Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians

a

steleyn.metry

7) 67,fperfectrcycle

th

ph;

ational

Advances in Applied Mathematics 32 (2004) 655–668

www.elsevier.com/locate/yaam

Enumeration of perfect matchings of graphswith reflective symmetry by Pfaffians

Weigen Yana,b,∗ and Fuji Zhangb

a Department of Mathematics, Jimei University, Xiamen 361021, PR Chinab Department of Mathematics, Xiamen University, Xiamen 361005, PR China

Received 25 August 2002; accepted 15 January 2003

Abstract

The Pfaffian method enumerating perfect matchings of plane graphs was discovered by KaWe use this method to enumerate perfect matchings in a type of graphs with reflective symwhich is different from the symmetric graphs considered in [J. Combin. Theory Ser. A 77 (199MATCH—Commun. Math. Comput. Chem. 48 (2003) 117]. Here are some of our results: (1) IG isa reflective symmetric plane graph without vertices on the symmetry axis, then the number ofmatchings ofG can be expressed by a determinant of order|G|/2, where|G| denotes the numbeof vertices ofG. (2) If G contains no subgraph which is, after the contraction of at most oneof odd length, an even subdivision ofK2,3, then the number of perfect matchings ofG × K2 canbe expressed by a determinant of order|G|. (3) LetG be a bipartite graph without cycles of leng4s, s ∈ 1,2, . . .. Then the number of perfect matchings ofG × K2 equals

∏(1+ θ2)mθ , where the

product ranges over all non-negative eigenvaluesθ of G andmθ is the multiplicity of eigenvalueθ .Particularly, ifT is a tree then the number of perfect matchings ofT × K2 equals

∏(1 + θ2)mθ ,

where the product ranges over all non-negative eigenvaluesθ of T andmθ is the multiplicity ofeigenvalueθ . 2003 Elsevier Inc. All rights reserved.

Keywords:Perfect matchings; Pfaffian orientation; Skew adjacency matrix; Symmetric graph; Bipartite graEven subdivision; Nice cycle; Outerplanar graph

This work is supported by NSFC (19971071) and FJCEF and the Doctoral Program Foundation of NEducation Department of China.

* Corresponding author.E-mail address:[email protected] (W. Yan).

0196-8858/$ – see front matter 2003 Elsevier Inc. All rights reserved.doi:10.1016/S0196-8858(03)00097-6

656 W. Yan, F. Zhang / Advances in Applied Mathematics 32 (2004) 655–668

ne-

xt

ph

n

t,

n

gsartite

tlyhatron thes

d

esulte

yddding),ant

1. Introduction

By a simple graphG = (V (G),E(G)) we mean a finite undirected graph, that is, owith no loops or parallel edges, with the vertex-setV (G) = v1, v2, . . . , vn and the edgesetE(G) = e1, e2, . . . , em, if not specified. A setM of edges inG is a matching if everyvertex ofG is incident with at most one edge inM; it is a perfect matching if every verteof G is incident with exactly one edge inM. We denote byM(G) the number of perfecmatchings ofG. If M is a perfect matching ofG, anM-alternating cycle inG is a cyclewhose edges are alternately inE(G)\M andM. LetG be a graph. We say that a cycleC ofG is niceif G−C contains a perfect matching, whereG−C denotes the induced subgraof G obtained fromG by deleting the vertices ofC. Let G = (V (G),E(G)) be a graphallowed to have loops but have no parallel edges, and letGe be an arbitrary orientatioof G. Theskew adjacency matrixof Ge, denoted byA(Ge), is defined as follows:

A(Ge) = (aij )n×n, aij =

1 if (vi, vj ) ∈ E(Ge),

−1 if (vj , vi) ∈ E(Ge) andi = j,

0 otherwise.

Generally speaking, skew adjacency matrixA(Ge) is not a skew symmetric matrix. In facA(Ge) is a skew symmetric matrix if and only ifG contains no loops.

Let G be a simple graph and letG0,G1,G2, . . . ,Gk be graphs such thatG0 = G and,for eachi > 0, Gi can be obtained fromGi−1 by subdividing an edge twice. ThenGk issaid to be aneven subdivisionof G. Throughout this paper,G × H denotes the Cartesiaproduct of two graphsG andH .

Let G be a simple graph. We sayG hasreflective symmetryif it is invariant under thereflection across some straight linel (the symmetry axis). In [3] Ciucu gave a matchinfactorization theorem of the number of perfect matchings of a symmetric plane bipgraph in which there are some vertices lying on the symmetry axisl but no edges crossingl.Ciucu’s theorem expresses the number of perfect matchings ofG in terms of the producof the number of perfect matchings of two subgraphs ofG each one of which has nearhalf the number of vertices ofG. On the other hand, in [23] Zhang and Yan proved tif a bipartite graphG without nice-cycles of length 4s, s ∈ 1,2, . . . was invariant undethe reflection across some plane (or straight line) and there were no vertices lyingsymmetry plane (or axis) thenM(G) = |detA(G+)|, whereG+ is a graph having loopwith half the number of vertices ofG, andA(G+) is the adjacency matrix ofG+.

In this paper, we will consider first a symmetric plane graphG (which does not neeto be bipartite) in which there are no vertices lying on the symmetry axisl. In otherwords,G is invariant under reflection across some symmetry straight line. Our first ris to prove that the numberM(G) of perfect matchings ofG can be expressed by thdeterminant of the skew adjacency matrix of some orientationGe of a graphG whichcontains loops and has half the number of vertices ofG. This result leads naturallto a corollary: if G is an outerplanar graph (a graph isouterplanar if it is planar anembeddable into the plane such that all vertices lie on the outer face of the embethen the number of perfect matchings ofG × K2 can be expressed by the determinof the skew adjacency matrix of some orientationGe of a graphG obtained fromG by

W. Yan, F. Zhang / Advances in Applied Mathematics 32 (2004) 655–668 657

phngth,

the

t

erfect

at

nd.

ted

nial

the

ch

adding a loop to every vertex ofG. Our second result is to prove that if a simple graG contains no subgraph which is, after the contraction of at most one cycle of odd lean even subdivision ofK2,3, thenM(G × K2) can be expressed by the determinant ofskew adjacency matrix of some orientationGe of a graphG obtained fromG by addinga loop to every vertex ofG. Our third result is to prove that ifG is a bipartite graph withoucycles of length 4s, s ∈ 1,2, . . ., then the number of perfect matchings ofG × K2 equals∏

(1 + θ2)mθ , where the product ranges over all non-negative eigenvaluesθ of G andmθ

is the multiplicity of eigenvalueθ .The starting point of this paper is Kasteleyn’s Pfaffian method for enumerating p

matchings [14,15]. IfD is an orientation of a simple graphG andC is a cycle of evenlength, we say thatC is oddly orientedin D if C contains odd number of edges thare directed inD in the direction of each orientation ofC. We say thatD is a Pfaffianorientationof G if every nice cycle of even length ofG is oddly oriented inD. It is wellknown that if a graphG contains no subdivision ofK3,3 thenG has a Pfaffian orientatio(see Little [17]). McCuaig [19], McCuaig etal. [20], and Robertson et al. [21] founa polynomial-time algorithm to show whether a bipartite graph has a Pfaffian orientation

Lemma 1 [15,18].LetGe be a Pfaffian orientation of a graphG. Then

M2(G) = detA(Ge),

whereA(Ge) is the skew adjacency matrix ofGe.

Lemma 2 [18]. Let G be a plane graph, andGe an orientation ofG such that everyboundary face—except possibly the infiniteface—has an odd number of edges orienclockwise. ThenGe is a Pfaffian orientation.

Lemma 3 [14,15,18] (Kasteleyn’s theorem).Every plane graph has a Pfaffian orientatiosatisfying the condition in Lemma2. Such an orientation can be constructed in polynomtime.

Lemma 4 [18]. Let G be any simple graph with even number of vertices, andGe anorientation ofG. Then the following three properties are equivalent:

(1) Ge is a Pfaffian orientation.(2) Every nice cycle of even length inG is oddly oriented inGe.(3) If G contains a perfect matching, then for some perfect matchingF , everyF -alter-

nating cycle is oddly oriented inGe.

Lemma 5 [10]. If G is a simple graph containing no subgraph which is, aftercontraction of at most one cycle ofodd length, an even subdivision ofK2,3, thenG hasan orientation under which every cycle of evenlength is oriented oddly. Furthermore, suan orientation is a Pfaffian orientation ofG.


ef even

rix

rcsgults

ts

t

e

ft

lyingists

The above lemma was given by Fischer andLittle [10] in 2002. In fact, they gave a morgeneral characterization of graphs that have an orientation under which every cycle olength has a prescribed parity.

Now let us recall the relationship between the terms of the determinant of a real matA = (aij )n×n and the 1-factors of its associated weighted digraph. LetA be a matrix ofordern. A digraphD(A) with n vertices labelled by the integers from 1 ton is defined asfollows. If aij = 0 then there is an arc from vertexi to vertexj with associated weightaij

in D(A), where 0< i, j n. Clearly, in this digraph loops are allowed and different awith the same head and tail are not allowed. A 1-factor ofD(A) is defined to be a spanninsubgraph ofD(A) which is regular of in-degree and out-degree 1. The following reswere first exploited by König [16] and were developed by Coates [4].

Lemma 6 [2,7]. LetA be a matrix of ordern andΩ be the set of1-factors ofD(A). Then

detA =∑h∈Ω

(−1)Lhf (h),

where the summation ranges over every1-factor ofD(A), Lh is the number of componenwith even number of vertices ofh andf (h) is the product of the weights of arcs inh.

Lemma 7 [8,9,11,24].Let G be a bipartite graph with2n vertices and letA andM(G)

denote the adjacency matrix and the number of perfect matchings ofG, respectively.ThendetA = (−1)n[M(G)]2 if and only if G has no nice cycles of length4s, wheres ∈ 1,2, . . ..

2. Main results

Let G be a simple connected graph with the symmetry axis (or plane)l and assume thathere are no vertices lying onl (we considerl to be vertical). Then the set of edges ofG

crossingl forms an edge cutB of G. If we delete the edges ofB from G, two isomorphicsubgraphs (theleft and right half of G) are obtained (see Fig. 1), denoted byL(G) andR(G), respectively.

Now we define a graphG obtained from the left half ofG and use the determinant of thskew adjacency matrix of some orientation ofG to calculate the numberM(G) of perfectmatchings ofG. In fact,G is obtained fromG by adding a loop to each vertex in the lehalf of G which is an end vertex of an edge in edge cutB. By this definition the graphGobtained from the graph indicated in Fig. 1 is pictured in Fig. 2.

Theorem 8. Let G be a symmetric connected plane simple graph with no verticeson the symmetry axisl, and let G be the graph described above. Then there exan orientationGe of G such that the numberM(G) of perfect matchings ofG equals|detA(Ge)|, whereA(Ge) denotes the skew adjacency matrix ofGe.

Proof. By Kasteleyn’s theorem (Lemma 3) the left halfL(G) of G has a Pfaffianorientation satisfying conditions in Lemma 2. We denote this orientation byL(G)e. If


ff,

tation

e

.

(a) (b)

Fig. 1. (a) A symmetric graphG. (b) The left halfL(G) of a symmetric graphG.

Fig. 2. The graphG obtained fromG in Fig. 1.

we reverse the orientation of each arc ofL(G)e, then we can obtain an orientation ofR(G)

satisfying conditions in Lemma 2, denoted byR(G)e. Hence,R(G)e is the converse oL(G)e. It is clear thatR(G)e is a Pfaffian orientation ofR(G). For example, the left halL(G) of G in Fig. 1 has a Pfaffian orientationL(G)e satisfying conditions in Lemma 2which is pictured in Fig. 3(a). Figure 3(b) shows the corresponding Pfaffian orienR(G)e of R(G) obtained from the orientationL(G)e .

Hence, we have oriented every edge ofG except the edges in edge cutB in G whichcross the symmetry axisl. For every edge inB, let the direction be from the left to thright. Hence, we obtain an orientation ofG, denoted byGe. For the graphG in Fig. 1 andthe orientationsL(G)e andR(G)e of the left halfL(G) and the right halfR(G) of G inFig. 3, by the above definition, the corresponding orientationGe of G is showed in Fig. 4


.ise,

ix

(a) (b)

Fig. 3. (a) A Pfaffian orientationL(G)e of L(G). (b) Pfaffian orientationR(G)e of R(G).

Fig. 4. An orientationGe of G obtained fromR(G)e andL(G)e in Fig. 3.

It is clear that every boundary face inGe intersected byl is oddly oriented clockwiseNote that every boundary face inGe—except the faces intersected by symmetry axland possibly the infinite face—has an odd number of edges oriented clockwise. Hencthe orientationGe of G satisfies conditions in Lemma 2. Consequently,Ge is a Pfaffianorientation ofG. By a suitable labelling of vertices ofGe, the skew adjacency matrA(Ge) of Ge has the following form:

A(Ge) =[

A R

−R −A

],

whereA is the skew adjacency matrix ofL(G)e . Hence, we haveAT = −A. By ourassumption, the reflection interchanges each pair of end vertices of the edges inB. Thus,R is represented to be a diagonal matrix. By Lemma 1 we have


f

tsf

e

f

y

narn

[M(G)

]2 = det

[A R

−R −A

]= det

[A + R A + R

−R −A

]= det

[A + R 0−R R − A

]= det(A + R)det(R − A).

Note that det(R − A) = det(R − A)T = det(R + A). Hence, we prove that

M(G) = ∣∣det(A + R)∣∣.

For the graphG described above, letGe be the orientation ofG obtained from theorientationL(G)e by adding a loop to each vertex inL(G)e which is an end vertex oan edge in edge cutB. It is clear thatA + R equals the skew adjacency matrixA(Ge) ofGe. The theorem is thus proved.Corollary 9. Let G be a simple graph. IfG is an outerplanar graph then there exisa Pfaffian orientationGe of G such that the numberM(G × K2) of perfect matchings oG × K2 equalsdet(A(Ge) + I), whereA(Ge) is the skew adjacency matrix ofGe andI isthe identity matrix.

Proof. Noting thatG is an outerplanar graph, it is clear thatG × K2 is a plane graph (se[13]). Note thatG × K2 can be obtained as follows. Take two copies ofG, denoted byG1with vertex-setV (G1) = v′

1, v′2, . . . , v

′n andG2 with vertex-setV (G2) = v′′

1, v′′2, . . . , v′′

n(we say thatG1 and G2 are the left half and right half of G × K2, respectively), andadd an edgev′

iv′′i between every pair of corresponding verticesv′

i andv′′i of G1 andG2,

respectively. It is clear that the resulting graph isG × K2 and all edges addedv′iv

′′i (for

1 i n) between the left half and the right half ofG × K2 form a perfect matching oG × K2, denoted byM. It is obvious thatM is an edge cut ofG × K2. Hence,G × K2is a symmetric plane graph which has no vertices lying on the symmetry axisl crossedby edges in the edge cutM. Note that the left half of graphG × K2 is isomorphic toG.Then graphG × K2 described above is the one obtained fromG by adding a loop to eververtex ofG. Hence, by Theorem 8 there exists a Pfaffian orientationGe of G such that

M(G × K2) = ∣∣det(A(Ge) + I

)∣∣ = det(A(Ge) + I

).

The corollary is thus proved.Example 10. Let G be ann × 2 grid graph pictured in Fig. 5(a), which is an outerplagraph. It is obvious that the orientationGe of G showed in Fig. 5(b) is a Pfaffian orientatioof G. The skew adjacency matrixA(Ge) of Ge has the following form:

A(Ge) =[

A I

−I −A

], whereA =

0 1−1 0 1

−1 0 1. . .

. . .. . .

−1 0 1

n×n

.


es

r offornder

rms

tionnt

one

(a) (b)

Fig. 5. (a) Ann × 2 grid graphG. (b) A Pfaffian orientation of ann × 2 grid graphG.

By Corollary 9, we have

M(G × K2) = M(Pn × K2 × K2) = M(Pn × C4) = det(A(Ge) + I

),

where Pn is a path withn vertices andI is an n × n identity matrix. Note that theeigenvalues ofA are 2i cos(kπ)/(n + 1) (k = 1,2, . . . , n) (see [7]). Hence, the eigenvaluof matrixA(Ge) + I are

1± i

√1+ 4 cos2

kπ

n + 1(k = 1,2, . . . , n).

So we have

M(G × K2) = M(Pn × C4) = det(A(Ge) + I

)=

n∏k=1

(1+ i

√1+ 4 cos2

kπ

n + 1

)(1− i

√1+ 4 cos2

kπ

n + 1

)

=n∏

k=1

(2+ 4 cos2

kπ

n + 1

)= 2n

n∏k=1

(1+ 2 cos2

kπ

n + 1

).

Given an outerplanar graphG, Corollary 9 gives a method to calculate the numbeperfect matchings ofG × K2. Generally speaking, this method does not always holdthe case whenG is not an outerplanar graph. A natural problem to be asked is that uwhich condition for a graphG the approach in Corollary 9 can be used efficiently. In teof Lemma 5 (a result from Fischer and Little [10]), we obtain the following theorem.

Theorem 11. If G is a simple graph containing no subgraph which is, after the contracof at most one cycle of odd length, an even subdivision ofK2,3, then there exists a PfaffiaorientationGe of G, under which every cycle of even length is oddly oriented, such thathe number of perfect matchings ofG × K2 equals

det(A(Ge) + I

),

whereA(Ge) denotes the skew adjacency matrix ofGe andI is the identity matrix.

Proof. Noting thatG contains no subgraph which is, after the contraction of at mostcycle of odd length, an even subdivision ofK2,3, then by Lemma 5,G admits an orientation


ed,s

fise

ve

farcs

en

g

under which every cycle of even length is oddly oriented, denoted byGe. Obviously,Ge

is a Pfaffian orientation ofG. In order to obtain a Pfaffian orientation ofG × K2, we takea copy ofGe, denoted byGe

1. If we reverse the orientation of each arc ofGe, then wecan obtain an orientation ofG under which every cycle of even length is oddly orientdenoted byGe

2. Hence,Ge2 is the converse ofGe

1. Note thatG × K2 can be obtained afollows. Take two copies ofG, denoted byG1 with vertex-setV (G1) = v′

1, v′2, . . . , v

′n

and G2 with vertex-setV (G2) = v′′1, v′′

2, . . . , v′′n (we say thatG1 and G2 are the left

half and right half ofG × K2, respectively), and add an edgev′iv

′′i between every pair o

corresponding verticesv′i andv′′

i of G1 andG2, respectively. Then the resulting graphG×K2. It is obvious that all edges addedv′

iv′′i (for 1 i n) between the left half and th

right half of G × K2 form a perfect matching ofG × K2, denoted byM, andGe1 (or Ge

2)is an orientation ofG1 (or G2). If we define the direction of every edge inM is from theleft to the right, then an orientation ofG×K2 is obtained, denoted by(G×K2)

e. Now weprove the following claim.

Claim. The orientation(G × K2)e described above is a Pfaffian orientation ofG × K2.

Note thatM is a perfect matching ofG × K2, then by Lemma 4 we only need to prothat everyM-alternating cycle is oddly oriented in(G × K2)

e . Let C be anM-alternatingcycle ofG × K2. ThenC has the following form

v′i1v′′i1v′′i2v′i2

· · ·v′′i2m

v′i2m

v′i1,

wherev′ijv′′ij

∈ M for 1 j 2m, and v′ij

| 1 j 2m ∈ V (G1) and v′′ij

| 1 j 2m ∈ V (G2). If m = 1, it is clear thatC is oriented oddly in(K2 × G)e. Supposem > 1. By the definition ofG × K2, v′

i1v′i2

· · ·v′i2m

v′i1

is a cycle of even length inG1 andv′′i1v′′i2

· · ·v′′i2m

v′′i1

is a cycle of even length inG2, denoted byC1 andC2, respectively. Forthe sake of convenience, we say the direction of arc(v′

i1, v′′

i1) in C is positive (negative, i

otherwise). Suppose thatm′1 andm′

2 denote the number of positive arcs and negativein C which are in the right half of(K2 × G)e, then we havem′

1 + m′2 = m. Similarly, if we

suppose thatm1 andm2 denote the number of positive arcs and negative arcs inC whichare in the left half of(K2 × G)e, then we havem1 + m2 = m. By the definitions ofGe

1andGe

2, the number of positive and negative arcs in the orientation ofC1 arem1 + m′2 and

m2 + m′1, respectively. SinceGe

1 is a Pfaffian orientation under which every cycle of evlength is oddly oriented inGe

1, m1 + m′2 is odd. Similarly, we have thatm2 + m′

1 is odd.Note that the number of positive arcs in the orientation ofC equalsm + m1 + m′

1, hencewe only need to prove thatm + m1 + m′

1 is odd. This is immediate from the followinstatements which are proved above:

m1 + m2 = m, m′1 + m′

2 = m, m1 + m′2 is odd, m2 + m′

1 is odd.

The claim thus follows.By a suitable labelling of vertices of(G×K2)

e, the skew adjacency matrix of(G×K2)e

has the following form


y

n

ngs fornplanar

stoon

a-

A((G × K2)

e) =

[A(Ge) I

−I −A(Ge)

],

whereA(Ge) is the skew adjacency matrix ofGe andI is the identity matrix. Hence, bLemma 1 we have

[M(G × K2)

]2 = detA((G × K2)

e) = det

[A(Ge) I

−I −A(Ge)

]

= det

[A(Ge) + I A(Ge) + I

−I −A(Ge)

]= det

[A(Ge) + I 0

−I I − A(Ge)

]

= det(A(Ge) + I

)det

(I − A(Ge)

) = det(A(Ge) + I

)det

(I − A(Ge)

)T

= det(A(Ge) + I

)det

(I + A(Ge)

) = [det

(A(Ge) + I

)]2.

Hence, we have proved

M(G × K2) = ∣∣det(A(Ge) + I

)∣∣.SinceA(Ge) is a skew symmetric matrix, we have

det(A(Ge) + I

) =t∏

j=1

(1+ iθj )(1− iθj ) =t∏

j=1

(1+ θ2

j

),

where±iθj for 1 j t are the non-zero eigenvalues ofA(Ge) andi2 = −1. So we have

M(G × K2) = ∣∣det(A(Ge) + I

)∣∣ = det(A(Ge) + I

).

The theorem is thus proved.Remark 12. If every cycle of even length inG is oriented oddly, then that orientatioworks for the statement of Theorem 11.

Note that the above theorem can be used to count the number of perfect matchisome planar graph as well as some nonplanar graphs. We give an example of the nograph as follows.

Example 13. Let G1 = K5, andV (G1) = 1,2,3,4,5. Note thatK5 contains subgraphK2,3, hence G1 (= K5) does not satisfy conditions in Theorem 11. In orderget a nonplanar graph satisfying conditions in Theorem 11, we choose a pentagC(1−2−3−4−5−1) in G1 and subdivide every edge ofC once. Then we obtain a nonplnar graph with 10 vertices, denoted byG.

Claim. Every cycle of even length in the orientation ofG showed in Fig.6 is orientedoddly.


r

s

d

rk 12,

t

Fig. 6. An orientationGe of G under which every cycle of even length is oddly oriented.

Note that the cycle of length ten inG is C(1−6−2−7−3−8−4−9−5−10−1), it isobvious that this cycle is oriented oddly inGe. There are five cycles of length fouin G: C(1−3−8−4−1), C(2−4−9−5−2), C(3−5−10−1−3), C(4−1−6−2−4), andC(5−2−7−3−5). These cycles are oriented oddly inGe. Similarly, every cycle of lengthsix in G is oriented oddly inGe. Hence, claim holds.

By Theorem 11 or Remark 12, we have

M(G × K2) = det(A(Ge) + I

),

whereA(Ge) is the skew adjacency matrix andI is a 10× 10 identity matrix.By using computer software Matlab, we can get easily that

det(A(Ge) + I

) = 400.

Hence, the number of perfect matchings ofG × K2 equals 400.

Theorem 14. Let G be a bipartite graph without cycles of length4s, s ∈ 1,2, . . .. Thenthe number of perfect matchings ofG×K2 equals

∏(1+ θ2)mθ , where the product range

over all non-negative eigenvalues ofG andmθ is the multiplicity of eigenvalueθ .

Proof. SinceG is a bipartite graph,G × K2 is a bipartite graph. Assume the vertices inG

have been properly colored, with colors white and black. LetGe be the orientation obtaineby directing each edge so that it points from the white to the black vertices. Note thatG

contains no cycles of length 4s, s ∈ 1,2, . . ., henceGe is a Pfaffian orientation ofGunder which every cycle of even length is oddly oriented. By Theorem 11 and Remawe have

M(G × K2) = det(A(Ge) + I

).

So we only need to prove that det(A(Ge) + I) equals∏

(1 + θ2)mθ , where the producranges over all non-negative eigenvalues ofG andmθ is the multiplicity of eigenvalueθ .


in

g

f

e

with

Let A be the adjacency matrix ofG, and letφ(G,x) be the characteristic polynomial ofG.SinceG is a bipartite graph, we may assume that

φ(G,x) = xn − a1xn−2 + a2x

n−4 + · · · + (−1)iaixn−2i + · · · + (−1)rarx

n−2r , (1)

wheren and r are the number of vertices ofG and the maximum number of edgesa matching ofG (see [7]). Note that(−1)iai equals the sum of all principal minors ofA

of order 2i, hence(−1)iai equals the sum of detA(H) over all induced subgraphsH of G

with 2i vertices, whereA(H) is the adjacency matrix of subgraphH .

Claim 1. ai is equal to the sum ofM(H)2 over all induced subgraphsH of G with 2i

vertices, whereM(H) denotes the number of perfect matchings ofH .

SinceG is a bipartite graph without cycles of length 4s, s ∈ 1,2, . . ., by Lemma 7every principal minor of order 2i in A equals(−1)iM(H)2, whereH is the correspondininduced subgraph ofG. Hence,(−1)iai is equal to the sum of(−1)iM(H)2 over allinduced subgraphsH of G with 2i vertices. Claim 1 thus follows.

Claim 2. For the orientationGe of G, we have

det(xI − A(Ge)

) = xn + a1xn−2 + a2x

n−4 + · · · + aixn−2i + · · · + arx

n−2r . (2)

Note that the coefficient ofxn−k in det(xI − A(Ge)) is equal to the sum o(−1)k detA(He) over all induced subdigraphsHe of Ge with k vertices, whereA(He)

is the skew adjacency matrix of subdigraphHe. Let H be the underlying graph ofHe

and letA(H) be the adjacency matrix ofH . Note thatG contains no cycles of length 4s,s ∈ 1,2, . . ., by Lemma 6 we can see easily that

detA(H) =

(−1)k/2 detA(He) if k is even,0 otherwise.

Hence, the coefficient ofxn−k in det(xI − A(Ge)) is equal to 0 ifk is odd, and the sumof (−1)k/2detA(H) over all subgraphH of G if k is even, respectively. Note that(−1)iai

equals the sum of detA(H) over all induced subgraphsH of G with 2i vertices, hence whave

det(xI − A(Ge)

) = xn + a1xn−2 + a2x

n−4 + · · · + aixn−2i + · · · + arx

n−2r .

Claim 2 is thus proved.Both of Claims 1 and 2 show thatθ is an eigenvalue ofA(G) (= A) with multiplicity mθ

if and only if iθ is an eigenvalue ofA(Ge) with multiplicity mθ , wherei2 = −1. Hence,the theorem follows from the fact that the spectrum of a bipartite graph is symmetricrespect to 0, a result by Coulson and Rushbrooke [6] (see also Biggs [1]).


t

ce, by

s oflues

bovephorbit

ofs.

there

als

cy

o-

Fig. 7. A bipartite graphG without cycles of length 4s, s ∈ 1,2, . . ..

Example 15. Let G be the graph showed in Fig. 7, which has 4N + 2 vertices. Note thaG has the following eigenvalues

1, −1, and1

2

(±1±

√9+ 8 cos

πj

N + 1

), j = 1,2, . . . ,N,

where all four combinations of signs have to be considered (see [5,12,22]). HenTheorem 14 the number of perfect matchings ofG × K2 equals

2N∏

j=1

[1+ 1

4

(1+

√9+ 8 cos

πj

N + 1

)2]

N∏j=1

[1+ 1

4

(−1+

√9+ 8 cos

πj

N + 1

)2]

= 2N+1N∏

j=1

(5+ 6 cos

πj

N + 1+ 2 cos2

πj

N + 1

).

Corollary 16. Let T be a tree withn vertices. Then the number of perfect matchingT × K2 equals

∏(1+ θ2)mθ , where the product ranges over all non-negative eigenva

of T andmθ is the multiplicity of eigenvalueθ .

The corollary is immediate from Theorem 14. The weight of every edge of the agraphsG equals one. For the case whenG is a weighted graph, we say a weighted grais symmetric if its underlying graph is symmetric and the weights are constant on theof reflection. The weight of one perfect matchingM is defined to the product of weightsedges contained inM. We denote byM(G) the sum of weights of all perfect matchingThen we can extend Theorem 8 as follows.

Theorem 17. LetG be a symmetric connected plane simple weighted graph such thatare no vertices lying on the symmetry axisl. Suppose thatG is the graph obtained fromGby adding a loop to each vertex in the left half ofG which is an end vertex of an edgee inthe edge cutB intersected by the symmetry axisl and the weight of the loop added equthe weight of edgee. Then there exists an orientationGe of G such that the numberM(G)

of perfect matchings ofG equals|detA(Ge)|, whereA(Ge) denotes the skew adjacenmatrix ofGe.

Remark 18. Similarly, for the case whenG is a weighted graph we can extend Therems 11 and 14 and Corollary 9.


fereemore

ry

9)

Soc. 36

erlin,

72)

ital

of

retical

k,

ted

nn.

rgh

ry,

pl.

Acknowledgments

We thank Professor C.H.C. Little for telling us Example 13. We also thank the refor many valuable suggestions and help in much details to make this paper to bepleasant to read.

References

[1] N. Biggs, Algebraic Graph Theory,Cambridge Univ. Press, Cambridge, 1993.[2] W.K. Chen, Applied Graph Theory, North-Holland, Amsterdam, 1971.[3] M. Ciucu, Enumeration of perfect matchings in graphs with reflective symmetry, J. Combin. Theo

Ser. A 77 (1997) 67–97.[4] C.L. Coates, Flow-graph solution of linear algebraic equations, IRE Trans. Circuit Theory CT 6 (195

170–187.[5] C.A. Coulson, Exited electronic levels in conjugated molecules, Proc. Phys. Soc. 60 (1948) 257–269.[6] C.A. Coulson, G.S. Rushbrooke, Note on the method of molecular orbits, Proc. Cambridge Philos.

(1940) 193–200.[7] D. Cvetkovic, M. Doob, H. Sachs, Spectra of Graphs, in: Theory and Application, 2nd ed., DVW, B

1982, pp. 228–259, Chapter 8.[8] D. Cvetkovic, I. Gutman, N. Trinajstic, Kekule structures and topology, Chem. Phys. Lett. 16 (3) (19

614–616.[9] M.S.J. Dewar, H.C. Longuet-Higgins, The correspondence between the resonance and molecular orb

theories, Proc. Roy. Soc. London Ser. A 214 (1952) 482–493.[10] I. Fischer, C.H.C. Little, Even circuits of prescribed clockwise parity, preprint.[11] A. Graovac, I. Gutman, N. Trinajstic, T. Živkovic, Graph theory and molecular orbitals. Applications

Sachs Theorem, Theoret. Chim. Acta 26 (1972) 67–78.[12] E. Heilbronner, Die Eigenwerte von LCAO-MO’s in homologen Reihen, Helv. Chim. Acta 37 (1954) 921–

935.[13] W. Imrich, S. Klavžar, Product Graphs, Structure and Recognition, Wiley, New York, 2000.[14] P.W. Kasteleyn, Dimer statistics andphase transition, J. Math. Phys. 4 (1963) 287–293.[15] P.W. Kasteleyn, in: F. Harary (Ed.), Graph Theory and Crystal Physics, Graph Theory and Theo

Physics, Academic Press, 1967, pp. 43–110.[16] D. König, Über Graphen und ihre Anwendungen auf Determinantentheorie und Mengenlehre, Math. Ann. 77

(1916) 453–465.[17] C.H.C. Little, A characterization of convertible(0,1)-matrices, J. Combin. Theory 18 (1975) 187–208.[18] L. Lovász, M. Plummer, Matching Theory, in: Ann. Discrete Math., vol. 29, North-Holland, New Yor

1986.[19] W. McCuaig, Pólya’s permanent problem, preprint.[20] W. McCuaig, N. Robertson, P.D. Seymour, R. Thomas, Permanents, Pfaffian orientations, and even direc

circuits (Extended abstract), in: Proc. Symp. on the Theory of Computing (STOC’97), 1997.[21] N. Robertson, P.D. Seymour, R. Thomas, Permanents,Pfaffian orientations, and even directed circuits, A

of Math. 150 (1999) 929–975.[22] D.E. Rutherford, Some continuant determinants arising in physics and chemistry, Proc. Roy. Soc. Edinbu

Sect. A 62 (1947) 229–236.[23] F.J. Zhang, W.G. Yan, Enumeration of perfect matchings in a type of graphs with reflective symmet

MATCH—Commun. Math. Comput. Chem. 48 (2003) 117–124.[24] F.J. Zhang, H.P. Zhang, A note on the number of perfect matchings of bipartite graphs, Discrete Ap

Math. 73 (1997) 275–282.

Documents

Enumeration of perfect matchings of graphs with reflective symmetry by Pfaffians