The Spanning Trees Formulas in a Class of Double Fixed-Step Loop Networks

A double fixed-step loop network, Cp,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 2q−1.