连通图的SDD谱半径及能量的界

Spectral Radius and Energy Bounds of Symmetric Division Deg Matrix for Connected Graphs

设G是n阶连通图,顶点集V(G)={v_(1),v_(2),…,v_(n)}.顶点v_(i)的度用d_(i)表示.G的SDD邻接矩阵A_(SDD)(G)是一个n阶方阵,其中当顶点v_(i)和v_(j)邻接时,它的第(i,j)元素为d_(i)/d_(j)+d_(j)/d_(i),否则为0.图G的SDD谱半径和能量是它的SDD邻接矩阵的谱半径和能量.本文利用基本不等式、Cauchy-Schwarz不等式等的放缩,得到图的SDD谱半径的一些上、下界,并给出达到这些界的极图,也获得了SDD能量的一些上、下界.

Let G be a connected graph of order n with vertex set V(G)={v_(1),v_(2),…,v_(n)}.For v_(i)∈V(G),the degree of the vertex v_(i),is denoted by d i.The symmetric division deg adjacency matrix A SDD(G)of G is the matrix A SDD(G)of order n,where(i,j)-entry is equal to d_(i)/d_(j)+d_(j)/d_(i) if the vertices v_(i) and v_(j) are adjacent and 0 otherwise.The symmetric division deg spectral radius and symmetric division deg energy of G are the spectral radius and energy of its symmetric division deg adjacency matrix,respectively.In this paper we obtained some upper and lower bounds for spectral radius of the symmetric division deg in terms of some fundamental inquality,Cauchy-Schwarzi inquality and other inequalities,and characterize corresponding extremal graphs.And we also obtained some upper and lower bounds for the symmetric division deg energy.