  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.
