Counting rooted spanning forests for circulant foliation over a graph

Abstract

In this paper, we present a new method to produce explicit formulas for the number of rooted spanning forests f(n) for the infinite family of graphs Hn = Hn(G1, G2, . . . , Gm) obtained as a circulant foliation over a graph H on m vertices with fibersG1, G2, . . . , Gm. Each fiber Gi = Cn(si,1, si,2, . . . , si,ki) of this foliation is the circulant graph on n vertices with jumps si,1, si,2, . . . , si,ki . This family includes the family of generalized Petersen graphs, I-graphs, sandwiches of circulant graphs, discrete torus graphs and others. The formulas are expressed through Chebyshev polynomials. We prove that the number of rooted spanning forests can be represented in the form f(n) = p f(H)a(n)2, where a(n) is an integer sequence and p is a prescribed natural number depending on the number of odd elements in the set of si,j. Finally, we find an asymptotic formula for f(n) through the Mahler measure of the associated Laurent polynomial.