Resistance Distances and Kirchhoff Indices Under Graph Operations

Abstract

The resistance distance between any two vertices of a connected graph $G$ is defined as the net effective resistance between them in the 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. In this paper, two unary graph operations on $G$ are taken into consideration, with the resulted graphs being denoted by $RT(G)$ and $H(G)$ . Using electrical network approach and combinatorial approach, we derive explicit formulae for resistance distances and Kirchhoff indices of $RT(G)$ and $H(G)$ . It turns out that resistance distances and Kirchhoff indices of $RT(G)$ and $H(G)$ could be expressed in terms of resistance distances and graph invariants of $G$ . Our result generalizes the previously known result on the Kirchhoff index of $RT(G)$ for a regular graph $G$ to the Kirchhoff index of $RT(G)$ for an arbitrary graph $G$ .

Keywords

• Resistance distance
• Kirchhoff index
• multiplicative degree-Kirchhoff index
• additive degree-Kirchhoff index
• Foster’s formula