Some Properties on Estrada Index of Folded Hypercubes Networks

Jia-Bao Liu,Xiang-Feng Pan,Jinde Cao

doi:10.1155/2014/167623

Abstract

LetGbe a simple graph withnvertices and letλ1,λ2,…,λnbe the eigenvalues of its adjacency matrix; the Estrada indexEEGof the graphGis defined as the sum of the termseλi, i=1,2,…,n. Then-dimensional folded hypercube networksFQnare an important and attractive variant of then-dimensional hypercube networksQn, which are obtained fromQnby adding an edge between any pair of vertices complementary edges. In this paper, we establish the explicit formulae for calculating the Estrada index of the folded hypercubes networksFQnby deducing the characteristic polynomial of the adjacency matrix in spectral graph theory. Moreover, some lower and upper bounds for the Estrada index of the folded hypercubes networksFQnare proposed.

Highlights

Complex networks have become an important area of multidisciplinary research involving mathematics, physics, social sciences, biology, and other theoretical and applied sciences
An interconnection network is usually modelled by a connected graph G = (V, E), where V denotes the set of processors and E denotes the set of communication links between processors in networks
We present some explicit formulae for calculating the Estrada index of FQn

Summary

Introduction

Complex networks have become an important area of multidisciplinary research involving mathematics, physics, social sciences, biology, and other theoretical and applied sciences. From the Taylor the kth spectral expansion of ex, we have the following important relation between the Estrada index and the spectral moments of G: EE. At this point one should recall [4] that Mk(G) is equal to the number of self-returning walks of length k of the graph. As an important variant of Qn, the folded hypercubes networks FQn, proposed by Amawy and Latifi [8], are the graphs obtained from Qn by adding an edge between any pair of vertices complementary addresses.

Notations and Some Preliminaries

Main Results

Conclusions