Kemeny's constant and the Kirchhoff index for the cluster of highly symmetric graphs

TL;DR

This paper derives explicit formulas for Kemeny's constant and Kirchhoff index for a specific class of highly symmetric graph clusters, enhancing understanding of their structural properties.

Contribution

It provides new closed-form expressions for these indices for the cluster of two highly symmetric graphs, linking them to the original graphs' parameters.

Findings

Formulas for Kemeny's constant and Kirchhoff index for graph clusters

Conditions for a graph to be highly symmetric

Insights into the structure of symmetric graph clusters

Abstract

We find closed form formulas for the Kemeny's constant and the Kirchhoff index for the cluster $G_1\{G_2\}$ of two highly symmetric graphs $G_1$, $G_2$, in terms of the parameters of the original graphs. We also discuss some necessary conditions for a graph to be highly symmetric.