The Spectra And Kirchhoff Index Of Enhanced Hypercubes

Spectral graph theory is an important research field in algebraic graph theory.Spec-tral graph theory mainly investigate the relationship between the structural properties and adjacency spectra,Laplacian spectra or signless Laplacian spectra of graphs.Spectral graph theory has very important applications in chemistry,theoretical physics,quantum mechanics and computer science.Let{e₁,e₂,...,e_n}be the standard and orthometric basis of abelian group Z₂ⁿ,which can be also viewed as a linear space of dimension n over the Galois field GF（2ⁿ）,and?_k=e_k+e_k+1+···+e_nfor some 1≤k≤n-1,then we have?_k∈Z₂ⁿ.It is well known that the so called enhanced hypercube Q_n,k（1≤k≤n-1）is just the Cayley graph Cay（Z₂ⁿ,S）,where S={e₁,...,e_n,?_k}.LetΓbe a connected graph,an electrical network is obtained by replacing each edge of a graphΓwith a fixed resistance.The node in electrical network can be regarded as the vertex of graphΓ.The resistance distance between any two vertices v_iand v_jof graphΓ,denoted by r_ij,is defined to be the effective resistance between nodes i and j.The Kirchhoff index Kf（Γ）is the sum of the resistance distances between all the pairs of vertices inΓ.This paper is divided into three chapters.In the first chapter,we firstly introduce the research background of spectral graph theory and resistance distance;secondly,some useful definitions and symbols are introduced;finally,some known results of Kirchhoff index are listed.The second chapter is divided into two sections.In the first section,some lemmas about character and Kirchhoff index are listed;in the second section,we have characterised the adjacency spectrum and Laplacian spectrum of enhanced hypercube.In the third chapter,we calculated the Kirchhoff index of the enhanced hypercube and its related networks by using the results obtained above.This chapter is divided into two sections.In the first section,the Kirchhoff index of the enhanced hypercube is calculated and some properties about it are obtained.In the second section,we obtain the formulae for the Kirchhoff index of l(Q_n,k),s(Q_n,k)and t(Q_n,k),respectively.