On the Tutte and Matching Polynomials for Complete Graphs

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 integers, and show that for every mu is an element of N the sequence t(a,b)(n) is ultimately periodic modulo mu provided a not equal 1 mod mu and b not equal 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. This gives also 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.