On the Kirchhoff Indices and the Number of Spanning Trees of Möbius Phenylenes Chain and Cylinder Phenylenes Chain

Abstract

Let denote a molecular graph of linear phenylene. Then the Möbius phenylenes chain is the graph obtained from the by identifying the opposite lateral edges in reversed way, whereas the cylinder phenylene chain is the graph obtained from the by identifying the opposite lateral edges in ordered way. In this paper, by the decomposition theorem of Laplacian polynomial and the relationship between the roots and the coefficients of the characteristic polynomial, explicit closed-form formulae of the Kirchhoff index (resp. the number of spanning trees) of and are obtained.