Upper and Lower Bounds for the Kirchhoff Index of the n-Dimensional Hypercube Network

Jia-Bao Liu,Jinde Cao,Fuad E Alsaadi,Zhi-Yu Shi,Jing Zhao

doi:10.1155/2020/5307670

Abstract

The hypercube Qn is one of the most admirable and efficient interconnection network due to its excellent performance for some practical applications. The Kirchhoff index KfG is equal to the sum of resistance distances between any pairs of vertices in networks. In this paper, we deduce some bounds with respect to Kirchhoff index of hypercube network Qn.

Highlights

Network is usually modelled by a connected graphL(G) D(G) − A(G)
We recall some basic definition in graph theory. e hypercube network Qn may be constructed from the family of subsets of a set with a binary string of length n, by making a vertex for each possible subset and joining two vertices by an edge whenever the corresponding subsets differ in a single binary string. e hypercube network Qn admits several definitions of which one is stated as below [15]
E result of eorem 3 is obtained by directly calculating the eigenvalues of the Laplacian matrix of the hypercube network, which is different from the technique in [23]

Summary

Introduction

For other notations and graph theoretical terminologies that not state here, we follow [1]. Various parameters are always used to characterize and describe the complex networks of which the fundamental one is named as the distance dij, concerned as the shortest path between the vertices i and j in networks. Denote rij the resistance distance between two arbitrary vertices i and j in electrical networks by replacing every edge by a unit resistor [3,4,5,6,7]. E Kirchhoff index Kf(G) of networks is defined as. The rest of the context is summarized.

Definition and Preliminaries

Main Results

Further Discussion