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 [21]. 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.

27pages, 3 figures

References in corpus (2)

• On two-dimensional Bessel functions
• Exponential-type Inequalities Involving Ratios of the Modified Bessel Function of the First Kind and their Applications

Cited by in corpus (6)

• On complexity of cyclic coverings of graphs
• Asymptotic Analysis Of Determinant Of Discrete Laplacian
• An explicit prime geodesic theorem for discrete tori and the hypergeometric functions
• Spanning trees in directed circulant graphs and cycle power graphs
• A formula for the number of spanning trees in circulant graphs with non-fixed generators and discrete tori
• On rationality of generating function for the number of spanning trees in circulant graphs