In the present paper, we investigate a family of circulant graphs with non-fixed jumps $$G_n=C_{βn}(s_1, \ldots,s_k,α_1n,\ldots,α_\ell n),\, 1\le s_1<\ldots<s_k\le[\frac{βn}{2}],\, 1\le α_1< \ldots<α_\ell\le[\fracβ{2}].$$ Here $n$ is an arbitrary large natural number and integers $s_1, \ldots,s_k,α_1, \ldots,α_\ell$ 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 $βs_k-1$ 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 $τ(n)=p \,n \,a(n)^2,$ 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 Laurent polynomials differing by a constant from $2k-\sum\limits_{i=1}^k(z^{s_i}+z^{-s_i}).$