Extremal polyphenyl chains with respect to the Kirchhoff index

Abstract

Let G be a simple connected graph. The Kirchhoff index of a graph is the sum of the resistance distance between all vertex pairs in the graph G. The resistance distance in a graph is equivalent to the effective resistance between any node pairs in the electrical network obtained by replacing each edge in the graph G with a unit resistance. The polyphenyl chain $PL\left(S\right)$ is a graph with n hexagons where each hexagon can be reduced to a vertex and connect to a path. In this paper, we characterize polyphenyl chains with maximal and minimal Kirchhoff index.