On the Multiplicative Degree-Kirchhoff Indices and the Number of Spanning Trees of Linear Phenylene Chains

Let Ln6,4,4 denote a molecular graph of linear [n] phenylene, containing n hexagons and 2n−1 squares. In this paper, using the decomposition theorem of the normalized Laplacian characteristic polynomial, we obtain that the normalized Laplacian spectrum of Ln6,4,4 consisting of the eigenvalues of two symmetric tridiagonal matrices of order 4n is determined. By applying the relationship between the roots and coefficients of the characteristic polynomial of the above two matrices, an explicit closed-form formula of the multiplicative degree-Kirchhoff index (resp. the number of spanning trees) of Ln6,4,4 is derived.