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On the Matching Polynomials of Graphs with Small Number of Cycles of Even Length On the Matching Polynomials of Graphs with Small Number of Cycles of Even Length WEIGEN YAN,1,2 YEONG-NAN YEH,2 FUJI ZHANG3 1School of Sciences, Jimei University, Xiamen 361021, China 2Institute of Mathematics, Academia Sinica, Nankang, Taipei 11529, Taiwan 3Department of Mathematics, Xiamen University, Xiamen 361005, China Received 15 November 2004; accepted 16 March 2005 Published online 31 May 2005 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/qua.20670 ABSTRACT: Suppose that G is a simple graph. We prove that if G contains a small number of cycles of even length then the matching polynomial of G can be expressed in terms of the characteristic polynomials of the skew adjacency matrix A(Ge)ofan arbitrary orientation Ge of G and the minors of A(Ge). In addition to a formula previously discovered by Godsil and Gutman, we obtain a different formula for the matching polynomial of a general graph. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem 105: 124–130, 2005 Key words: matching polynomial; acyclic polynomial; matching generating polynomial; perfect matching; Hosoya index; Pfaffian orientation mer [23] for terminology and notation not defined 1. Introduction here. Suppose that G is a simple graph. Let ⌽(G)bethe n this article, we suppose that G ϭ (V(G), E(G)) set of all orientations of G. For an arbitrary orien- I is a finite simple graph with the vertex-set tation Ge (ʦ ⌽(G)) of G, the skew adjacency matrix ϭ ϭ e e V(G) {v1, v2,...,vn} and the edge-set E(G) {e1, of G , denoted by A(G ), is defined as follows: ϫ e2,...,e␧}. Denote by G H the Cartesian product ͑ ͒ ʦ ͑ e͒ of two graphs G and H and by In the unit matrix of 1ifvi, vj E G , e Ϫ ͑ ͒ ʦ ͑ e͒ A͑G ͒ ϭ ͑a ͒ ϫ , a ϭ ͭ 1ifvj, vi E G , order n, respectively. We refer to Lova´sz and Plum- ij n n ij 0 otherwise. Obviously, the skew adjacency matrix A(Ge)isa Correspondence to: Y.-N. Yeh; e-mail: [email protected]. edu.tw skew symmetric matrix. This work is supported by FJCEF (JA03131), FMSTF A set M of edges in G is a matching if every (2004J024), NSC93-2115-M001-002, and NSFC (10371102). vertex of G is incident with at most one edge in M; International Journal of Quantum Chemistry, Vol 105, 124–130 (2005) © 2005 Wiley Periodicals, Inc. MATCHING POLYNOMIALS OF GRAPHS it is a perfect matching if every vertex of G is yk. Godsil and Gutman [7] reported some properties incident with exactly one edge in M. Denote by of matching polynomial of a graph. Beezer and ␾ k(G) the number of k-element matchings in G.We Farrell [3] considered the matching polynomial of ␾ ϭ ␾ ϭ ␧ set 0(G) 1 by convention. Thus 1(G) is the the distance-regular graph. ␾ number of edges in G, and if n is even then n/2(G) The matching polynomial of a tree T equals its is the number of perfect matchings of G and will be characteristic polynomial det(xI Ϫ A), where A is denoted by ␾(G). The number of matchings of G is the adjacency matrix of T. A related question has called the Hosoya index (see Ref. [18]) and will be been posed in Ref. [2]: For any graph G, is it possi- ϭ ¥ ␾ denoted by Z(G), that is, Z(G) kՆo k(G). The ble to construct a Hermitian matrix H(G) such that following two polynomials m(G, x) and g(G, x) are the matching polynomial of G is the characteristic called the matching polynomial and matching gen- polynomial of H(G)? The matrix H(G) can be con- erating polynomial of G, respectively (see Ref. [23]): structed for certain graphs including unicyclic graphs and cacti (the progress in this field can be n/2 found in Refs. [9–12, 14, 15, 17, 26, 27, 29]). But, in ͑ ͒ ϭ ͸ ͑Ϫ ͒k␾ ͑ ͒ nϪ2k m G, x 1 k G x , general, H(G) does not exist [14]. The matching kϭ0 polynomial of an outerplanar graph can be deter- n/2 mined in polynomial time (see Refs. [6, 23]). But the ͑ ͒ ϭ ͸ ␾ ͑ ͒ k g G, x k G x . computation of the matching polynomial of general kϭ0 graphs is NP-complete [23]. Let G be a graph with ϭ adjacency matrix A (aij)nϫn.IfS is a subset of ϭ n Ϫ Ϫ2 ϭ Then m(G, x) x g(G, x ) and Z(G) g(G, 1). E(G), let AS be the matrix such that The matching polynomial is a crucial concept in the topological theory of aromaticity [1, 15, 30, 31] ͑ ͒ ʦ͞ ͑ ͒ ϭ ͭaij if vi, vj S, and is also named the acyclic polynomial [15, 30, AS ij Ϫ ͑ ͒ ʦ aij if vi, vj S. 31]. Sometimes the matching polynomial is also referred to as the reference polynomial [1]. It takes Godsil and Gutman [8] showed that no great imagination to consider the idea of repre- senting a molecule by a graph, with the atoms as vertices and the bonds as edges, but it is surprising 1 m͑G, x͒ ϭ Ϫ␧ ͸ det͑xI Ϫ A ͒ 2 S that there can be an excellent correlation between SʕE͑G͒ the chemical properties of the molecule and suit- ably chosen parameters of the associated graph. and used this to deduce that if G is connected then One such parameter is the sum of the absolute the spectral radius of A is at least as large as the values of the zeros of the matching polynomial of largest zero of m(G, x), and that equality holds if the graph, and this has been related to properties of and only if G is a tree. Furthermore, Godsil and aromatic hydrocarbons. The roots of the matching Gutman [7] proved an interesting result as follows: polynomial correspond to energy levels of a mole- ␧ Let u ϭ (u , u ,...,u␧) ʦ {Ϫ1, ϩ1} . Let e , e ,..., cule and a reference structure. A new application in 1 2 1 2 e␧ be the edges of G and G be the weighted graph Hu¨ ckel theory was discussed in detail by Trinajstic´ u obtained from G by associating the weight u with [30]. Hosoya [18] used the matching polynomial in i the edge e (i ϭ 1,2,...,␧). Further let A(G )bethe chemical thermodynamics. For further applications i u adjacency matrix of G . Then of the matching polynomial in chemistry, see, e.g., u Trinajstic´’s book [31]. ͑ ͒ ϭ Ϫ␧ ͸ ͑ Ϫ ͑ ͒͒ The matching polynomial m(G, x) has also been m G, x 2 det xI A Gu , (1) independently discovered in statistical physics and u in a mathematical context. In statistical physics it ␧ was first considered by Heilman and Lieb in 1970 where the summation ranges over all 2 distinct [16] and soon after by Kunz [21] and Gruber and ␧-tuples u. Kunz [13]. Finally, Farrell [5] introduced what he So far, there exist no valid methods to calculate called the matching polynomial of a graph, this the matching polynomial of a graph except the time in a mathematical context. This is a polynomial outerplanar graph G, the matching polynomial of in two variables (say x and y) which results if, in the which can be determined in polynomial time (see definition of m(G, x), the term (Ϫ1)k is replaced by Exercise 8.5.4 in Ref. [23]). INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 125 YAN, YEH, AND ZHANG In next section, we prove that if G contains only obtained from G by deleting the vertices of C and small number of cycles of even length then the incident edges. matching polynomial of G can be expressed in If Ge is an orientation of a simple graph G and C terms of the characteristic polynomials of the skew is a cycle of even length, we say that C is oddly adjacency matrix A(Ge) of an arbitrary orientation oriented in Ge if C contains odd number of edges Ge of G and the minors of A(Ge). In our case, we that are directed in Ge in the direction of each reduce the running time of (1) by a factor of about orientation of C. We say that Ge is a Pfaffian orien- ␧Ϫ 1/2 k (where k is the number of cycles of even tation of G if every nice cycle of even length of G is length in G), because we only need to calculate 2k oddly oriented in Ge. determinants. In addition to formula (1) previously discovered by Godsil and Gutman, we obtain a Proposition 4 ([23]). Let Ge be a Pfaffian orienta- different formula for the matching polynomial of a tion of a graph G. Then general graph. In Section 3, some further problems are posed. ͓␾͑G͔͒2 ϭ det A͑Ge͒, e e 2. Main Results where A(G ) is the skew adjacency matrix of G . The method to enumerate perfect matchings of Proposition 1 ([6, 23]). Let G be a graph with ␧ plane graphs in Proposition 4 was found by the edges. The number of perfect matchings of G physicist Kasteleyn [19, 20], and is called the Pfaf- fian method. This method was generalized by Little [22]. McCuaig [24], McCuaig et al. [25], and Robert- 1 e ␾͑G͒ ϭ ␧ ͸ det͑A͑G ͒͒, son et al. [28] found a polynomial-time algorithm to 2 Geʦ⌽͑G͒ show whether a bipartite graph has a Pfaffian ori- entation.
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