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.

Here we study the normalized Laplacian characteristics polynomial (L-polynomial) for trees and specifically for starlike trees. We describe how the L-polynomial of a tree depends on some topological indices. For which, we also define the higher order general Randić indices for matching and which are different from higher order connectivity indices. Finally we provide the multiplicity of the eig...

An (N +N) evolutionary algorithm is considered for the problem of finding the maximum cardinality matching in a graph. It is shown that the performance of the evolutionary algorithm is the same as classical simulated annealing. That is, the evolutionary algorithm cannot find the maximum matching in polynomial average time for a family of bipartite graphs considered in this paper although there ...

Let G be a simple graph on n vertices. An r-matching in G is a set of r independent edges. The number of r-matchings in G will be denoted by p(G,r). We set p(G,0)=1 and define the matching polynomial of G by μ(G,x):= ∑bn/2c r=0 (−1)·p(G,r)·x and the signless matching polynomial of G by μ(G,x):= ∑bn/2c r=0 p(G,r)·x. It is classical that the matching polynomials of a graph G determine the matchin...

Given a graph G with n vertices, let p(G, j) denote the number of ways j mutually nonincident edges can be selected in G. The polynomial M(x) =∑[n/2] j=0 (−1) j p(G, j)xn−2 j , called the matching polynomial of G, is closely related to the Hosoya index introduced in applications in physics and chemistry. In this work we generalize this polynomial by introducing the number of disjoint paths of l...

We establish closed-form expansions for the universal edge elimination polynomial of paths and cycles and their generating functions. This includes closed-form expansions for the bivariate matching polynomial, the bivariate chromatic polynomial, and the covered components polynomial.