Minimal hexagonal chains with respect to the Kirchhoff index

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 of G is defined as the sum of resistance distances between all pairs of vertices. Hexagonal chains are graph representations of unbranched catacondensed benzenoid hydrocarbons. It was shown in Yang and Klein (2014) [30] that among all hexagonal chains with n hexagons, the linear chain Ln is the unique chain with maximum Kirchhoff index. However, for hexagonal chains with minimum Kirchhoff index, it was only claimed that the minimum Kirchhoff index is attained only when the hexagonal chain is an “all–kink” chain. In this paper, by standard techniques of electrical networks and comparison results on Kirchhoff indices of S,T-isomers, “all-kink” chains with maximum and minimum Kirchhoff indices are characterized. As a consequence, hexagonal chains with minimum Kirchhoff indices are singled out.