On the Normalized Laplacian Spectrum of the Linear Pentagonal Derivation Chain and Its Application

Abstract

A novel distance function named resistance distance was introduced on the basis of electrical network theory. The resistance distance between any two vertices u and v in graph G is defined to be the effective resistance between them when unit resistors are placed on every edge of G. The degree-Kirchhoff index of G is the sum of the product of resistance distances and degrees between all pairs of vertices of G. In this article, according to the decomposition theorem for the normalized Laplacian polynomial of the linear pentagonal derivation chain QPn, the normalize Laplacian spectrum of QPn is determined. Combining with the relationship between the roots and the coefficients of the characteristic polynomials, the explicit closed-form formulas for degree-Kirchhoff index and the number of spanning trees of QPn can be obtained, respectively. Moreover, we also obtain the Gutman index of QPn and we discovery that the degree-Kirchhoff index of QPn is almost half of its Gutman index.