Exact Results on Potts/Tutte Polynomials for Families of Networks with Edge and Vertex Inflations

Abstract

We derive exact relations between the Potts model partition function, or equivalently the Tutte polynomial, for a network (graph) $G$ and a network obtained from $G$ by (i) by replacing each edge (i.e., bond) of $G$ by two or more edges joining the same vertices, and (ii) by inserting one or more degree-2 vertices on edges of $G$. These processes are called edge and vertex inflation, respectively. The physical effects of these edge and vertex inflations are discussed. We also present exact calculations of these polynomials for families of networks obtained via the operation (ii) on a subset of the bonds of the network. Applications of these results include calculations of some network reliability polynomials. In addition, we evaluate our results to calculate various quantities of structural interest such as numbers of spanning trees, etc., and to determine their asymptotic behavior for large networks.