Asymptotics for the number of spanning trees in circulant graphs and degenerating d-dimensional discrete tori

TL;DR

This paper derives precise asymptotic formulas for the number of spanning trees in circulant graphs and degenerating discrete tori, linking number theory constants with continuous limits.

Contribution

It provides new asymptotic results for these graph families, answers an open question, and formulates a related conjecture using advanced analytical techniques.

Findings

Asymptotic formulas involve interesting number theory constants.

Answer to a question posed by Atajan, Yong, and Inaba.

Formulation of a new conjecture relating to Zhang, Yong, and Golin.

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.