Let be the number of spanning trees in graph . In this note, we explore the asymptotics of when is a circulant graph with given jumps.
The circulant graph is the -regular graph with vertices labeled , where node has the neighbors where all the operations are . We give a closed formula for the asymptotic limit as a function of . We then extend this by permitting some of the jumps to be linear functions of , i.e., letting , and be arbitrary integers, and examining While this limit does not usually exist, we show that there is some such that for , there exists such that limit (1) restricted to only congruent to modulo does exist and is equal to . We also give a closed formula for .
One further consequence of our derivation is that if go to infinity (in any arbitrary order), then Interestingly, this value is the same as the asymptotic number of spanning trees in the -dimensional square lattice recently obtained by Garcia, Noy and Tejel.