Kirchhoff indices of spiro and polyphenyl hexagonal 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 give the recurrences or explicit formulae for computing the Kirchhoff indices of spiro and polyphenyl hexagonal chains, which are graphs of a class of unbranched multispiro molecules and polycyclic aromatic hydrocarbons. We establish a relation between the Kirchhoff indices of a spiro hexagonal chain and its corresponding polyphenyl hexagonal chain, and determine the extremal values and characterize the extremal graphs with respect to the Kirchhoff index among all spiro and polyphenyl hexagonal chains with n hexagons.