Complexity of circulant graphs with non-fixed jumps, its arithmetic properties and asymptotics
Keywords:Spanning tree, circulant graph, Laplacian matrix, Chebyshev polynomial, Mahler measure
In the present paper, we investigate a family of circulant graphs with non-fixed jumps
Gn= Cβn(s1, ... , sk, α1n, ... , αln), 1 ≤ s1< ... < sk≤ [βn/2], 1 ≤ α1< ... < αl ≤ [β/2].
Here n is an arbitrary large natural number and integers s1, ... , sk, α1, ... , αl are supposed to be fixed.
First, we present an explicit formula for the number of spanning trees in the graph Gn. This formula is a product of βsk-1 factors, each given by the n-th Chebyshev polynomial of the first kind evaluated at the roots of some prescribed polynomial of degree sk. Next, we provide some arithmetic properties of the complexity function. We show that the number of spanning trees in Gn can be represented in the form τ(n) = p n a(n)2, where a(n) is an integer sequence and p is a prescribed natural number depending of parity of β and n. Finally, we find an asymptotic formula for τ(n) through the Mahler measure of the associated Laurent polynomials differing by a constant from 2k - ∑i = 1k(zsi+z−si).