Tricyclic Graphs with Minimal Kirchhoff Index

Shubo Chen1, Jianguang Yang1
1School of Mathematics and Computer Science, Hunan City University, Yiyang, Hunan 413000, P. R. China

Abstract

The resistance distance between two vertices of a connected graph \(G\) is defined as the effective resistance between them in the corresponding electrical network constructed from \(G\) by replacing each edge of \(G\) with a unit resistor. The Kirchhoff index \(Kf(G)\) is the sum of resistance distances between all pairs of vertices of the graph \(G\). In this paper, we determine the tricyclic graphs with the smallest and the second smallest Kirchhoff indices.