Chebyshev polynomials and spanning tree formulas for circulant and related graphs

The formulas for the number of spanning trees in circulant graphs

lim n→∞ T  C s1,s2,...,sk,⌊ n d1 ⌋+e1,⌊ n d2 ⌋+e2,...,⌊ n dl ⌋+el n  1 n , as a function of si, dj and ek, where T (G) is the number of spanning trees in graph G. In this paper we derive simple and explicit formulas for the number of spanning trees in circulant graphs C12l pn . Following from the formulas we show that lim n→∞ T  C1,a1n,a2n,...,aln pn  1 n

Further analysis of the number of spanning trees in circulant graphs

Let 1 s1<s2< · · ·<sk n/2 be given integers. An undirected even-valent circulant graph, C12k n , has n vertices 0, 1, 2, . . ., n− 1, and for each si (1 i k) and j (0 j n− 1) there is an edge between j and j + si (mod n). Let T (C12k n ) stand for the number of spanning trees of C12k n . For this special class of graphs, a general and most recent result, which is obtained in [Y.P. Zhang, X.Yong...

N ov 2 01 7 The number of spanning trees in circulant graphs , its arithmetic properties and asymptotic

In this paper, we develop a new method to produce explicit formulas for the number τ(n) of spanning trees in the undirected circulant graphs Cn(s1, s2, . . . , sk) and C2n(s1, s2, . . . , sk, n). Also, we prove that in both cases the number of spanning trees can be represented in the form τ(n) = p n a(n), where a(n) is an integer sequence and p is a prescribed natural number depending only of p...

An efficient approach for counting the number of spanning trees in circulant and related graphs

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

We consider the number of spanning trees in circulant graphs of βn vertices with generators depending linearly on n. The matrix tree theorem gives a closed formula of βn factors; while we derive a formula of β−1 factors. The spanning tree entropy of these graphs is then compared to the one of fixed generated circulant graphs.