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In this paper, we develop a new method to produce explicit formulas for thenumber $\tau(n)$ of spanning trees in the undirected circulant graphs$C_{n}(s_1,s_2,\ldots,s_k)$ and $C_{2n}(s_1,s_2,\ldots,s_k,n).$ Also, we provethat in both cases 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 aprescribed natural number depending on the parity of $n.$ Finally, we find anasymptotic formula for $\tau(n)$ through the Mahler measure of the associatedLaurent polynomial $L(z)=2k-\sum\limits_{i=1}^k(z^{s_i}+z^{-s_i}).$

The number of spanning trees in circulant graphs, its arithmetic properties and asymptotic