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

Abstract

An m-path cover Γ={Pℓ1,Pℓ2,…,Pℓr} of a simple graph G is a set of vertex disjoint paths of G, each with ℓk≤m vertices, that span G. With every Pℓ we associate a weight, ω(Pℓ), and define the weight of Γ to be ω(Γ)=∏k=1r‍ω(Pℓk). The m-path cover polynomial of G is then defined as ℙm(G)=∑Γ‍ω(Γ), where the sum is taken over all m-path covers Γ of G. This polynomial is a specialization of the path-cover polynomial of Farrell. We consider the m-path cover polynomial of a weighted path P(m-1,n) and find the (m+1)-term recurrence that it satisfies. The matrix form of this recurrence yields a formula equating the trace of the recurrence matrix with the m-path cover polynomial of a suitably weighted cycle C(n). A directed graph, T(m), the edge-weighted m-trellis, is introduced and so a third way to generate the solutions to the above (m+1)-term recurrence is presented. We also give a model for general-term linear recurrences and time-dependent Markov chains.