Abstract
A vertex|matching-partition (V|M) of a simple graph G is a spanning collection of vertices and independent edges of G. Let vertex v in V have weight w_v and edge e in M have weight w_e. Then the weight of V|M is w(V|M) = Prod_{v in V} w_v · Prod_{e in M} w_e. Define the vertex|matching-partition function of G as W(G) = Sum_{V|M} w(V|M). In this paper we study this function when G is a path and a cycle. We generate all orthogonal polynomials as vertex|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 are extended to the weighted cycle.
Recommended Citation
McSorley, John P., Feinsilver, Philip and Schott, Rene. "Generating Orthogonal Polynomials and their Derivatives using Vertex|Matching-Partitions of Graphs." Ars Combinatoria 87 (Jan 2008): 75-95.