On the Kirchhoff Index and the Number of Spanning Trees of Linear Phenylenes Chain

Let L-n(6,4,4) denote a molecular graph of linear [n] phenylene, containing n hexagons and 2n-1 squares. In this paper, according to the decomposition theorem of Laplacian polynomial, we obtain that the Laplacian spectrum of L-n(6,4,4) consists of the Laplacian spectrum of path P-4n and eigenvalues of a symmentric tridiagonal matrix of order 4n. By applying the relationship between the roots and coefficients of the characteristic polynomial of the above matrix, explicit closed formula of Kirchhoff index and the number of spanning trees of L-n(6,4,4) are derived in terms of the Laplacian spectrum.