On the Tutte and matching polynomials for complete graphs

TL;DR

This paper proves that the Tutte polynomial and related graph polynomials exhibit ultimately periodic behavior modulo any integer under certain conditions, extending to various graph polynomials and relying on modular recurrence relations.

Contribution

It establishes the periodicity of the Tutte polynomial sequence for complete graphs modulo any integer, generalizing to other graph polynomials definable in Monadic Second Order Logic.

Findings

Sequences are ultimately periodic modulo μ under specified conditions.

Results apply to a broad class of graph polynomials including independence and clique polynomials.

Dependent on the Specker-Blatter Theorem for modular recurrence relations.

Abstract

Let $T(G;X,Y)$ be the Tutte polynomial for graphs. We study the sequence $t_{a,b}(n) = T(K_n;a,b)$ where $a,b$ are non-negative integers, and show that for every $\mu \in \N$ the sequence $t_{a,b}(n)$ is ultimately periodic modulo $\mu$ provided $a \neq 1 \mod{\mu}$ and $b \neq 1 \mod{\mu}$. This result is related to a conjecture by A. Mani and R. Stones from 2016. The theorem is a consequence of a more general theorem which holds for a wide class of graph polynomials definable in Monadic Second Order Logic and some of its extensions, such as the the independence polynomial, the clique polynomial, etc. We also show similar results for the various substitution instances of the bivariate matching polynomial and the trivariate edge elimination polynomial $\xi(G;X,Y,Z)$ introduced by I. Averbouch, B. Godlin and the second author in 2008. All our results depend on the Specker-Blatter Theorem from 1981, which studies modular recurrence relations of combinatorial sequences which count the number of labeled graphs.