Further analysis of the number of spanning trees in circulant graphs

In this paper, we explore further properties of the numbers an from their combinatorial structures. Comparing with the previous work, the differences are that (1) in finding the coefficients of recurrence formulas for an, we avoid solving a system of linear equations with exponential size, but instead, we give explicit formulas; (2) we find the asymptotic functions and therefore we 'answer' the open problem posed in the conclusion of [Y.P. Zhang, X. Yong, M. Golin, The number of spanning trees in circulant graphs, Discrete Math. 223 (2000) 337-350]. As examples, we describe our technique and the asymptotics of the numbers.