The Normalized Laplacians, Degree-Kirchhoff Index and the Spanning Trees of Cylinder Phenylene Chain

Let PHn denote a linear phenylene chain with n hexagons and n – 1 quadrangles. Then the cylinder phenylene chain CPn is the graph obtained from the PHn by identifying the opposite lateral edges in ordered way. In this paper, using the normalized Laplacian decomposition theorem, the normalized Laplacian spectrum of CPn consisting of the eigenvalues of two symmetric quasi-tridiagonal matrices LΔ and LΛ of order 3n is determined. Together with the relationship between the roots and coefficients of the characteristic polynomials of the above two matrices, an explicit closed-form formulas of the degree-Kirchhoff index, the Kemeny’s constant and the number of spanning trees of CPn are obtained in terms of the index n.