Abstract It has long been known that the number of spanning trees in n node circulant graphs with fixed jumps satisfies a bounded order, constant coefficient, recurrence relation in n. The proof of this fact was algebraic (evaluating products of eigenvalues of the graphs ' adjacency matrices) and not combinatorial. In this paper we derive a straightforward combinatorial proof.

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 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 C

For a graph G, let fij be the number of spanning rooted forests in which vertex j belongs to a tree rooted at i. In this paper, we show that for a path graph, fij’s can be expressed as products of Fibonacci numbers; for a cycle graph, they are products of Fibonacci numbers and Lucas numbers. The doubly stochastic graph matrix is the matrix F = (fij)n×n f, where f is the total number of spanning rooted forests of G and n is the number of vertices in G. By the matrix forest theorem, F −1 = I + L, where L is the Laplacian matrix of G. F provides a proximity measure for graph vertices. We show that for the path graphs and the so-called T-caterpillars, some diagonal entries of F (which measure the self-connectivity of vertices) converge to φ −1 or to 1 − φ −1, where φ is the golden ratio, as the number of vertices goes to infinity. Thereby, in the asymptotic, the corresponding vertices can be metaphorically considered as “golden introverts ” and “golden extroverts, ” respectively. This metaphor is reinforced by a Markov chain interpretation of the doubly stochastic graph matrix, according to which F equals the overall transition matrix of a random walk with a random number of steps on G.

number of spanning trees in a class of double fixed-step loop networks

A double fixed-step loop network, ⃗ C p,q, is a digraph on n vertices 0, 1, 2,..., n − 1 and for each vertex i (0 < i ≤ n − 1), there are exactly two arcs leaving from vertex i to vertices i + p, i + q (mod n). In this paper, we first derive an expression formula of elementary symmetric polynomials as polynomials in sums of powers then, by using this, for any positive integers p, q, n with p < q < n, an explicit formula for counting the number of spanning trees in a class of double fixed-step loop networks with constant or nonconstant jumps. We allso find two classes of networks that share the same number of spanning trees and we, finally, prove that the number of spanning trees can be approximated by a formula which is based on the mth order Fibonacci numbers. In some special cases, our results generate the formulas obtained in [15],[19],[20]. And, compared with the previous work, the advantage is that, for any jumps p, q, the number of spanning trees can be calculated directly, without establishing the recurrence relation of order 2 q−1. 1 Introduction. A directed circulant graph, ⃗ Cs1,s2,...,sk, is a digraph on n vertices 0, 1, 2,..., n − 1 and for each vertex i (0 < i ≤ n − 1), there are k arcs from vertex i to vertices

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