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 graphsthis foliation is the circulant graph on n vertices with jumps s i,1 , s i,2 , . . . , s i,k i . 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 s i,j . Finally, we find an asymptotic formula for f (n) through the Mahler measure of the associated Laurent polynomial.