Generating orthogonal polynomials and their derivatives using vertex| matching-partitions of graphs

A vertex vertical bar matching-partition (VIM) of a simple graph G is a spanning collection of vertices and independent edges of G. Let vertex v is an element of V have weight w(v) and edge e is an element of M have weight w(e).Then the weight of V vertical bar M is w(V vertical bar M) = Pi(v is an element of V) w(v)center dot Pi(e is an element of M) w(e). Define thevertex vertical bar matching-partition function of G as W(G) = Sigma(V vertical bar M) w(V vertical bar M). In this paper we study this function when G is a path and a cycle. We generate all orthogonal polynomials as vertex vertical bar matching-partition functions of suitably labelled paths and indicate how to find their derivatives in some cases. Here Taylor's Expansion is used and an application to associated polynomials is given. We also give a combinatorial interpretation of coefficients in the case of multiplicative and additive weights. Results axe extended to the weighted cycle.