Complexity of circulant graphs with non-fixed jumps, its arithmetic properties and asymptotics

Alexander Mednykh, Ilya Mednykh

Abstract


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+zsi).


Keywords


Spanning tree, circulant graph, Laplacian matrix, Chebyshev polynomial, Mahler measure

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DOI: https://doi.org/10.26493/1855-3974.2530.e7c

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications