Summary: Let \(T(G)\) be the number of spanning trees in graph \(G\). In this note, we explore the asymptotics of \(T(G)\) when \(G\) is a circulant graph with given jumps.
The circulant graph \(C_n^{s_1,s_2,\dots,s_k}\) is the \(2k\)-regular graph with \(n\) vertices labeled \(0,1,2,\dots,n-1\), where node \(i\) has the \(2k\) neighbors \(i\pm s_1,i\pm s_2,\dots,i\pm s_k\) where all the operations are \(\pmod n\). We give a closed formula for the asymptotic limit \(\lim_{n\to\infty} T(C_n^{s_1,s_2,\dots,s_k})^{\frac1n}\) as a function of \(s_1,s_2,\dots,s_k\). We then extend this by permitting some of the jumps to be linear functions of \(n\), i.e., letting \(s_i\), \(d_i\) and \(e_i\) be arbitrary integers, and examining
\[ \lim_{n\to\infty} T\left(C_n^{s_1,s_2,\dots,s_k,\lfloor\frac{n}{d_1}\rfloor+ e_1,\lfloor \frac{n}{d_2}\rfloor+ e_2,\dots, \lfloor\frac{n}{d_l}\rfloor+e_l}\right)^{\frac1n}. \]
While this limit does not usually exist, we show that there is some \(p\) such that for \(0\leq q<p\), there exists \(c_q\) such that limit (1) restricted to only \(n\) congruent to \(q\) modulo \(p\) does exist and is equal to \(c_q\). We also give a closed formula for \(c_q\).
One further consequence of our derivation is that if \(s_i\) go to infinity (in any arbitrary order), then
\[ \lim_{s_1,s_2,\dots,s_k\to\infty}\;\lim_{n\to\infty} T(C_n^{2_1,s_2,\dots,s_k})^{\frac1n}= 4\exp \left[ \int_0^1 \int_0^1\cdots \int_0^1 \ln\bigg(\sum_{i=1}^k \sin^2\pi x_i\bigg)\, dx_1dx_2\cdots dx_k\right]. \]
Interestingly, this value is the same as the asymptotic number of spanning trees in the \(k\)-dimensional square lattice recently obtained by Garcia, Noy and Tejel.