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

Abstract

Let $F(x)=\sum _{n=1}^{\infty }\tau (n)x^n$ be the generating function for the number $\tau (n)$ of spanning trees in the circulant graphs $C_n(s_1,s_2,\dots ,s_k).$ 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 $C_{2n}(s_1,s_2,\dots ,s_k,n).$ We illustrate the obtained results by a series of examples.

11 pages