Ordering connected graphs by their Kirchhoff indices

TL;DR

This paper characterizes extremal graphs with respect to Kirchhoff index after edge deletions from complete graphs, providing bounds and identifying graphs with top maximal indices.

Contribution

It offers a complete characterization of extremal graphs with maximum Kirchhoff indices after edge deletions from complete graphs, including bounds and specific graph classifications.

Findings

Sharp upper bounds on Kirchhoff indices for certain graph classes

Complete classification of graphs with top nine maximal Kirchhoff indices for large n

Identification of extremal graphs after edge deletions from complete graphs

Abstract

The Kirchhoff index $Kf(G)$ of a graph $G$ is the sum of resistance distances between all unordered pairs of vertices, which was introduced by Klein and Randi\'c. In this paper we characterized all extremal graphs with Kirchhoff index among all graphs obtained by deleting $p$ edges from a complete graph $K_n$ with $p\leq\lfloor\frac{n}{2}\rfloor$ and obtained a sharp upper bound on the Kirchhoff index of these graphs. In addition, all the graphs with the first to ninth maximal Kirchhoff indices are completely determined among all connected graphs of order $n>27$.