Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Maximum Multiplicity of a Root of the Matching Polynomial of a Tree and Minimum Path Cover. | Maximum Multiplicity of a Root of the Matching Polynomial of a Tree and Minimum Path Cover Cheng Yeaw Ku Department of Mathematics National University of Singapore Singapore 117543 matkcy@ . Wong Institute of Mathematical Sciences University of Malaya 50603 Kuala Lumpur Malaysia kbwong@ Submitted Nov 18 2008 Accepted Jun 26 2009 Published Jul 2 2009 Mathematics Subject Classification 05C31 05C70 Abstract We give a necessary and sufficient condition for the maximum multiplicity of a root of the matching polynomial of a tree to be equal to the minimum number of vertex disjoint paths needed to cover it. 1 Introduction All the graphs in this paper are simple. The vertex set and the edge set of a graph G are denoted by V G and E G respectively. A matching of a graph G is a set of pairwise disjoint edges of G. Recall that for a graph G on n vertices the matching polynomial p G x of G is given by p G x -1 kp G k xn 2k k 0 where p G k is the number of matchings with k edges in G. Let mult ớ G denote the multiplicity of Ỡ as a root of p G x . The following results are well known. The proofs can be found in 2 Theorem on . Theorem . The maximum multiplicity of a root of the matching polynomial p G x is at most the minimum number of vertex disjoint paths needed to cover the vertex set of G. THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 R81 1 Consequently Theorem . If G has a Hamiltonian path then all roots of its matching polynomial are simple. The above is the source of motivation for our work. It is natural to ask when does equality holds in Theorem . In this note we give a necessary and sufficient condition for the maximum multiplicity of a root of the matching polynomial of a tree to be equal to the minimum number of vertex disjoint paths needed to cover it. Before stating the main result we require some terminology and basic properties of matching polynomials. It is well known that the roots of the matching polynomial are real. .
