The Kirchhoff Index of Folded Hypercubes and Some Variant Networks

Jiabao Liu,Xiang-Feng Pan,Yi Wang,Jinde Cao

doi:10.1155/2014/380874

Abstract

Then-dimensional folded hypercubeFQnis an important and attractive variant of then-dimensional hypercubeQn, which is obtained fromQnby adding an edge between any pair of vertices complementary edges.FQnis superior toQnin many measurements, such as the diameter ofFQnwhich is⌈n/2⌉, about a half of the diameter in terms ofQn. The Kirchhoff indexKf(G)is the sum of resistance distances between all pairs of vertices inG. In this paper, we established the relationships between the folded hypercubes networksFQnand its three variant networksl(FQn),s(FQn), andt(FQn)on their Kirchhoff index, by deducing the characteristic polynomial of the Laplacian matrix in spectral graph theory. Moreover, the explicit formulae for the Kirchhoff indexes ofFQn,l(FQn),s(FQn), andt(FQn)were proposed, respectively.

Highlights

It is well known that interconnection networks play an important role in parallel communication systems
An interconnection network is usually modelled by connected graphs G = (V, E), where V denotes the set of processors and E denotes the set of communication links between processors in networks
We proposed the formula for calculating the Kirchhoff index, denoted by Kf(l(FQn)), on the line graph of folded hypercubes l(FQn)

Summary

Introduction

It is well known that interconnection networks play an important role in parallel communication systems. As an important variant of Qn, the folded hypercube networks FQn, proposed by El-Amawy and Latifi [18], are the graph obtained from Qn by adding an edge between any pair of vertices complementary addresses. FQn and three variant networks l(FQn), s(FQn), and t(FQn) on their Kirchhoff index, by deducing the characteristic polynomial of the Laplacian matrix in spectral graph theory. Recall the definitions of n-dimensional folded hypercubes networks FQn as follows [18]. The folded hypercubes FQn can be constructed from Qn by adding an edge to every pair of vertices with complementary addresses. We present the corresponding calculated formulae for the Kirchhoff index of the hypercubes networks Qn and its three-variant networks l(FQn), s(FQn), and t(FQn) in this paper.

Notations and Some Preliminaries

Main Results

Conclusions