In this paper we obtain precise asymptotics for certain families of graphs,namely circulant graphs and degenerating discrete tori. The asymptotics containinteresting constants from number theory among which some can be interpreted ascorresponding values for continuous limiting objects. We answer one questionformulated in a paper from Atajan, Yong and Inaba in [1] and formulate aconjecture in relation to the paper from Zhang, Yong and Golin [21]. A crucialingredient in the proof is to use the matrix tree theorem and express thecombinatorial laplacian determinant in terms of Bessel functions. Anon-standard Poisson summation formula and limiting properties of thetafunctions are then used to evaluate the asymptotics.
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