverthe eigenvalue 0 has a multiplicity of N + 1 and theremaining 2N eigenvalues have at least N + 2 distinctnamely the maximummultiplicity of these eigenvalues shallnot exceed N minus 1 -erefore the maximum multiplicity ofC4-EPN1rsquos eigenvalues is N + 1 (the eigenvalue is 0) -usthe minimum number of driver nodes of the system cor-responding to the graph C4-EPN1 is as follows

ND N + 1 (30)

By the analytical method of the preceding part we canobtain the recursive relations of characteristic polynomial ofC4-EPN2 and K4-EPN (see (31) and (32)) as follows

SN λ3 minus 4λ1113872 1113873SNminus 1 minus λ4SNminus 2 λN+1fN(λ) (31)

SN (λ + 1) λ2 minus λ minus 41113872 1113873SNminus 1 minus (λ + 1)4SNminus 2

(λ + 1)N+2

fN(λ + 1)(32)

Using a method similar to that of the C3-EPN andC4-EPN1 it is easy to get that the minimum number ofdriver nodes of C4-EPN2 and K4-EPN shown in Table 1

4 Characterization of Driver Nodes

-e topology of the complex network determines thecontrollability and the minimum driver nodes canquantitatively reflect the controllability of the network

structures -erefore the minimum driver nodes are thekey factors in the structural controllability analysis Infact evaluating the exact controllability of complexnetworks is to obtain the minimum driver nodes ofnetworks

Using the method based on the PBH rank criterion[11 20] we can characterize the minimum driver node set indetail -e condition of the full rank of the matrixλI minus A B1113858 1113859 is

(1) Matrix λI minus A avoids zero vector(2) Matrix λI minus A avoids repetitive vector(3) Matrix λI minus A avoids correlation vector

Because of the symmetry of graph structures the min-imum driver nodes set is not unique and its number ismainly determined by three factors such as zero row vectorrepetitive row vector and correlation row vector of [λI minus A]

Let zero vector have x rows repetitive row vector have ygroups in which each group has r1 r2 ry rows and thecorrelation row vectors have z groups in which each grouphas c1 c2 cz rows then the number of the minimumdriver nodes ND and the number of groups obtained NC areshown in Table 2

Based on the above conditions we design a correspondingalgorithm to obtain the minimum driver nodes set -roughsimulation analysis of Matlab platform we compute thecontrollability and describe all sets ofminimal driver nodes ofC3-EPN C4-EPN and K4-EPN(N 4 5) respectively theyare shown in Tables 3 4 and 5 H-EPNs of N 4 are shownin Figure 4

-e simulation results show that in order to fullycontrol these extended-path networks the driver nodesselected are usually nodes with small degree rather thanthe maximum degree nodes -e point is consistent withthe conclusion in the literature [11] And we can obtainfrom Table 3 that the C3-EPN has a fixed number ofminimum driver node set which can be fully controlled byminimal cost Namely the C3-EPN has higher control-lability In addition although the structures of theC4-EPN1 and C4-EPN2 are different we find that thecontrollability of the two networks is exactly the same indifferent network scales

Table 1 ND of extended-path networks

Networks Number of cells n ND

C3-EPN N 2N + 1 2C4-EPN1 N 3N + 1 N + 1C4-EPN2 N 3N + 1 N + 1K4-EPN N 3N + 1 N + 2

Table 2 ND and NC of three kinds of situations

Case ND NC

Zero row vector x 1Repetitive row vector 1113936

y

i1(ri minus 1) 1113937y

i1(Cri minus 1ri

)

Correlation row vector z 1113937zj1(C1

cj)

Journal of Chemistry 5

5 Conclusion

To summarize we mainly study the controllability of severaltypes of the extended-path molecular graph networks -estructural characteristics of these networks are linearly con-nected by a cycle or complete graph For these special network

structures we derive the general formula of the characteristicpolynomial of the networkrsquos adjacent matrix Furthermore theexact controllability of these kinds of networks is analyzed andsome results are obtained In addition in order to directly reflectthe concrete minimum driver node set of each network wedetermine the distribution of the nodes by simulation

Table 3 Controllability of C3-EPN C4-EPN and K4-EPN (N 4 5)

Networks Number of cell (N) Number of nodes (n) Minimum driver node (ND) Controllability (NDn)

C3-EPN 4 9 2 02222C4-EPN1 4 13 5 03846C4-EPN2 4 13 5 03846K4-EPN 4 13 6 04615C3-EPN 5 11 2 01818C4-EPN1 5 16 6 03750C4-EPN2 5 16 6 03750K4-EPN 5 16 7 04375

Table 4 Minimum driver nodes set of C3-EPN C4-EPN and K4-EPN (N 4)

Networks Minimum driver node setC3-EPN (n1 n2)| n1 isin 1 2 n2 isin 8 9

C4-EPN1

(1 n1 n2 n3 n4)|

n1 isin 2 3 n2 isin 5 6 n3 isin 8 9 n4 isin 11 12 or(n1 n2 n3 n4 13)| n1 isin 2 3

n2 isin 5 6 n3 isin 8 9 n4 isin 11 12

C4-EPN2 ( n1 10 11 12 13)| n1 isin 1 3 or (1 2 3 4 n1)| n1

isin 11 13

K4-EPN

(n1 n2 n3 n4 n5 n6)| n1 isin 1 2 3 n2 isin

1 2 3 n1ne n2 n3 isin 5 6 n4 isin 8 9 n5isin 11 12 13 n6 isin 11 12 13 n5ne n6

Table 5 Minimum driver node set of C3-EPN C4-EPN and K4-EPN (N 5)

Networks Minimum driver node setC3-EPN (n1 n2)| n1 isin 1 2 n2 isin 10 11

C4-EPN1 (1 n1 n2 n3 n4 n5)| n1 isin 2 3 n2 isin 5 6 n3 isin

8 9 n4 isin 11 12 n5 isin 14 15 n3 isin 8 9 n4 isin 11 12 n5 isin 14 15

C4-EPN2 (n1 12 13 14 15 16)| n1 isin 1 3 or (1 2 3 4 5 n1)| n2 isin 14 16

K4-EPN(n1 n2 n3 n4 n5 n6 n7)| n1 isin 1 2 3 n2 isin 1 2 3 n1ne n2 n3 isin 5 6 n4 isin 8 9

n5 isin 11 12 n6 isin 14 15 16 n7 isin 14 15 16 n6ne n7

1

2

3

4

5 9

6

7

8

(a)

1

2 3

4

5 96

7

8

10

11 12

13

(b)

1

2

3

4

5

96

8

7 10

11

12

13

(c)

1

2

34

5

967

8

10

11

12

13

(d)

Figure 4 C3-EPN C4-EPN and K4-EPN (N 4)

6 Journal of Chemistry

Data Availability

All data models and codes generated or used during thestudy are available from the corresponding author uponrequest

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

-is work was supported by the Nature Science Foundationfrom Qinghai Province (No 2018-ZJ-718) the Tibetan In-formation Processing and Machine Translation Key Labo-ratory of Qinghai Province and the Key Laboratory ofTibetan Information Processing Ministry of Education

References

[1] A Graovac I Gutman and N Trinajstic Topological Ap-proach to the Chemistry of Conjugated Molecules Springer-Verlag Berlin Germany 1977

[2] A Lombardi and M Hornquist ldquoControllability analysis ofnetworksrdquo Physical Review E vol 75 no 5 Article ID 0561102007

[3] X F Wang and G R Chen ldquoPinning control of scale-freedynamical networksrdquo Physica A Statistical Mechanics and itsApplications vol 310 no 3-4 pp 521ndash531 2002

[4] B Liu T Chu L Wang and G Xie ldquoControllability of aleader-follower dynamic network with switching topologyrdquoIEEE Transactions on Automatic Control vol 53 no 4pp 1009ndash1013 2008

[5] A Rahmani M Ji M Mesbahi and M Egerstedt ldquoCon-trollability of multi-agent systems from a graph-theoreticperspectiverdquo SIAM Journal on Control and Optimizationvol 48 no 1 pp 162ndash186 2009

[6] G-R Chen ldquoProblems and challenges in control theory undercomplex dynamical network environmentsrdquoActa AutomaticaSinica vol 39 no 4 pp 312ndash321 2013

[7] G Chen ldquoPinning control and synchronization on complexdynamical networksrdquo International Journal of Control Au-tomation and Systems vol 12 no 2 pp 221ndash230 2014

[8] L Hou S Y Lao Y D Xiao et al ldquoRecent progressioncontrollability of complex networkrdquo Acta Physica Sinicavol 64 no 18 pp 188901_1ndash188901_11 2015 (in Chinese)

[9] C T Lin ldquoStructural controllabilityrdquo IEEE Trans AutomaticControl vol 19 no 3 pp 201ndash208 1974

[10] Y-Y Liu J-J Slotine and A-L Barabasi ldquoControllability ofcomplex networksrdquo Nature vol 473 no 7346 pp 167ndash1732011

[11] Z Z Yuan C Zhao Z R Di et al ldquoExact controllability ofcomplex networksrdquo Nature Communications vol 4 no 1p 2447 2013

[12] L P Wu J K Zhou Z Z Yuan et al ldquoExact controllability ofa star-type networkrdquo in Proceedings of the 27th ChineseControl and Decision Conference (2015 CCDC) QingdaoChina May 2015 pp 5187-5191 in Chinese

[13] Z Yuan C Zhao W-X Wang Z Di and Y-C Lai ldquoExactcontrollability of multiplex networksrdquoNew Journal of Physicsvol 16 no 10 Article ID 103036 2014

[14] J Li Z Yuan Y Fan W-X Wang and Z Di ldquoControllabilityof fractal networks an analytical approachrdquo EPL (EurophysicsLetters) vol 105 no 5 Article ID 58001 2014

[15] J J Slotine and W Li Applied Nonlinear Control Prentice-Hall Upper Saddle River NJ USA 1991

[16] R E Kalman ldquoMathematical description of linear dynamicalsystemsrdquo Journal of the Society for Industrial and AppliedMathematics Series A Control vol 1 no 2 pp 152ndash192 1963

[17] A Shafieinejad F Hendessi and F Fekri ldquoStar-structurenetwork coding for multiple unicast sessions in wireless meshnetworksrdquo Wireless Personal Communications vol 72 no 4pp 2185ndash2214 2013

[18] A Jain A Vyas and H Yadav ldquoAnalysis of ethernet load onvarious nodes of star network using opnetrdquo InternationalJournal of Computer Science and Applications (TIJCSA) vol 3no 1 2014

[19] X J Li and J M Xu ldquoFault-tolerance of (n k) star networkrdquoAppliedMathematics and Computation vol 248 pp 525ndash5302014

[20] X K Chen Y Lu and L Y Chen ldquoNetwork structuralcontrollability analysis based on PBHrdquo Journal of PLA Uni-versity of Science and Technology (Natural Science Edition)

Journal of Chemistry 7

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2018

Bioinorganic Chemistry and ApplicationsHindawiwwwhindawicom Volume 2018

SpectroscopyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Medicinal ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Biochemistry Research International

Hindawiwwwhindawicom Volume 2018

Enzyme Research

Hindawiwwwhindawicom Volume 2018

Journal of

SpectroscopyAnalytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

MaterialsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

BioMed Research International Electrochemistry

International Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

B3N+1 λ3 minus 3λ1113872 1113873B3Nminus 2 minus λ4B3Nminus 5 (26)

Now let us define SN B3N+1 SNminus 1 B3Nminus 2 andSNminus 2 B3Nminus 5 then the equation (26) can be formalized as

SN λ3 minus 3λ1113872 1113873SNminus 1 minus λ4SNminus 2 (27)

According to the above definitions and equation (27) wehave S0 B1 λ and S1 B4 λ2(λ2 minus 4) so

S2 λ3 minus 3λ1113872 1113873S1 minus λ4S0 λ3 λ4 minus 8λ2 + 121113872 1113873 λ3f2(λ)

(28)

where fi(x) represents the i-th polynomial about x forwhich the lowest order is 0 Fox example f0(λ) 1f1(λ) λ2 minus 4 and f2(λ) λ4 minus 8λ2 + 12 Obviously wecan get remaining items formalized as

S3 λ3 minus 3λ1113872 1113873S2 minus λ4S1 λ4f3(λ)

S4 λ3 minus 3λ1113872 1113873S3 minus λ4S2 λ5f4(λ)

SN λ3 minus 3λ1113872 1113873SNminus 1 minus λ4SNminus 2 λN+1fN(λ)

(29)

Using a simple induction and computation we can easilyget (29) -at is to say the lowest orders of polynomials S0S1 S2 and SN are 1 2 3 and N + 1 respectively andthe multiplicity of the eigenvalue 0 of SN is N + 1 Since aconnected graph with diameter D has at least D + 1 distincteigenvalues noticing that the network C4-EPN1 withN cellshas 3N + 1 nodes and the diameter is N + 2 we have thatC4-EPN1 has at least N + 3 distinct eigenvalues Howeverthe eigenvalue 0 has a multiplicity of N + 1 and theremaining 2N eigenvalues have at least N + 2 distinctnamely the maximummultiplicity of these eigenvalues shallnot exceed N minus 1 -erefore the maximum multiplicity ofC4-EPN1rsquos eigenvalues is N + 1 (the eigenvalue is 0) -usthe minimum number of driver nodes of the system cor-responding to the graph C4-EPN1 is as follows

ND N + 1 (30)

By the analytical method of the preceding part we canobtain the recursive relations of characteristic polynomial ofC4-EPN2 and K4-EPN (see (31) and (32)) as follows

SN λ3 minus 4λ1113872 1113873SNminus 1 minus λ4SNminus 2 λN+1fN(λ) (31)

SN (λ + 1) λ2 minus λ minus 41113872 1113873SNminus 1 minus (λ + 1)4SNminus 2

(λ + 1)N+2

fN(λ + 1)(32)

Using a method similar to that of the C3-EPN andC4-EPN1 it is easy to get that the minimum number ofdriver nodes of C4-EPN2 and K4-EPN shown in Table 1

4 Characterization of Driver Nodes

-e topology of the complex network determines thecontrollability and the minimum driver nodes canquantitatively reflect the controllability of the network

structures -erefore the minimum driver nodes are thekey factors in the structural controllability analysis Infact evaluating the exact controllability of complexnetworks is to obtain the minimum driver nodes ofnetworks

Using the method based on the PBH rank criterion[11 20] we can characterize the minimum driver node set indetail -e condition of the full rank of the matrixλI minus A B1113858 1113859 is

(1) Matrix λI minus A avoids zero vector(2) Matrix λI minus A avoids repetitive vector(3) Matrix λI minus A avoids correlation vector

Because of the symmetry of graph structures the min-imum driver nodes set is not unique and its number ismainly determined by three factors such as zero row vectorrepetitive row vector and correlation row vector of [λI minus A]

Let zero vector have x rows repetitive row vector have ygroups in which each group has r1 r2 ry rows and thecorrelation row vectors have z groups in which each grouphas c1 c2 cz rows then the number of the minimumdriver nodes ND and the number of groups obtained NC areshown in Table 2

Based on the above conditions we design a correspondingalgorithm to obtain the minimum driver nodes set -roughsimulation analysis of Matlab platform we compute thecontrollability and describe all sets ofminimal driver nodes ofC3-EPN C4-EPN and K4-EPN(N 4 5) respectively theyare shown in Tables 3 4 and 5 H-EPNs of N 4 are shownin Figure 4

-e simulation results show that in order to fullycontrol these extended-path networks the driver nodesselected are usually nodes with small degree rather thanthe maximum degree nodes -e point is consistent withthe conclusion in the literature [11] And we can obtainfrom Table 3 that the C3-EPN has a fixed number ofminimum driver node set which can be fully controlled byminimal cost Namely the C3-EPN has higher control-lability In addition although the structures of theC4-EPN1 and C4-EPN2 are different we find that thecontrollability of the two networks is exactly the same indifferent network scales

Table 1 ND of extended-path networks

Networks Number of cells n ND

C3-EPN N 2N + 1 2C4-EPN1 N 3N + 1 N + 1C4-EPN2 N 3N + 1 N + 1K4-EPN N 3N + 1 N + 2

Table 2 ND and NC of three kinds of situations

Case ND NC

Zero row vector x 1Repetitive row vector 1113936

y

i1(ri minus 1) 1113937y

i1(Cri minus 1ri

)

Correlation row vector z 1113937zj1(C1

cj)

Journal of Chemistry 5

5 Conclusion

To summarize we mainly study the controllability of severaltypes of the extended-path molecular graph networks -estructural characteristics of these networks are linearly con-nected by a cycle or complete graph For these special network

structures we derive the general formula of the characteristicpolynomial of the networkrsquos adjacent matrix Furthermore theexact controllability of these kinds of networks is analyzed andsome results are obtained In addition in order to directly reflectthe concrete minimum driver node set of each network wedetermine the distribution of the nodes by simulation

Table 3 Controllability of C3-EPN C4-EPN and K4-EPN (N 4 5)

Networks Number of cell (N) Number of nodes (n) Minimum driver node (ND) Controllability (NDn)

C3-EPN 4 9 2 02222C4-EPN1 4 13 5 03846C4-EPN2 4 13 5 03846K4-EPN 4 13 6 04615C3-EPN 5 11 2 01818C4-EPN1 5 16 6 03750C4-EPN2 5 16 6 03750K4-EPN 5 16 7 04375

Table 4 Minimum driver nodes set of C3-EPN C4-EPN and K4-EPN (N 4)

Networks Minimum driver node setC3-EPN (n1 n2)| n1 isin 1 2 n2 isin 8 9

C4-EPN1

(1 n1 n2 n3 n4)|

n1 isin 2 3 n2 isin 5 6 n3 isin 8 9 n4 isin 11 12 or(n1 n2 n3 n4 13)| n1 isin 2 3

n2 isin 5 6 n3 isin 8 9 n4 isin 11 12

C4-EPN2 ( n1 10 11 12 13)| n1 isin 1 3 or (1 2 3 4 n1)| n1

isin 11 13

K4-EPN

(n1 n2 n3 n4 n5 n6)| n1 isin 1 2 3 n2 isin

1 2 3 n1ne n2 n3 isin 5 6 n4 isin 8 9 n5isin 11 12 13 n6 isin 11 12 13 n5ne n6

Table 5 Minimum driver node set of C3-EPN C4-EPN and K4-EPN (N 5)

Networks Minimum driver node setC3-EPN (n1 n2)| n1 isin 1 2 n2 isin 10 11

C4-EPN1 (1 n1 n2 n3 n4 n5)| n1 isin 2 3 n2 isin 5 6 n3 isin

8 9 n4 isin 11 12 n5 isin 14 15 n3 isin 8 9 n4 isin 11 12 n5 isin 14 15

C4-EPN2 (n1 12 13 14 15 16)| n1 isin 1 3 or (1 2 3 4 5 n1)| n2 isin 14 16

K4-EPN(n1 n2 n3 n4 n5 n6 n7)| n1 isin 1 2 3 n2 isin 1 2 3 n1ne n2 n3 isin 5 6 n4 isin 8 9

n5 isin 11 12 n6 isin 14 15 16 n7 isin 14 15 16 n6ne n7

1

2

3

4

5 9

6

7

8

(a)

1

2 3

4

5 96

7

8

10

11 12

13

(b)

1

2

3

4

5

96

8

7 10

11

12

13

(c)

1

2

34

5

967

8

10

11

12

13

(d)

Figure 4 C3-EPN C4-EPN and K4-EPN (N 4)

6 Journal of Chemistry

Data Availability

All data models and codes generated or used during thestudy are available from the corresponding author uponrequest

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

-is work was supported by the Nature Science Foundationfrom Qinghai Province (No 2018-ZJ-718) the Tibetan In-formation Processing and Machine Translation Key Labo-ratory of Qinghai Province and the Key Laboratory ofTibetan Information Processing Ministry of Education

References

[1] A Graovac I Gutman and N Trinajstic Topological Ap-proach to the Chemistry of Conjugated Molecules Springer-Verlag Berlin Germany 1977

[2] A Lombardi and M Hornquist ldquoControllability analysis ofnetworksrdquo Physical Review E vol 75 no 5 Article ID 0561102007

[3] X F Wang and G R Chen ldquoPinning control of scale-freedynamical networksrdquo Physica A Statistical Mechanics and itsApplications vol 310 no 3-4 pp 521ndash531 2002

[4] B Liu T Chu L Wang and G Xie ldquoControllability of aleader-follower dynamic network with switching topologyrdquoIEEE Transactions on Automatic Control vol 53 no 4pp 1009ndash1013 2008

[5] A Rahmani M Ji M Mesbahi and M Egerstedt ldquoCon-trollability of multi-agent systems from a graph-theoreticperspectiverdquo SIAM Journal on Control and Optimizationvol 48 no 1 pp 162ndash186 2009

[6] G-R Chen ldquoProblems and challenges in control theory undercomplex dynamical network environmentsrdquoActa AutomaticaSinica vol 39 no 4 pp 312ndash321 2013

[7] G Chen ldquoPinning control and synchronization on complexdynamical networksrdquo International Journal of Control Au-tomation and Systems vol 12 no 2 pp 221ndash230 2014

[8] L Hou S Y Lao Y D Xiao et al ldquoRecent progressioncontrollability of complex networkrdquo Acta Physica Sinicavol 64 no 18 pp 188901_1ndash188901_11 2015 (in Chinese)

[9] C T Lin ldquoStructural controllabilityrdquo IEEE Trans AutomaticControl vol 19 no 3 pp 201ndash208 1974

[10] Y-Y Liu J-J Slotine and A-L Barabasi ldquoControllability ofcomplex networksrdquo Nature vol 473 no 7346 pp 167ndash1732011

[11] Z Z Yuan C Zhao Z R Di et al ldquoExact controllability ofcomplex networksrdquo Nature Communications vol 4 no 1p 2447 2013

[12] L P Wu J K Zhou Z Z Yuan et al ldquoExact controllability ofa star-type networkrdquo in Proceedings of the 27th ChineseControl and Decision Conference (2015 CCDC) QingdaoChina May 2015 pp 5187-5191 in Chinese

[13] Z Yuan C Zhao W-X Wang Z Di and Y-C Lai ldquoExactcontrollability of multiplex networksrdquoNew Journal of Physicsvol 16 no 10 Article ID 103036 2014

[14] J Li Z Yuan Y Fan W-X Wang and Z Di ldquoControllabilityof fractal networks an analytical approachrdquo EPL (EurophysicsLetters) vol 105 no 5 Article ID 58001 2014

[15] J J Slotine and W Li Applied Nonlinear Control Prentice-Hall Upper Saddle River NJ USA 1991

[16] R E Kalman ldquoMathematical description of linear dynamicalsystemsrdquo Journal of the Society for Industrial and AppliedMathematics Series A Control vol 1 no 2 pp 152ndash192 1963

[17] A Shafieinejad F Hendessi and F Fekri ldquoStar-structurenetwork coding for multiple unicast sessions in wireless meshnetworksrdquo Wireless Personal Communications vol 72 no 4pp 2185ndash2214 2013

[18] A Jain A Vyas and H Yadav ldquoAnalysis of ethernet load onvarious nodes of star network using opnetrdquo InternationalJournal of Computer Science and Applications (TIJCSA) vol 3no 1 2014

[19] X J Li and J M Xu ldquoFault-tolerance of (n k) star networkrdquoAppliedMathematics and Computation vol 248 pp 525ndash5302014

[20] X K Chen Y Lu and L Y Chen ldquoNetwork structuralcontrollability analysis based on PBHrdquo Journal of PLA Uni-versity of Science and Technology (Natural Science Edition)

Journal of Chemistry 7

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2018

Bioinorganic Chemistry and ApplicationsHindawiwwwhindawicom Volume 2018

SpectroscopyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Medicinal ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Biochemistry Research International

Hindawiwwwhindawicom Volume 2018

Enzyme Research

Hindawiwwwhindawicom Volume 2018

Journal of

SpectroscopyAnalytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

MaterialsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

BioMed Research International Electrochemistry

International Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

5 Conclusion

To summarize we mainly study the controllability of severaltypes of the extended-path molecular graph networks -estructural characteristics of these networks are linearly con-nected by a cycle or complete graph For these special network

structures we derive the general formula of the characteristicpolynomial of the networkrsquos adjacent matrix Furthermore theexact controllability of these kinds of networks is analyzed andsome results are obtained In addition in order to directly reflectthe concrete minimum driver node set of each network wedetermine the distribution of the nodes by simulation

Table 3 Controllability of C3-EPN C4-EPN and K4-EPN (N 4 5)

Networks Number of cell (N) Number of nodes (n) Minimum driver node (ND) Controllability (NDn)

C3-EPN 4 9 2 02222C4-EPN1 4 13 5 03846C4-EPN2 4 13 5 03846K4-EPN 4 13 6 04615C3-EPN 5 11 2 01818C4-EPN1 5 16 6 03750C4-EPN2 5 16 6 03750K4-EPN 5 16 7 04375

Table 4 Minimum driver nodes set of C3-EPN C4-EPN and K4-EPN (N 4)

Networks Minimum driver node setC3-EPN (n1 n2)| n1 isin 1 2 n2 isin 8 9

C4-EPN1

(1 n1 n2 n3 n4)|

n1 isin 2 3 n2 isin 5 6 n3 isin 8 9 n4 isin 11 12 or(n1 n2 n3 n4 13)| n1 isin 2 3

n2 isin 5 6 n3 isin 8 9 n4 isin 11 12

C4-EPN2 ( n1 10 11 12 13)| n1 isin 1 3 or (1 2 3 4 n1)| n1

isin 11 13

K4-EPN

(n1 n2 n3 n4 n5 n6)| n1 isin 1 2 3 n2 isin

1 2 3 n1ne n2 n3 isin 5 6 n4 isin 8 9 n5isin 11 12 13 n6 isin 11 12 13 n5ne n6

Table 5 Minimum driver node set of C3-EPN C4-EPN and K4-EPN (N 5)

Networks Minimum driver node setC3-EPN (n1 n2)| n1 isin 1 2 n2 isin 10 11

C4-EPN1 (1 n1 n2 n3 n4 n5)| n1 isin 2 3 n2 isin 5 6 n3 isin

8 9 n4 isin 11 12 n5 isin 14 15 n3 isin 8 9 n4 isin 11 12 n5 isin 14 15

C4-EPN2 (n1 12 13 14 15 16)| n1 isin 1 3 or (1 2 3 4 5 n1)| n2 isin 14 16

K4-EPN(n1 n2 n3 n4 n5 n6 n7)| n1 isin 1 2 3 n2 isin 1 2 3 n1ne n2 n3 isin 5 6 n4 isin 8 9

n5 isin 11 12 n6 isin 14 15 16 n7 isin 14 15 16 n6ne n7

1

2

3

4

5 9

6

7

8

(a)

1

2 3

4

5 96

7

8

10

11 12

13

(b)

1

2

3

4

5

96

8

7 10

11

12

13

(c)

1

2

34

5

967

8

10

11

12

13

(d)

Figure 4 C3-EPN C4-EPN and K4-EPN (N 4)

6 Journal of Chemistry

Data Availability

All data models and codes generated or used during thestudy are available from the corresponding author uponrequest

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

-is work was supported by the Nature Science Foundationfrom Qinghai Province (No 2018-ZJ-718) the Tibetan In-formation Processing and Machine Translation Key Labo-ratory of Qinghai Province and the Key Laboratory ofTibetan Information Processing Ministry of Education

References

[1] A Graovac I Gutman and N Trinajstic Topological Ap-proach to the Chemistry of Conjugated Molecules Springer-Verlag Berlin Germany 1977

[2] A Lombardi and M Hornquist ldquoControllability analysis ofnetworksrdquo Physical Review E vol 75 no 5 Article ID 0561102007

[3] X F Wang and G R Chen ldquoPinning control of scale-freedynamical networksrdquo Physica A Statistical Mechanics and itsApplications vol 310 no 3-4 pp 521ndash531 2002

[4] B Liu T Chu L Wang and G Xie ldquoControllability of aleader-follower dynamic network with switching topologyrdquoIEEE Transactions on Automatic Control vol 53 no 4pp 1009ndash1013 2008

[5] A Rahmani M Ji M Mesbahi and M Egerstedt ldquoCon-trollability of multi-agent systems from a graph-theoreticperspectiverdquo SIAM Journal on Control and Optimizationvol 48 no 1 pp 162ndash186 2009

[6] G-R Chen ldquoProblems and challenges in control theory undercomplex dynamical network environmentsrdquoActa AutomaticaSinica vol 39 no 4 pp 312ndash321 2013

[7] G Chen ldquoPinning control and synchronization on complexdynamical networksrdquo International Journal of Control Au-tomation and Systems vol 12 no 2 pp 221ndash230 2014

[8] L Hou S Y Lao Y D Xiao et al ldquoRecent progressioncontrollability of complex networkrdquo Acta Physica Sinicavol 64 no 18 pp 188901_1ndash188901_11 2015 (in Chinese)

[9] C T Lin ldquoStructural controllabilityrdquo IEEE Trans AutomaticControl vol 19 no 3 pp 201ndash208 1974

[10] Y-Y Liu J-J Slotine and A-L Barabasi ldquoControllability ofcomplex networksrdquo Nature vol 473 no 7346 pp 167ndash1732011

[11] Z Z Yuan C Zhao Z R Di et al ldquoExact controllability ofcomplex networksrdquo Nature Communications vol 4 no 1p 2447 2013

[12] L P Wu J K Zhou Z Z Yuan et al ldquoExact controllability ofa star-type networkrdquo in Proceedings of the 27th ChineseControl and Decision Conference (2015 CCDC) QingdaoChina May 2015 pp 5187-5191 in Chinese

[13] Z Yuan C Zhao W-X Wang Z Di and Y-C Lai ldquoExactcontrollability of multiplex networksrdquoNew Journal of Physicsvol 16 no 10 Article ID 103036 2014

[14] J Li Z Yuan Y Fan W-X Wang and Z Di ldquoControllabilityof fractal networks an analytical approachrdquo EPL (EurophysicsLetters) vol 105 no 5 Article ID 58001 2014

[15] J J Slotine and W Li Applied Nonlinear Control Prentice-Hall Upper Saddle River NJ USA 1991

[16] R E Kalman ldquoMathematical description of linear dynamicalsystemsrdquo Journal of the Society for Industrial and AppliedMathematics Series A Control vol 1 no 2 pp 152ndash192 1963

[17] A Shafieinejad F Hendessi and F Fekri ldquoStar-structurenetwork coding for multiple unicast sessions in wireless meshnetworksrdquo Wireless Personal Communications vol 72 no 4pp 2185ndash2214 2013

[18] A Jain A Vyas and H Yadav ldquoAnalysis of ethernet load onvarious nodes of star network using opnetrdquo InternationalJournal of Computer Science and Applications (TIJCSA) vol 3no 1 2014

[19] X J Li and J M Xu ldquoFault-tolerance of (n k) star networkrdquoAppliedMathematics and Computation vol 248 pp 525ndash5302014

[20] X K Chen Y Lu and L Y Chen ldquoNetwork structuralcontrollability analysis based on PBHrdquo Journal of PLA Uni-versity of Science and Technology (Natural Science Edition)

Journal of Chemistry 7

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2018

Bioinorganic Chemistry and ApplicationsHindawiwwwhindawicom Volume 2018

SpectroscopyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Medicinal ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Biochemistry Research International

Hindawiwwwhindawicom Volume 2018

Enzyme Research

Hindawiwwwhindawicom Volume 2018

Journal of

SpectroscopyAnalytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

MaterialsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

BioMed Research International Electrochemistry

International Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

Data Availability

All data models and codes generated or used during thestudy are available from the corresponding author uponrequest

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

-is work was supported by the Nature Science Foundationfrom Qinghai Province (No 2018-ZJ-718) the Tibetan In-formation Processing and Machine Translation Key Labo-ratory of Qinghai Province and the Key Laboratory ofTibetan Information Processing Ministry of Education

References

[1] A Graovac I Gutman and N Trinajstic Topological Ap-proach to the Chemistry of Conjugated Molecules Springer-Verlag Berlin Germany 1977

[2] A Lombardi and M Hornquist ldquoControllability analysis ofnetworksrdquo Physical Review E vol 75 no 5 Article ID 0561102007

[3] X F Wang and G R Chen ldquoPinning control of scale-freedynamical networksrdquo Physica A Statistical Mechanics and itsApplications vol 310 no 3-4 pp 521ndash531 2002

[4] B Liu T Chu L Wang and G Xie ldquoControllability of aleader-follower dynamic network with switching topologyrdquoIEEE Transactions on Automatic Control vol 53 no 4pp 1009ndash1013 2008

[5] A Rahmani M Ji M Mesbahi and M Egerstedt ldquoCon-trollability of multi-agent systems from a graph-theoreticperspectiverdquo SIAM Journal on Control and Optimizationvol 48 no 1 pp 162ndash186 2009

[6] G-R Chen ldquoProblems and challenges in control theory undercomplex dynamical network environmentsrdquoActa AutomaticaSinica vol 39 no 4 pp 312ndash321 2013

[7] G Chen ldquoPinning control and synchronization on complexdynamical networksrdquo International Journal of Control Au-tomation and Systems vol 12 no 2 pp 221ndash230 2014

[8] L Hou S Y Lao Y D Xiao et al ldquoRecent progressioncontrollability of complex networkrdquo Acta Physica Sinicavol 64 no 18 pp 188901_1ndash188901_11 2015 (in Chinese)

[9] C T Lin ldquoStructural controllabilityrdquo IEEE Trans AutomaticControl vol 19 no 3 pp 201ndash208 1974

[10] Y-Y Liu J-J Slotine and A-L Barabasi ldquoControllability ofcomplex networksrdquo Nature vol 473 no 7346 pp 167ndash1732011

[11] Z Z Yuan C Zhao Z R Di et al ldquoExact controllability ofcomplex networksrdquo Nature Communications vol 4 no 1p 2447 2013

[12] L P Wu J K Zhou Z Z Yuan et al ldquoExact controllability ofa star-type networkrdquo in Proceedings of the 27th ChineseControl and Decision Conference (2015 CCDC) QingdaoChina May 2015 pp 5187-5191 in Chinese

[13] Z Yuan C Zhao W-X Wang Z Di and Y-C Lai ldquoExactcontrollability of multiplex networksrdquoNew Journal of Physicsvol 16 no 10 Article ID 103036 2014

[14] J Li Z Yuan Y Fan W-X Wang and Z Di ldquoControllabilityof fractal networks an analytical approachrdquo EPL (EurophysicsLetters) vol 105 no 5 Article ID 58001 2014

[15] J J Slotine and W Li Applied Nonlinear Control Prentice-Hall Upper Saddle River NJ USA 1991

[16] R E Kalman ldquoMathematical description of linear dynamicalsystemsrdquo Journal of the Society for Industrial and AppliedMathematics Series A Control vol 1 no 2 pp 152ndash192 1963

[17] A Shafieinejad F Hendessi and F Fekri ldquoStar-structurenetwork coding for multiple unicast sessions in wireless meshnetworksrdquo Wireless Personal Communications vol 72 no 4pp 2185ndash2214 2013

[18] A Jain A Vyas and H Yadav ldquoAnalysis of ethernet load onvarious nodes of star network using opnetrdquo InternationalJournal of Computer Science and Applications (TIJCSA) vol 3no 1 2014

[19] X J Li and J M Xu ldquoFault-tolerance of (n k) star networkrdquoAppliedMathematics and Computation vol 248 pp 525ndash5302014

[20] X K Chen Y Lu and L Y Chen ldquoNetwork structuralcontrollability analysis based on PBHrdquo Journal of PLA Uni-versity of Science and Technology (Natural Science Edition)

Journal of Chemistry 7

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2018

Bioinorganic Chemistry and ApplicationsHindawiwwwhindawicom Volume 2018

SpectroscopyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Medicinal ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Biochemistry Research International

Hindawiwwwhindawicom Volume 2018

Enzyme Research

Hindawiwwwhindawicom Volume 2018

Journal of

SpectroscopyAnalytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

MaterialsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

BioMed Research International Electrochemistry

International Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

ls

Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

Submit your manuscripts atwwwhindawicom

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2018

Bioinorganic Chemistry and ApplicationsHindawiwwwhindawicom Volume 2018

SpectroscopyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Medicinal ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Biochemistry Research International

Hindawiwwwhindawicom Volume 2018

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