Matching Polynomial Of Graph Research Articles

Generalizations of the Matching Polynomial to the Multivariate Independence Polynomial

We generalize two main theorems of matching polynomials of undirected simple graphs, namely, real-rootedness and the Heilmann-Lieb root bound. Viewing the matching polynomial of a graph $G$ as the independence polynomial of the line graph of $G$, we determine conditions for the extension of these theorems to the independence polynomial of any graph. In particular, we show that a stability-like property of the multivariate independence polynomial characterizes claw-freeness. Finally, we give and extend multivariate versions of Godsil's theorems on the divisibility of matching polynomials of trees related to $G$.

Higher-order matching polynomials and d-orthogonality

We show combinatorially that the higher-order matching polynomials of several families of graphs are d-orthogonal polynomials. The matching polynomial of a graph is a generating function for coverings of a graph by disjoint edges; the higher-order matching polynomial corresponds to coverings by paths. Several families of classical orthogonal polynomials—the Chebyshev, Hermite, and Laguerre polynomials—can be interpreted as matching polynomials of paths, cycles, complete graphs, and complete bipartite graphs. The notion of d-orthogonality is a generalization of the usual idea of orthogonality for polynomials and we use sign-reversing involutions to show that the higher-order Chebyshev (first and second kinds), Hermite, and Laguerre polynomials are d-orthogonal. We also investigate the moments and find generating functions of those polynomials.

Multivariate matching polynomials of cyclically labelled graphs

We consider the matching polynomials of graphs whose edges have been cyclically labelled with the ordered set of t labels { x 1 , … , x t } . We first work with the cyclically labelled path, with first edge label x i , followed by N full cycles of labels { x 1 , … , x t } , and last edge label x j . Let Φ i , N t + j denote the matching polynomial of this path. It satisfies the ( τ , Δ ) -recurrence: Φ i , N t + j = τ Φ i , ( N − 1 ) t + j − Δ Φ i , ( N − 2 ) t + j , where τ is the sum of all non-consecutive cyclic monomials in the variables { x 1 , … , x t } and Δ = ( − 1 ) t x 1 ⋯ x t . A combinatorial/algebraic proof and a matrix proof of this fact are given. Let G N denote the first fundamental solution to the ( τ , Δ ) -recurrence. We express G N (i) as a cyclic binomial using the symmetric representation of a matrix, (ii) in terms of Chebyshev polynomials of the second kind in the variables τ and Δ , and (iii) as a quotient of two matching polynomials. We extend our results from paths to cycles and rooted trees.

On the matching polynomials of graphs with small number of cycles of even length

AbstractSuppose 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) of an 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, 2005

The matching polynomial of a regular graph

The matching polynomial of a graph has coefficients that give the number of matchings in the graph. For a regular graph, we show it is possible to recover the order, degree, girth and number of minimal cycles from the matching polynomial. If a graph is characterized by its matching polynomial, then it is called matching unique. Here we establish the matching uniqueness of many specific regular graphs; each of these graphs is either a cage, or a graph whose components are isomorphic to Moore graphs. Our main tool in establishing the matching uniqueness of these graphs is the ability to count certain subgraphs of a regular graph.