Counting spanning trees in cobordism of two circulant graphs

We consider a family of graphs $H_n(s_1,\dots,s_k;t_1,\dots,t_\ell)$ that is a generalisation of the family of $I$-graphs, which, in turn, includes the generalized Petersen graphs. We present an explicit formula for the number $\tau(n)$ of spanning trees in these graphs in terms of the Chebyshev polynomials and find its asymptotics. Also, we show that the number of spanning trees 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 integer depending on the number of even elements in the sequence $s_1,\dots,s_k,t_1,\dots,t_\ell$ and the parity of $n$.