On the Kirchhoff Indices and the Numbers of Spanning Trees of Two Types of Molecular Graphs

Abstract

Let Ln be the linear phenylene. Then let be the molecular graph obtained from Ln by connecting each pair of non-adjacent vertices on each 4-cycle of Ln by an edge. In this paper, according to the decomposition theorem of Laplacian polynomial, we study the Laplacian spectra of Ln and respectively. By applying the relationship between the roots and coefficients of the characteristic polynomial of Ln (resp. ), explicit closed formulae of Kirchhoff index and the number of spanning trees of Ln and are, respectively, derived in terms of the corresponding Laplacian spectrum. Furthermore, it is surprising to find that the Kirchhoff index of Ln is approximately to one half of its Wiener index, whereas the Kirchhoff index of is approximately to of its Wiener index.