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

In the present paper, we investigate a family of circulant graphs with non-fixed jumps $$G_n=C_{\beta n}(s_1, \ldots,s_k,\alpha_1n,\ldots,\alpha_\ell n),\, 1\le s_1<\ldots<s_k\le[\frac{\beta n}{2}],\, 1\le \alpha_1< \ldots<\alpha_\ell\le[\frac{\beta}{2}].$$ Here $n$ is an arbitrary large natural number and integers $s_1, \ldots,s_k,\alpha_1, \ldots,\alpha_\ell$ 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 $\tau(n)=p \,n \,a(n)^2,$ 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 $2k-\sum\limits_{i=1}^k(z^{s_i}+z^{-s_i}).$