Extremal octagonal chains with respect to the Kirchhoff index

TL;DR

This paper characterizes the extremal octagonal chains that maximize or minimize the Kirchhoff index, a measure based on resistance distances in electrical network models of graphs.

Contribution

It provides a complete characterization of the extremal octagonal chains concerning the Kirchhoff index, a novel analysis for this class of graphs.

Findings

Identified octagonal chains with maximum Kirchhoff index.

Identified octagonal chains with minimum Kirchhoff index.

Extended the understanding of resistance-based indices in complex graphs.

Abstract

Let $G$ be a connected graph. The resistance distance between any two vertices of $G$ is equal to the effective resistance between them in the corresponding electrical network constructed from $G$ by replacing each edge with a unit resistor. The Kirchhoff index is defined as the sum of resistance distances between all pairs of the vertices. These indices have been computed for many interesting graphs, such as linear polyomino chain, linear/M\"{o}bius/cylinder hexagonal chain, and linear/M\"{o}bius/cylinder octagonal chain. In this paper, we characterized the maximum and minimum octagonal chains with respect to the Kirchhoff index.