Discrete Applied Mathematics 166 (2014) 210–214

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Discrete Applied Mathematics

journal homepage: www.elsevier.com/locate/dam

An infinite family of graphs with a facile count ofperfect matchingsVladimir R. Rosenfeld a,b, Douglas J. Klein a,∗

a Mathematical Chemistry Group, Department of Marine Sciences, Texas A&M University at Galveston, Galveston, TX 77553–1675, USAb Instituto de Ciencias Matematicas (ICMAT) CSIC, C/ Nicolás Cabrera, no 13–15, Campus Cantoblanco UAM, 28049 Madrid, Spain

a r t i c l e i n f o

Article history:Received 8 February 2013Received in revised form 5 September 2013Accepted 24 September 2013Available online 29 October 2013

Keywords:Cartesian product of graphsk-matchingPerfect matchingMatching polynomialPermanentContracted permanentHosoya indexDegree sequenceImmanant

a b s t r a c t

Given a graph G = (V , E), let the triangulation G△= (V△, E△) of G be the graph obtained

from G by supplementing each uv ∈ E with a new vertex w along with new edges uw andwv (while retaining uv). Let dv be the degree of a vertex v ∈ V and let G be a tree T . Thenit is proved that the count of perfect matchings of the Cartesian product of T△ with K2 isgiven as the product of factors dv + 1 over all v ∈ V . Also under favorable conditions, thedegree sequence of T△

× K2 is reconstructed via factorization of the number of its perfectmatchings. Previously introduced degree product polynomials play a helpful role.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Counting perfect matchings of an arbitrary graph is an N P -complete problem [9]. Mathematicians and programmerscontinue to elaborate new combinatorial approaches and algorithmswhichmay bemore efficient than already known ones,often for a particular type of graph. This work requires also graphs for which the count of perfect matchings is facile enoughto be used for checking their working hypotheses.

We found an infinite family of required graphs. Every tree generates such a graph. By construction, all these graphs havetriangles. In a more complete and rigorous form, the graphs will be introduced in the main section, to which we turn now.

2. The main part

We consider finite graphs which may have simple loops, but no multiple edges. Let G = (V , E) be a graph with thevertex set V and edge set E(|V | = n, |E| = m). The Cartesian product G × K2 of graphs G and K2 is constructed fromtwo copies of G by joining with an edge each vertex v of the one copy G = (V , E) to the corresponding vertex v′ of theother copy G′

= (V ′, E ′) which is isomorphic to G = (V , E), under the correspondence V ∋ v ↔ v′∈ V ′. Accordingly,

|V (G × K2)| = 2n, |E(G × K2)| = 2m + n, and n new connecting edges comprise a perfect matchingM which is a cut set inG × K2.

∗ Corresponding author. Tel.: +1 409 740 4538; fax: +1 409 740 4787.E-mail addresses: rosenfev@tamug.edu, vladimir_rosenfeld@yahoo.com (V.R. Rosenfeld), kleind@tamug.edu (D.J. Klein).

0166-218X/$ – see front matter© 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.dam.2013.09.010

V.R. Rosenfeld, D.J. Klein / Discrete Applied Mathematics 166 (2014) 210–214 211

The positive-sign matching polynomial of graph G is:

α+(G; x) :=

⌊n/2⌋k=0

p(G; k)xn−2k, (1)

where p(G; k) denotes the number of k-matchings in G, that is, the number of ways k independent edges can be chosen inG; p(G; 0) := 1; and, for even n, p(G; n/2) = µ(G) is the number of perfect matchings, covering all n vertices of G. We denoterespectively by P (G) and M(G) the set of all perfect matchings and the set of all matchings, of G, which may be empty;P (G) ⊆ M(G) and |P (G)| = µ(G).

The negative-sign matching polynomial of graph G is an alternative to (1):

α−(G; x) :=

⌊n/2⌋k=0

(−1)kp(G; k)xn−2k. (2)

From (1) and (2), it evidently follows that

α+(G; x) = inα−(G; −ix) and α−(G; x) = inα+(G; −ix). (3)

The following is due to Corollary 6(a) of [1]:

Proposition 1. Let ‘‘×’’ denote the Cartesian product of graphs. Then,

α+(G × K2; x) =

U⊆V (G)

α+(G − U; x)

2. (4)

Proof. Corollary 6(a) of [1] gives a similar negative-sign result for α−(G; x):

α−(G × K2; x) =

U⊆V (G)

(−1)|U|α−(G − U; x)

2. (5)

Hence, using (3) or, simply, substituting a sign ‘‘+’’ for all signs ‘‘−’’ therein, we immediately arrive at (4). �

The following corollary will play a technical role. Viz.:

Corollary 1.1. Let ‘‘×’’ denote the Cartesian product of graphs. Then,

µ(G × K2) =

U⊆V (G)

[µ(G − U)]2 . (6)

By definition, the adjacency matrix A(G × K2) of G × K2 is

A(G × K2) =

A InIn A

, (7)

where A is the adjacency matrix of G, while In is an n × n unit matrix.Recall that the permanent [10] of a square matrix B = [brs]nr, s=1 is defined as

perB :=

σ∈Sn

nr=1

brσ r , (8)

where σ r is the image of an index r under the action of the permutation σ of the symmetric group Sn; and the summationranges over all elements σ ∈ Sn.

For broader interests, we introduce here also the contracted permanent:

per∗B :=

σ∈Tn

nr=1

brσ r , (9)

where Tn is the subset of all permutations of Sn without permutation-cycles of odd length ≥3. To address a practicalimport of per∗, first recall [3] that, for all bipartite graphs G with adjacency matrix A(G), per[A(G)] is the square of thenumber µ(G) of the perfect matchings of G, while for nonbipartite graphs just the inequality [µ(G)]2 ≤ per[A(G)] holds.As demonstrated below, the contracted permanent per∗[A(G)] equals the square of the number of perfect matchings ofG : [µ(G)]2 = per∗[A(G)].

212 V.R. Rosenfeld, D.J. Klein / Discrete Applied Mathematics 166 (2014) 210–214

Note that both per and per∗ are particular cases of a more general matrix function. Let λ label a group theoretic characterχλ of the symmetric group Sn – which is to say that χλ is a conjugacy-class function (from Sn to R). The immanant of an n×nmatrix B = [ars]nr, s=1 associated with a character χλ is defined as the expression

Immλ(B) :=

σ∈Sn

χλ(σ )

nr=1

brσ r . (10)

This follows Littlewood and Richardson [8] and Littlewood [7], though they required the λ to label an irreduciblerepresentation. Most of the general properties Littlewood and Richardson note for their immanants (e.g., a generalizedLaplace expansion) extend to those with a reducible representation, such as for our per∗.

The determinant is a special case of the immanant, where χλ is the alternating character sgn, of Sn, defined by the parityof a permutation. The permanent is the case where χλ is the trivial character, which is identically equal to 1.

At this juncture, we recall that, due to the graph-theoretical context of this discussion, we can also define per[A(G)] usingpurely graphical combinatorial arguments. A (full) Sachs cover of vertices of a graph G is a cover of (all) vertices thereof withits disjoint j-cycles (1 ≤ j ≤ n) (where 1-cycles, if any, are loops, and 2-cycles are edges). By virtue of the famous Sachstheorem, which is also ascribed to Coulson and Harary (see Theorems 1.2◦ and 1.3◦ in Section 1.4 of [3]), we have

per[A(G)] =

γ∈C(G)

2c(γ ), (11)

where γ stands for a full Sachs cover of the vertices of G with disjoint unoriented j-cycles (1 ≤ j ≤ n); c(γ ) is the numberof cycles of lengths ≥3 in γ .

Similarly, we equivalently characterize

per∗[A(G)] =

γ ∗∈C∗(G)

2c(γ ∗), (12)

where C∗(G) is the set of all Sachs covers of vertices, of G, containing only unoriented cycles of even lengths and/or loops(the latter, if any, correspond to the fixed points of permutations σ , on the R.H.S. of (9)); c(γ ∗) is the number of all cyclesof even lengths ≥4, in γ ∗. Here, note that, in principle, (11) and (12) can directly be deduced from (8) and (9), respectively;analogously, there could be derived all permanental versions of the Sachs theorem that are presented in Section 1.4 of [3].

Originally (see Section 1.4 in [3]), the number 2 in (11) was introduced just as the number of all possible circularorientations of edges of an arbitrary cycle of G. Here, anticipating the treatment of the cycles of even lengths only, we mayalternatively interpret the same number 2 as the number of ways in which an arbitrary cycle of even length ≥4, of G, canbe decomposed into ordered pairs of edge-disjoint perfect matchings thereof (that is, say one pair is (M1,M2) and the other(M2,M1)).

Here, we state:

Lemma 2. Let µ(G) be the number of perfect matchings of a loopless graph G. Then,

[µ(G)]2 = |P × P | =

γ ∗∈C∗(G)

2c(γ ∗). (13)

Proof. The first equality in (13) is obvious; therefore, we need to prove only the second one. Considering that G has no loop,2c(γ ∗) equals the number of all ordered pairs of perfect matchings, of G, into which a respective full Sachs cover γ ∗ can bedecomposed. Since the final summation of (3) on the third side of (13) ranges just over the entire subset C∗(G) of such coversγ ∗, it precisely corresponds to the ordered pairs of perfect matchings, of G, and no other. This completes the proof. �

Now, we can prove the following general result:

Theorem 3. Let µ(G) and per∗[A(G)] be the number of perfect matchings of a loopless graph G and the contracted permanent ofits adjacency matrix, respectively. Then,

µ(G) =

per∗[A(G)]. (14)

Proof. First, emphasize that just covers γ ∗∈ C∗ and no covers γ ∈ C(G) \ C∗(G) make a nonzero contribution to the final

sum of (13). From (12), it follows that this is equal to per∗[A(G)]; but this same sum of (13) is also equal to [µ(G)]2. All thistogether gives [µ(G)]2 = per∗[A(G)]; whence the proof is immediate. �

A contracted permanental polynomial π∗(G; x) of a graph G may be defined by analogy with the usual permanentalpolynomial as:

π∗(G; x) := per∗(A(G) + xI), (15)

where I is the unit matrix.

V.R. Rosenfeld, D.J. Klein / Discrete Applied Mathematics 166 (2014) 210–214 213

A useful common corollary of Corollary 1.1 and Theorem 3 is:

Corollary 3.1. Let G × K2 be the Cartesian product of a loopless graph G and K2. Then,

µ(G × K2) = per∗[A(G) + I]. (16)

Proof. Recall Corollary 1.1. By the definitions of per∗ and π∗(G; x), and by virtue of Theorem 3, the R.H.S. of (6) transformsto give an equivalent form

µ(G × K2) =

U⊆V (G)

per∗(G − U), (17)

which in turn equivalently transforms to

µ(G × K2) = π∗(G; 1), (18)

since the sum on the R.H.S. of (17) ranges over all 2n vertex subgraphs of G, and this sum equals to π∗(G; 1). Whence theproof follows. �

The Hosoya index Z(G) [5] is defined as

Z(G) :=

⌊n/2⌋k=0

p(G; k) ≡ α+(G, 1). (19)

The next result is useful in a wide context:

Proposition 4. Let G be a loopless graph without cycles of even lengths, and let G × K2 be the Cartesian product of G and K2.Then,

µ(G × K2) = Z(G). (20)

Proof. Indeed, with such G, without cycles of even lengths, π∗(G, 1) on the R.H.S. of (18) equals α+(G; 1), which gives theproof. �

Corollary 4.1. Let G be a loopless graph without cycles of even lengths, and let G×K2 be the Cartesian product of G and K2. Then,

per∗[A(G × K2)] = [Z(G)]2. (21)

Let the triangulation H△ of H be the graph [11] obtained from graph H such that for each uv ∈ E(G) there is added asingle new degree-2 vertex w attached by new edges to u and to v. That is, H△ is obtained from H changing each edge uv ofH into a triangle uwv with w the new vertex associated with uv. Yan and Yeh [11] proved the following theorem:

Theorem 5. Let H△ be the graph obtained from a loopless graph H as above. Then,

Z(H△) =

n(H)j=1

(dj + 1), (22)

where dj is the degree of vertex j in H .Since the calculation of Z(H△) in (22) is done with little cost, (20) and (21) become important. Indeed, one can take G

to be H△ (without even cycles) and consider the Cartesian product H△× K2; however, this is possible only if H is a tree T ,

because any cycle of an odd length sharing an edge with a triangle generates a cycle of an even length, which is forbiddenin G, in this context.

Proposition 4 and Theorem 5 have in common the following target corollary:

Proposition 6. Let T△ be the graph obtained from a tree T as above. Then,

µ(T△× K2) =

n(T )j=1

(dj + 1), (23)

where dj is the degree of vertex j in T .

Remark. T△× K2 is the graph obtained by gluing lateral (nonbasal) edges of trigonal prisms such that no prism is attached

to more than two others. Such graphs may be of use in chemistry as models of novel nanomolecules.

214 V.R. Rosenfeld, D.J. Klein / Discrete Applied Mathematics 166 (2014) 210–214

Now, recall two equivalent degree-product polynomials [6] reflecting all the information contained in the degree sequenceof a graph H . Namely,

P+(H; x) :=

n(H)j=1

(x + dj);

P−(H; x) :=

n(H)j=1

(x − dj). (24)

Utilizing the degree-product polynomials, we state also:

Proposition 7. Let H△ be the graph obtained from H as above. Then,

Z(H△) = P+(H; 1) = (−1)n(H)P−(H; −1). (25)

In fact, Yan and Yeh [11] defined the polynomialn(H)

j=1 (1 + xdj) which can easily be related to P+(H; x) and P−(H; x).

Corollary 7.1. Let H△ be the graph obtained from H as above. Then

Z(H△) = 2−2m(H)−n(H)P+(H△; 2) = 2−2m(H)−n(H)(−1)m(H)+n(H)P−(H△

; −2). (26)

The proof rests with the reader as well as that of deriving a similar expression for Z(T△) in terms of the vertex degreesof T△

× K2 (with H△ reduced to T△). Already (25) and (26) show the utility of the polynomials (24).Note that, if every dj ≤ 4 (as in graphs of organic molecules), the factorization of the product on the R.H.S. of (23) may

produce just 2, 3, 4, 5 as factors. As has been specially shown [2,4], knowing n(T ) and m(T ) for such a case allows oneto reconstruct the entire degree sequence of T from the values of P+(T ; 1), as appear on the R.H.S. of (23). Here we note(a modestly more general result):

Proposition 8. If T is a tree of n vertices, which are confined to be of degrees 1, 2, 3, 4, 5, or any ρ − 1 with ρ a prime ≥7,then µ(T△

× K2) and the degree sequence of T uniquely determine one another.

3. Conclusions

Simple formulations for perfect-matching (or Kekulé-structure) counts have been obtained for different classes of graphs.The classes generated from trees (via Cartesian product, or triangulation followed by Cartesian product) are neatly treated,retaining some of the simplicity of the acyclic tree. Such graphs might conceivably be of chemical relevance in connectionwith dendrimers and other hyperbranched polymer structures.

We also hope to better investigate relationships found for the product of degrees and degree-product polynomials [2,4,6]to see if these may be used in further generalizations of the (Cartesian products of) graphs.

Acknowledgment

The authors acknowledge the support of the Welch Foundation of Houston, Texas, via grant BD–0894.

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