A formula for the number of spanning trees in circulant graphs with non-fixed generators and discrete tori

We consider the number of spanning trees in circulant graphs of $\beta n$ vertices with generators depending linearly on $n$. The matrix tree theorem gives a closed formula of $\beta n$ factors, while we derive a formula of $\beta-1$ factors. Using the same trick, we also derive a formula for the number of spanning trees in discrete tori. Moreover, the spanning tree entropy of circulant graphs with fixed and non-fixed generators is compared.Comment: 8 pages, 2 figure

On the number of spanning trees in random regular graphs

Let $d \geq 3$ be a fixed integer. We give an asympotic formula for the expected number of spanning trees in a uniformly random $d$-regular graph with $n$ vertices. (The asymptotics are as $n\to\infty$, restricted to even $n$ if $d$ is odd.) We also obtain the asymptotic distribution of the number of spanning trees in a uniformly random cubic graph, and conjecture that the corresponding result holds for arbitrary (fixed) $d$. Numerical evidence is presented which supports our conjecture.Comment: 26 pages, 1 figure. To appear in the Electronic Journal of Combinatorics. This version addresses referee's comment

The asymptotic number of spanning trees in circulant graphs

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 Cns1, s2, ..., sk is the 2 k-regular graph with n vertices labeled 0, 1, 2, ..., n - 1, where node i has the 2 k neighbors i ± s1, i ± s2, ..., i ± sk where all the operations are (mod n). We give a closed formula for the asymptotic limit limn → ∞ T (Cns1, s2, ..., sk)frac(1, n) as a function of s1, s2, ..., sk. We then extend this by permitting some of the jumps to be linear functions of n, i.e., letting si, di and ei be arbitrary integers, and examining under(lim, n → ∞) T (underover(C, n, s1, s2, ..., sk, ⌊ frac(n, d1) ⌋ + e1, ⌊ frac(n, d2) ⌋ + e2, ..., ⌊ frac(n, dl) ⌋ + el))frac(1, n) . While this limit does not usually exist, we show that there is some p such that for 0 ≤ q < p, there exists cq such that limit (1) restricted to only n congruent to q modulo p does exist and is equal to cq. We also give a closed formula for cq. One further consequence of our derivation is that if si go to infinity (in any arbitrary order), then under(lim, s1, s2, ..., sk → ∞) under(lim, n → ∞) T (underover(C, n, s1, s2, ..., sk))frac(1, n) = 4 exp [∫01 ∫01 ⋯ ∫01 ln (underover(∑, i = 1, k) sin2 π xi) d x1 d x2 ⋯ d xk] . 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. © 2009 Elsevier B.V. All rights reserved

The asymptotic number of spanning trees in circulant graphs (Extended Abstract)

Let T(G) be the number of spanning trees in graph G. In this note we explore the asymptotics of T(G) for circulant graphs. The circulant graph Cs1,s2,···,sk n is the 2k regular graph with n vertices labelled 0,1,2, · · ·,n − 1, where node i has the 2k neighbors, (0 ≤ i ≤ n − 1) adjacent to vertices i + s1,i + s2, · · ·,i + sk mod n. In this note we give a closed formula for the asymptotic limit limn→ ∞ T(C s1,s2,···,sk n) 1 n as a function of s1,s2,...,sn. We then extend this by permitting the si to be linear functions of n, i.e., we give a closed formula for lim n→ ∞ T n s1,s2,...,sk, ⌊ d ⌋+e1,