Algebraic formulas for counting spanning trees in regular multigraphs

The authors show that the number of spanning directed trees in a regular multigraph does not depend on the selection of a reference point and on trees orientation with respect to that point. A general formula for counting the number of spanning trees in such graphs is introduced. Closed form formulae for counting spanning trees in directed and undirected Cayley graphs are given. The results obtained are applicable, for example, in the analysis of electrical networks, and in the synthesis of optimally reliable networks. In the later application the problem of finding the maximally reliable network can be replaced, under certain conditions, by the problem of finding a graph with the number of spanning trees maximized