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

Abstract

In the present paper, we investigate a family of circulant graphs with non-fixed jumps $$G_n=C_{\beta n}(s_1,\dots ,s_k,{\alpha }_{1}n,\dots ,{\alpha }_{\ell }n),1\le s_1<\dots <s_k\le [\frac{\beta n}{2}],1\le {\alpha }_1<\dots <{\alpha }_{\ell }\le [\frac{\beta }{2}].$$ Here $n$ is an arbitrary large natural number and integers $s_1,\dots ,s_k,{\alpha }_1,\dots ,{\alpha }_{\ell }$ are supposed to be fixed. First, we present an explicit formula for the number of spanning trees in the graph $G_n.$ This formula is a product of $\beta s_k-1$ factors, each given by the $n$-th Chebyshev polynomial of the first kind evaluated at the roots of some prescribed polynomial of degree $s_k.$ Next, we provide some arithmetic properties of the complexity function. We show that the number of spanning trees in $G_n$ can be represented in the form $\tau (n)=pna(n{)}^2,$ where $a(n)$ is an integer sequence and $p$ is a prescribed natural number depending of parity of $\beta$ and $n.$ Finally, we find an asymptotic formula for $\tau (n)$ through the Mahler measure of the Laurent polynomials differing by a constant from $2k-\sum _{i=1}^k(z^{s_i}+z^{-s_i}).$

17 pages. arXiv admin note: text overlap with arXiv:1711.00175