Resistance distance, Kirchhoff index, and Kemeny's constant in flower graphs

TL;DR

This paper derives a general formula for resistance distance in flower graphs, a family formed by cyclically connecting multiple copies of a base graph, and provides exact expressions for Kirchhoff index and Kemeny's constant for specific cases.

Contribution

It introduces a new class of graphs called flower graphs and develops a general resistance distance formula applicable to them, with explicit results for certain base graphs.

Findings

Derived a general resistance distance formula for flower graphs.

Obtained exact formulas for Kirchhoff index and Kemeny's constant for specific flower graphs.

Provided bounds on Kirchhoff index and Kemeny's constant for general flower graphs.

Abstract

We obtain a general formula for the resistance distance (or effective resistance) between any pair of nodes in a general family of graphs which we call flower graphs. Flower graphs are obtained from identifying nodes of multiple copies of a given base graph in a cyclic way. We apply our general formula to two specific families of flower graphs, where the base graph is either a complete graph or a cycle. We also obtain bounds on the Kirchhoff index and Kemeny's constant of general flower graphs using our formula for resistance. For flower graphs whose base graph is a complete graph or a cycle, we obtain exact, closed form expressions for the Kirchhoff index and Kemeny's constant.