Asymptotics for the Number of Spanning Trees in Circulant Graphs and Degenerating d-Dimensional Discrete Tori

Abstract

In this paper we obtain precise asymptotics for certain families of graphs, namely circulant graphs and degenerating discrete tori. The asymptotics contain interesting constants from number theory among which some can be interpreted as corresponding values for continuous limiting objects. We answer one question formulated in a paper from Atajan, Yong, and Inaba in [1] and formulate a conjecture in relation to the paper from Zhang, Yong, and Golin [23]. A crucial ingredient in the proof is to use the matrix tree theorem and express the combinatorial Laplacian determinant in terms of Bessel functions. A non-standard Poisson summation formula and limiting properties of theta functions are then used to evaluate the asymptotics.