The minimum Kirchhoff index of phenylene chains

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 vertices. Recently, Yang and Wang determined the maximum Kirchhoff index of phenylene chains and proposed a conjecture about the minimum Kirchhoff index. In this note, we characterize the minimum phenylene chains with respect to the Kirchhoff index, thereby prove the conjecture.