The number of rooted forests in circulant graphs

Abstract

In this paper, we develop a new method to produce explicit formulas for the number f G (n) of rooted spanning forests in the circulant graphs G = C n (s 1 , s 2 , . . . , s k ) and G = C 2n (s 1 , s 2 , . . . , s k , n). These formulas are expressed through Chebyshev polynomials. We prove that in both cases the number of rooted spanning forests can be represented in the form f G (n) = p a(n) 2 , where a(n) is an integer sequence and p is a certain natural number depending on the parity of n. Finally, we find an asymptotic formula for f G (n) through the Mahler measure of the associated Laurent polynomial P (z) = 2k + 1 − k i=1 (z si + z −si ).

Cyclic coverings of graphs. Counting rooted spanning forests and trees, Kirchhoff index, and Jacobians

“…First of all, consider the case of circulant graphs with vertices of even degree. In this case the following two theorems proved in [33] are valid. In their statements the graph is not assumed to be connected.…”

“…Arithmetic properties of the number of rooted spanning forests. In the case of circulant graphs with vertices of odd degree the following result holds ( [33], Theorem 7.1). Theorem 9.3.…”

“…The following theorem describes the arithmetic properties of the number of rooted spanning forests in circulant graphs with vertices of odd degree ( [33], Theorem 5.2). Theorem 9.4.…”