The minimum Kirchhoff index of phenylene chains

TL;DR

This paper proves a conjecture by characterizing phenylene chains with the minimum Kirchhoff index, which measures overall electrical resistance in molecular graph structures.

Contribution

It provides a complete characterization of phenylene chains with the minimum Kirchhoff index, confirming a previously proposed conjecture.

Findings

Identified the structure of phenylene chains with minimum Kirchhoff index

Confirmed the conjecture about the minimum Kirchhoff index

Enhanced understanding of resistance properties in molecular 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. Recently, Yang and Wang determined the maximum Kirchhoff index of phenylene chains, and they proposed a conjecture about the minimum Kirchhoff index. In this note, we characterized the minimum phenylene chains with respect to the Kirchhoff index. This proves the conjecture.