Table of Contents
International Journal of Combinatorics
Volume 2014, Article ID 258017, 13 pages
http://dx.doi.org/10.1155/2014/258017
Research Article

The -Path Cover Polynomial of a Graph and a Model for General Coefficient Linear Recurrences

Department of Mathematics, Southern Illinois University, Carbondale, IL 62901-4408, USA

Received 30 June 2013; Accepted 18 November 2013; Published 12 January 2014

Academic Editor: Jiang Zeng

Copyright © 2014 John P. McSorley and Philip Feinsilver. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An -path cover of a simple graph is a set of vertex disjoint paths of , each with vertices, that span . With every we associate a weight, , and define the weight of to be . The -path cover polynomial of is then defined as where the sum is taken over all -path covers of . This polynomial is a specialization of the path-cover polynomial of Farrell. We consider the -path cover polynomial of a weighted path and find the -term recurrence that it satisfies. The matrix form of this recurrence yields a formula equating the trace of the recurrence matrix with the -path cover polynomial of a suitably weighted cycle . A directed graph, , the edge-weighted -trellis, is introduced and so a third way to generate the solutions to the above -term recurrence is presented. We also give a model for general-term linear recurrences and time-dependent Markov chains.