On the Laplacians for Strong Product Graphs Based on Polyacene Graphs

Abstract

Let Hn be the molecular graph of a linear polyacene chain. In this paper, we introduce a kind of linear polyacene cylinder graph (Formula presented.) which is the strong product of two automorphism polyacene graphs Hn and (Formula presented.) Next, we obtain that the Laplacian spectrum of (Formula presented.) consists of the eigenvalues of symmetric matrices LA and LS of order (Formula presented.) according to the decomposition theorem of Laplacian characteristic polynomial. Furthermore, by applying the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices, we study the Kirchhoff index and the number of spanning trees, and derive their explicit expressions in terms of n for the graph (Formula presented.).