Some Bounds for the Kirchhoff Index of Graphs

Yujun Yang

doi:10.1155/2014/794781

Abstract

The resistance distance between two vertices of a connected graphGis defined as the effective resistance between them in the corresponding electrical network constructed fromGby replacing each edge ofGwith a unit resistor. The Kirchhoff index ofGis the sum of resistance distances between all pairs of vertices. In this paper, general bounds for the Kirchhoff index are given via the independence number and the clique number, respectively. Moreover, lower and upper bounds for the Kirchhoff index of planar graphs and fullerene graphs are investigated.

Highlights

Let G be a connected graph with vertices labeled as 1, 2, . . . , n
It is reasonable to consider the effective resistance between any two vertices of G, and the novel concept of resistance distance [1] rij(G) between any two vertices i and j of G is defined as the effective resistance between them
The most famous one is the Wiener index W(G) [3], which is known as the sum of distances between all pairs of vertices

Summary

Introduction

Let G be a connected graph with vertices labeled as 1, 2, . . . , n. There are general bounds that are given in terms of various graph structural parameters like the number of vertices, the number of edges, the matching number, the chromatic number, the maximum degree, and the number of spanning trees [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20], and bounds for some special interesting classes of graphs, such as circulant graphs, unicyclic graphs, and bicyclic graphs [21,22,23,24,25,26,27] Along this line, we consider the relation between the Kirchhoff index and the independence number as well as the clique number, and bounds are obtained for the Kirchhoff index of graphs via the two graph invariants. For more information on the Kirchhoff index of graphs, the readers are referred to the most recent papers [28,29,30,31,32,33,34,35,36] and references therein

General Bounds

Planar Graphs

Fullerene Graphs