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. Yan and Zhang [32, 33] used the Pfaffian where the sum is over all orientations of G. method to obtain some formulas of the number of Proposition 1 is equivalent to the following: perfect matchings and the permanental polynomi- als for a type of symmetric graphs. Proposition 2 ([23]). Let G be a graph and Ge a random orientation of G, obtained by orienting Proposition 5 ([32]). Let T be a tree with n verti- e each edge in G independently of the others with ces, and T be an arbitrary orientation. Then 1, e probability 1/2 in either direction. Let A(G )bethe 2,..., n are eigenvalues of A(T) if and only if i 1, e skew adjacency matrix of G. Then the expected i 2,..., i n are eigenvalues of A(T ), where A(T) value of det A(Ge)is(G). and A(Te) are the adjacency matrix of T and the skew adjacency matrix of Te, respectively, and Proposition 3 ([6]). Let e ϭ uv be an edge of a i2 ϭϪ1. graph G. Then Proposition 6 ([32]). Suppose G is a graph with n m͑G, x͒ ϭ m͑G Ϫ e, x͒ Ϫ m͑G Ϫ u Ϫ v, x͒, vertices. If G admits an orientation Ge under which every cycle of even length is oddly oriented in Ge ϫ ϭ ϩ e where G Ϫ e (resp. G Ϫ u Ϫ v) denotes the subgraph then (G K2) det(In A(G )), where In is a unit of G by deleting edge e (resp. vertices u and v and matrix of order n. the incident edges). To give a valid method to calculate the matching Lemma 7. Suppose that G is a simple graph with even number of vertices having no cycles of even polynomial, the matching generating polynomial e and the Hosoya index for the graph containing length and G is an arbitrary orientation of G. Then small number of cycles of even length, we need to introduce the following terminology. 1, if G has perfect matchings, det͑A͑Ge͒͒ ϭ ͭ Let G be a graph. If M is a perfect matching of G, 0, otherwise. an M-alternating cycle in G is a cycle whose edges M and M. We say that a cycle Proof. Note that G contains no cycles of evenگ(are alternately in E(G C of G is nice if G Ϫ C contains a perfect matching, length. The number of perfect matchings of G where G Ϫ C denotes the induced subgraph of G equals one or zero. Otherwise, suppose that G con-
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tains two perfect matchings F1 and F2. Then the the vertex-set of which is V(Mj). It is easy to see that 1 2 ʦ ⌿ 1 symmetric difference of F1 and F2 has at least one is a surjection. Suppose Mj , Mj 1 and Mj 2 alternating cycle, a contradiction. By the definition Mj . Note that G contain no cycles of even length. It e 1 2 of the Pfaffian orientation, G is a Pfaffian orienta- is not difficult to see that G[Mj ] G[Mj ]. This tion of G. Hence, by Proposition 4, the square of the shows that is a injection. So we have proved that number of perfect matchings of G equals det- the number j(G)ofj-element matchings of G (A(Ge)). So the lemma has been proved. equals the number of induced subgraphs with 2j vertices of G, each of which has perfect matchings. Theorem 8. Suppose that G is a simple graph with Hence we have n vertices having no cycles of even length and Ge is an arbitrary orientation of G. Then n/2 ͑ ϩ ͑ e͒͒ ϭ ͑ ͑ e͒͒ nϪ2j det xIn A G ͫ det A H ͬx ϭ ϩ e 1. m(G, x) det(xIn iA(G )), jϭ0 He ϭ ϩ ͌ e 2. g(G, x) det(In xA(G )), n/2 ϭ ϩ e ϭ ͑ ͒ nϪ2j ϭ n ͑ Ϫ2͒ 3. Z(G) det(In A(G )), j G x x g G, x . jϭ0 where i2 ϭϪ1, and A(Ge) is the skew adjacency e matrix of G . Using the above result, we can show that
Proof. For an arbitrary orientation Ge (ʦ ⌽(G)) of ͑ ͒ ϭ ͑ ϩ ͑ e͒͒ G, we have m G, x det xI iA G , g͑G, x͒ ϭ det͑I ϩ ͱxA͑Ge͒͒, Z͑G͒ ϭ det͑I ϩ A͑Ge͒͒. n ͑ ϩ ͑ e͒͒ ϭ ͑ ͑ e͒͒ nϪk det xIn A G ͫ det A H ͬx , Hence the theorem has been proved. kϭ0 He
Remark 9. By Exercise 3.2.3 in Bondy and Murty where the second sum is over all induced subdi- [4], a graph G contains no cycles of even length if graphs with k vertices of Ge. Note that if k is odd and only if every block of G is either a cycle of odd then det(A(He)) ϭ 0, hence length or a K2.
n/2 By Propositions 5 and 6 and Theorem 8, we have e e nϪ j the following: det͑xI ϩ A͑G ͒͒ ϭ ͫ det͑A͑H ͒͒ͬx 2 , (2) jϭ0 He Corollary 10. If G is a graph with n vertices con- taining no cycles of even length then (G ϫ K ) ϭ where the second sum ranges over all induced sub- 2 Z(G) ϭ det(I ϩ A(Ge)) for any orientation Ge of G, digraphs with 2j vertices of Ge. Because there exist n where A(Ge) is the skew adjacency matrix of Ge. no cycles of even length in G, the underlying graph of He contains no cycles of even length. Hence, by Corollary 11. If T is a tree with n vertices, then the Lemma 7, matching polynomial of T equals the characteristic polynomial of T. 1, if H has perfect matchings, det͑A͑He͒͒ ϭ ͭ 0, otherwise. Corollary 12. Suppose G is a graph with n verti- ces, which has exactly one cycle C of even length. e Then ¥ e det(A(H )) in Eq. (2) equals the number of Јϭ e H Let e vsvt be an edge in C and G an arbitrary induced subgraphs with 2j vertices of G, each of e ϭ orientation of G, and let A(G ) (aij)nϫn be the skew which has perfect matchings. adjacency matrix of Ge. Then ⌿ ⌿ Let 1 and 2 denote the set of j-element match- ings of G and that of induced subgraphs with 2j m͑G x͒ ϭ ͑xI ϩ iB ͒ Ϫ ͑xI Ϫ ϩ iB ͒ vertices of G which have perfect matchings, respec- , det n 1 det n 2 2 , tively. Suppose :⌿ 3 ⌿ is a map such that 1 2 g͑G, x͒ ϭ det͑I ϩ ͱxB ͒ ϩ x det͑I Ϫ ϩ ͱxB ͒, ϭ n 1 n 2 2 (Mj) G[Mj] for an arbitrary j-element matching ͑ ͒ ϭ ͑ ϩ ͒ ϩ ͑ ϩ ͒ Mj of G, where G[Mj] is the induced subgraph of G, Z G det In B1 det InϪ2 B2 ,
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 127 YAN, YEH, AND ZHANG
ϭ ϭ where B1 (bij)nϫn, bij equals zero if (i, j) (s, t)or method calculating the matching polynomial is e (t, s) and ai,j otherwise, B2 is the minor of A(G )by valid. deleting the sth and tth rows and columns. Theorem 15. Suppose that G is a simple graph Proof. Note that both of graphs G Ϫ eЈ and G Ϫ with n vertices and edges. Then the matching Ϫ vs vt contain no cycles of even length. Hence, by polynomial of G Theorem 8, 1 ͑ Ϫ Ј ͒ ϭ ͑ ϩ ͒ ͑ ͒ ϭ ͑ ϩ ͑ e͒͒ m G e , x det xIn iB1 , m G, x det xI iA G , 2 n ͑ Ϫ Ϫ ͒ ϭ ͑ ϩ ͒ Geʦ⌽͑G͒ m G vs vt, x det xInϪ2 iB2 .
e By Proposition 3, m(G, x) ϭ m(G Ϫ eЈ, x) Ϫ m(G Ϫ where the sum ranges over all orientations G of G Ϫ and i2 ϭϪ1. vs vt). Hence
͑ ͒ ϭ ͑ ϩ ͒ Ϫ ͑ ϩ ͒ Proof. For an arbitrary orientation Ge (ʦ ⌽(G)) of m G, x det xIn iB1 det xInϪ2 iB2 . G, we have Ϫ Since m(G, x) ϭ xng(G, Ϫx 2) and Z(G) ϭ g(G, 1), the corollary follows. n Similarly, we can prove the following. ͑ ϩ ͑ e͒͒ ϭ ͑ ͑ e͒͒ nϪk det xIn A G ͫ det A H ͬx , kϭ0 He Corollary 13. Suppose G is a graph with n verti- ces, which has exactly two cycles C1 and C2 of even Јϭ Љϭ where the second sum is over all induced subdi- length. Let e vsvt and e vlvm be two edges in C1 e e graphs with k vertices of G . Note that if k is odd and C2, respectively. Let G be an arbitrary orienta- e e ϭ then det(A(H )) ϭ 0. Hence tion of G and A(G ) (aij)nϫn be the skew adjacency matrix of Ge. Then n/2 ͑ ϩ ͑ e͒͒ ϭ ͑ ͑ e͒͒ nϪ2j M͑G, x͒ ϭ det͑xI ϩ iB ͒ ϩ det͑xI Ϫ ϩ iB ͒ det xIn A G ͫ det A H ͬx , n 1 n 4 2 ϭ e Ϫ ͑ ϩ ͒ Ϫ ͑ ϩ ͒ j 0 H det xInϪ2 iB3 det xInϪ2 iB4 ,
͑ ͒ ϭ ͑ ϩ ͱ ͒ ϩ 2 ͑ ϩ ͱ ͒ where the second sum is over all induced subdi- g G, x det In xB1 x det InϪ4 xB2 graphs with 2j vertices of Ge. Thus we have ϩ ͑ ϩ ͱ ͒ ϩ ͑ ϩ ͱ ͒ x det InϪ2 xB3 x det InϪ2 xB4 , 1 ͑ ͒ ϭ ͑ ϩ ͒ ϩ ͑ ϩ ͒ e Z G det In B1 det InϪ4 B2 det͑xI ϩ A͑G ͒͒ 2 n ϩ ͑ ϩ ͒ ϩ ͑ ϩ ͒ Geʦ⌽͑G͒ det InϪ2 B3 det InϪ2 B4 , 1 n/2 ϭ ͑ ͑ e͒͒ nϪ2j e Ϫ ЈϪ ͫ det A H ͬx . where B1 is the skew adjacency matrix of G e 2 eʦ⌽͑ ͒ ϭ e Љ e Ϫ Ϫ Ϫ G G j 0 H e , B2 the skew adjacency matrix of G vs vt Ϫ e Ϫ ЈϪ vl vm, B3 the skew adjacency matrix of G e Ϫ e Ϫ Љ e el vm and B4 the skew adjacency matrix of G e Suppose that the underlying graph of H is H and Ϫ Ϫ Ј vs vt. the number of edges in H is . Then the subgraph ϪЈ G Ϫ E(H)ofG has 2 orientations. Denote the ϪЈ ⌽ Ϫ ϭ Remark 14. Similarly to that in Corollaries 12 and set ofϪ theseЈ 2 orientations by (G E(H)) Ј 2 Ј ⌽ Ϫ 13, by deleting one edge from each of the k cycles of {Mk}kϭ1 . For an arbitrary orientation Mk in (G Ј even length of G, we can express the matching E(H)), then H has 2 orientations. Hence we may Ͼ ⌽ polynomial of G with k (k 2, k is small) cycles of partition theϪЈ set (G) of all orientations of G into ⌽ ϭ ഫ2 Ј even length in terms of characteristic polynomials (G) kϭ1 Mk*, where Mk* is the set of those 2 Ј of 2k skew adjacency matrices, which are either the orientations of G in which H has 2 orientations e e Ϫ Ј skew adjacency matrix A(G ) of an orientation G of and the orientation of the subgraph G E(H)isMk. G or the minors of A(Ge). Because k is small, this Hence, by Proposition 1, we have
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1 n/2 1 ͑ ͑ e͒͒ nϪ2j ͑ ϩ ͑ e͒͒ ͫ det A H ͬx det ixIn A G 2 2 Geʦ⌽͑G͒ jϭ0 He Geʦ⌽͑G͒ n Ϫ2 n 2 2ϪЈ n/2 ϭ ͑ix͒ g͑G, ͑ix͒ ͒, m͑G, x͒ ϭ x g͑G, Ϫx ͒ 1 ϭ ͑ ͑ e͒͒ nϪ2j ͫ det A H ͬx 2 1 kϭ1 HeʦM* jϭ0 He ϭ ͑ Ϫ ͑ e͒͒ k det xI iA G 2 n ϪЈ Geʦ⌽͑G͒ 1 n/2 2 ϭ ͑ ͑ e͒͒ nϪ2j ͫ det A H ͬx 1 e 2 * ϭ ͑ ϩ ͑ ͒͒ jϭ0 He kϭ1 HeʦM det xIn iA G . k 2 Geʦ⌽͑G͒ ϪЈ 1 n/2 2 Ј nϪ2j ϭ ͓2 ͑H͔͒x 2 Thus we have finished the proof of the theorem. jϭ0 He kϭ1 By Theorem 15, we can prove the following. 1 n/2 n/2 ϭ ϪЈ͓ Ј͑ ͔͒ nϪ2j ϭ ͑ ͒ nϪ2j 2 2 H x ͫ H ͬx , Corollary 16. Suppose that G is a simple graph 2 ϭ e ϭ e j 0 H j 0 H with n vertices and edges. Then
where (H) denotes the number of perfect match- 1 e g͑G, x͒ ϭ det͑I ϩ ͱxA͑G ͒͒, ings of H. Now we prove the following claim. 2 n Geʦ⌽͑G͒
¥ e ϭ ¥ 1 Claim. H (H)( H (H)) equals j(G), the e Z(G)ϭ det͑I ϩA͑G ͒͒, number of j-element matchings in G. 2 n Geʦ⌽(G) Let ⌿ be the set of j-matchings of G and ⌼ equals ഫM(H), where M(H) denotes the set of perfect where g(G, x) and Z(G) denote the matching gener- matchings of H and the union ഫ ranges over all ating polynomial and the Hosoya index of G, re- induced subgraphs with 2j vertices of G. Note that പ ϭ A spectively. M(H1) M(H2) for any two induced subgraphs H and H with 2j vertices of G such that H H . 1 2 1 2 Remark 17. Proposition 1 is a special case of The- It is not difficult to see that there exists a bijection orem 15 (the case x ϭ 0). Hence Theorem 15 gener- between ⌿ and ⌼. The claim thus follows and then alizes the result of Proposition 1.
1 Remark 18. Though we obtain algebraic expres- ͑ ϩ ͑ e͒͒ det xIn A G sions for the matching polynomial by Theorem 15 2 eʦ⌽͑ ͒ G G and for the Hosoya index by Corollary 16, the com- 1 n/2 putation of these two formulae is not in polynomial ϭ ͑ ͑ e͒͒ nϪ2j ͫ det A H ͬx 2 time. Geʦ⌽͑G͒ jϭ0 He
n/2 n/2 ϭ ͑ ͒ nϪ2j ϭ ͑ ͒ nϪ2j ͫ H ͬx j G x 3. Conclusion jϭ0 He jϭ0 n/2 Theorem 15 and Corollary 16 assert that the ϭ n ͑ ͒͑ Ϫ2͒j ϭ n ͑ Ϫ2͒ ϩ e ϩ x j G x x g G, x , average values of det(xIn iA(G )), det(In jϭ0 ͌ e ϩ e xA(G )), and det(In A(G )) over the orientations of G are equal to m(G, x), g(G, x), and Z(G), respec- tively. Note that, by Propositions 1 and 2, the aver- that is, age value of det(A(Ge)) over the orientations of G is the number of perfect matchings of G. On the other 1 hand, it is well known (see p. 121 in Godsil [6]) that ͑ ϩ ͑ e͒͒ ϭ n ͑ Ϫ2͒ det xIn A G x g G, x . det( A(Ge)) is bounded above by the square of 2 eʦ⌽͑ ͒ G G the number of perfect matchings of G. Hence max{det(A(Ge)) Ϫ (G)} ϭ 2(G) Ϫ (G). Based on Because this it is natural to ask what is the upper bound for
INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 129 YAN, YEH, AND ZHANG
ϩ e Ϫ ϩ 6. Godsil, C. D. Algebraic Combinatorics; Chapman and Hall: det(In A(G )) Z(G), that is, what is max{det(In A(Ge)) Ϫ Z(G)}? For a graph with no cycles of even New York, 1993. ϩ 7. Godsil, C. D.; Gutman, I. J Graph Theory 1981, 5, 137–144. length, by Theorem 8 all determinants det(In A(Ge)) over the orientations of G are equal (to the 8. Godsil, C. D.; Gutman, I. In Algebraic Methods in Graph ϩ e Ϫ Theory, I; Lova´sz, L., So´s, V. T., Eds.; Colloq Math Soc Ja´nos Hosoya index Z(G)). Hence max{det(In A(G )) Bolyai 25; North-Holland: Amsterdam, 1981; pp 241–249. ϭ ϩ e Ϫ ϭ Z(G)} min{det(In A(G )) Z(G)} 0. 9. Graovac, A. Chem Phys Lett 1981, 82, 248–251. Note that the matching polynomial m(T, x)ofa 10. Graovac, A.; Babic, D. Z Naturforsch A 1985, 40, 66–72. tree T equals the characteristic polynomial of T, that 11. Graovac, A.; Polansky, O. E. MATCH Commun Math Chem is, m(T, x) can be expressed by the characteristic 1986, 21, 33–45. polynomial of the adjacency matrix of T. As that 12. Graovac, A.; Polansky, O. E. MATCH Commun Math Chem stated in the introduction of this article, the match- 1986, 21, 81–89. ing polynomials of certain graphs including unicy- 13. Gruber, C.; Kunz, H. Commun Math Phys 1971, 22, 133–161. clic graphs and cacti can be expressed in terms of 14. Gutman, I.; Graovac, A.; Mohar, B. MATCH Commun Math the characteristic polynomials of certain Hermitian Chem 1982, 13, 129–150. matrices [9–12, 14, 15, 17, 26, 29]. In this article, we 15. Gutman, I.; Milun, M.; Trinajstic´, N. J Am Chem Soc 1977, 99, express the matching polynomial of a graph G with 1692–1704. small number of cycles of even length in terms of 16. Heilmann, O. J.; Lieb, E. H. Phys Rev Lett 1970, 24, 1412– 1414. the characteristic polynomials of the skew adja- 17. Herndon, W. C.; Pa´rka´nyi, C. Tetrahedron 1978, 34, 3419– cency matrix of some orientation of G and its mi- 3425. nors. Yan et al. [33] use the characteristic poly- 18. Hosoya, H. Bull Chem Soc Jpn 1971, 44, 2332–2339. nomial of the skew adjacency matrix of some ori- 19. Kasteleyn, P. W. J Math Phys 1963, 4, 287–293. entation to express the permanental polynomial of 20. Kasteleyn, P. W. In Graph Theory and Theoretical Physics; a type of graphs. It is also interesting to find graphs Harary, F., Ed.; Academic Press: New York, 1967; pp 43–110. whose matching polynomial can be expressed in 21. Kunz, H. Phys Lett A 1970, 32, 311–312. terms of the characteristic polynomial of some ma- 22. Little, C. H. C. J Combinatorial Theory 1975, 18, 187–208. trices. 23. Lova´sz, L.; Plummer, M. Matching Theory; Annals of Dis- crete Mathematics 29; North-Holland: New York, 1986. 24. McCuaig, W. Po´lya’s permanent problem; Preprint. ACKNOWLEDGMENTS 25. McCuaig, W.; Robertson, N.; Seymour, P. D.; Thomas, R. Permanents, Pfaffian orientations, and even directed circuits We thank both referees for showing us some (Extended abstract); Proceedings of the Twenty-Ninth ACM references and some chemical illustrations. Thanks Symposium on the Theory of Computing (STOC); ACM, to one of the referees for telling us about some 1997. applications of the matching polynomial in Hu¨ ckel 26. Polansky, O. E.; Graovac, A. MATCH Commun Math Chem theory. 1986, 21, 47–79. 27. Polansky, O. E.; Graovac, A. MATCH Commun Math Chem 1986, 21, 93–113. 28. Robertson, N.; Seymour, P. D.; Thomas, R. Ann Math 1999, References 150, 929–975. 29. Schaad, L. J.; Hess, B. A.; Nation, J. B.; Trinajstc´, N.; Gutman, 1. Aihara, J. J Am Chem Soc 1976, 98, 2750–2758. I. Croat Chem Acta 1979, 52, 233–248. 2. Aihara, J. Bull Chem Soc Jpn 1979, 52, 1529–1530. 30. Trinajstic´, N. J Quantum Chem Symp 1997, 11, 469–477. 3. Beezer, R. A.; Farrell, E. J. J Math Math Sci 2000, 23, 89–97. 31. Trinajstic´, N. Chemical Graph Theory, Vol. 2; CRC Press: 4. Bondy, J. A.; Murty, U. S. R. Graph Theory with Applica- Boca Raton, FL, 1983. tions; American Elsevier: New York, 1976. 32. Yan, W. G.; Zhang, F. J. Adv Appl Math 2004, 32, 655–668. 5. Farrell, E. J. J Combinatorial Theory Ser B 1979, 27, 75–86. 33. Yan, W. G.; Zhang, F. J. J Math Chem 2004, 35, 175–188.
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