The Laplacian Spectrum, Kirchhoff Index, and the Number of Spanning Trees of the Linear Heptagonal Networks

This is the authors' abstract. We don't add key points for this paper.

Let H n be the linear heptagonal networks with 2 n heptagons. We study the structure properties and the eigenvalues of the linear heptagonal networks. According to the Laplacian polynomial of H n , we utilize the method of decompositions. Thus, the Laplacian spectrum of H n is created by eigenvalues of a pair of matrices: L A and L S of order numbers 5 n + 1 and 4 n + 1 n !/ r ! ( n − r )!, respectively. On the basis of the roots and coefficients of their characteristic polynomials of L A and L S , we get not only the explicit forms of Kirchhoff index but also the corresponding total number of spanning trees of H n .