The Generating Function is Rational for the Number of Rooted Forests in a Circulant Graph

Abstract

We consider the generating function \(\Phi \) for the number \(f_{\Gamma }(n) \) of rooted spanning forests in the circulant graph \(\Gamma \), where \(\Phi (x)= \sum _{n=1}^{\infty } f_{\Gamma }(n) x^n\) and either \(\Gamma =C_n(s_1,s_2,\dots ,s_k) \) or \(\Gamma =C_{2n}(s_1,s_2,\dots ,s_k,n) \). We show that \(\Phi \) is a rational function with integer coefficients that satisfies the condition \(\Phi (x)=-\Phi (1/x) \). We illustrate this result by a series of examples.