On rationality of generating function for the number of spanning trees in circulant graphs

A. D. Mednykh, I. A. Mednykh

Published 2018-11-09Version 1

Let $F(x)=\sum\limits_{n=1}^\infty\tau(n)x^n$ be the generating function for the number $\tau(n)$ of spanning trees in the circulant graphs $C_{n}(s_1,s_2,\ldots,s_k).$ We show that $F(x)$ is a rational function with integer coefficients satisfying the property $F(x)=F(1/x).$ A similar result is also true for the circulant graphs of odd valency $C_{2n}(s_1,s_2,\ldots,s_k,n).$ We illustrate the obtained results by a series of examples.