The expected values of Kirchhoff indices in the random polyphenyl and spiro chains

Abstract

The Kirchhoff index Kf ( G ) of a graph G is the sum of resistance distances between all pairs of vertices in G . In this paper, we obtain exact formulas for the expected values of the Kirchhoff indices of the random polyphenyl and spiro chains, which are graphs of a class of unbranched multispiro molecules and polycyclic aromatic hydrocarbons. Moreover, we obtain a relation between the expected values of the Kirchhoff indices of a random polyphenyl and its random hexagonal squeeze, and the average values for the Kirchhoff indices of all polyphenyl chains and all spiro chains with n hexagons, respectively.