Number of Spanning Trees of Cartesian and Composition Products of Graphs and Chebyshev Polynomials

Abstract

Enumerating all the spanning trees of a graph without duplication is one of the widely studied problems in electrical engineering and computer science literature. Since the number of spanning trees increases rapidly with the size of the graphs, a highly efficient method desired to compute all the spanning trees of a graph such as the method of eigenvalues of the Laplacian matrix. In this paper, using a connection between the eigenvalues of the Laplacian matrix and known properties of the Chebyshev polynomials, we have explicit formulas for counting the number of spanning trees of the Cartesian and Composition products of certain families graphs.