一类联图的距离谱半径以及盖理论

The Distance Spectral Radius of a Class of Joint Graphs and Majorization

令X=(n1,n2,…,nt),Y=(m1,m2,…,mt)是两个t维递减序列.如果对所有的j,1≤j≤t,都有∑i=1j、ni≥∑i=1j mi以及∑i=1t ni=∑i=1t mi,则称X可盖Y,记作X■Y.如果X≠Y,则记作X■Y.本文考虑联图G(n1,n2,…,nt;a)=(Kn1∪n2∪…∪Knt)∨Ka的谱半径,这里n1+n2+…+nt+a=n,(n1,n2,…,nt)是一个递减整数序列,2≤t≤n-a,且a≥1.完全图Knj称为联图G(n1,n2,…,nt;a)的一个分支.对联图G(n1,n2,…,nt;a),我们证明了λ(G(n1,n2,…,nt;a))<λ(G(m1,m2,…,mt;a)当且仅当(n1,n2,…,nt)■(m1,m2,…,mt),其中λ(G)表示图G的距离谱半径.此外,我们证明了在所有包含n个节点以及t个分支的联图中,联图G(X_(balance);a)具有最大谱半径,联图G(n-a-t+1,1,…,1;a)具有最小谱半径,其中X_(balance)是含有r项p=[n-a/t]和s项q=[n-a/t]的非增序列,rp+叫=?a;并给出了G(X_(balance))谱半径的上界和下界.

Let Bt={(z1,z2,…,zt)∈Zt:z1≥z2≥…zt≥1}and X=(n1,n2,…,nt),Y=(m1,m2…,mt)∈Bt.We say X majorizates Y,denoted by X■Y,if∑i=1t ni=∑i=1t mi and∑i=1j ni≥∑i=1j mi for all j,1≤j≤t.If X≠Y,we say X■Y.In this paper,we consider a class of joint graphs,denoted as G(n1,n2,…,nt:a)=(Kn1∪Kn2∪…∪Knt)∨Ka,where n1+n2+…+nt+a=n,(n1,n2,…,nt)∈Bt,2≤t≤n-a and a≥1.Each Knj is called a branch of G(n1,n2,…,nt;a).We show thatλ(G(n1,n2,…,nt;a))<λ(G(m1,m2,…,mt;a))if(n1,n2,…,nt)■(m1,m2,…,mt),whereλ(G)is the distance spectral radius of G.Furthermore,we prove that among this class of joint graphs with n vertices and t branches,graph G(X_(balance);a)has the maximum distance spectral radius and graph G(n-a-t+1,1,…,1;a)has the minimum distance spectral radius where X_(balance) is the non-increasing sequence with r terms of value p=[n-a/t],s terms of value q=[n-a/t]and rp+sq=n-a.For graph G(X_(balance);a),we give an upper and lower bound for its distance spectral radius.