Distance spectral radius of complete multipartite graphs and majorization

Abstract

Let G be a simple connected graph with vertices v1,…,vn. The distance matrix of G, denoted by D(G), is the n×n matrix whose (i,j)-entry is equal to d(vi,vj) (length of a shortest path between vi and vj). By the distance spectral radius of G that is denoted by μ(G), we mean the largest eigenvalue of the distance matrix of G. Let X=(m1,…,mt) and Y=(n1,…,nt), where m1≥⋯≥mt≥1 and n1≥⋯≥nt≥1 are integer. We say X majorizes Y and let X⪰MY, if for every j, 1≤j≤t−1, ∑i=1jmi≥∑i=1jni with equality if j=t. In this paper we find a relation between the majorization and the distance spectral radius of complete multipartite graphs. We show that if (m1,…,mt)⪰M(n1,…,nt) and (m1,…,mt)≠(n1,…,nt) then μ(Km1,…,mt)>μ(Kn1,…,nt), where Kn1,…,nt is the complete multipartite graph with t parts of size n1,…,nt. Using the above inequality we find that among all complete multipartite graphs with n vertices and t≥2 parts, the Turán graphs have the minimum distance spectral radius and the split graphs have the maximum distance spectral radius. Let t≥2 and n1,…,nt be some positive integers. We show thatn+a+b−4+(n+a+b)2−4ab(t+1)2≤μ(Kn1,…,nt)≤2n−t−2+(2n−2t+1)2+t2−12, where n=n1+⋯+nt, a=⌈nt⌉ and b=⌊nt⌋. In addition we investigate the equality in both sides. Finally we obtain that μ(Kn1,…,nt)≤2n−t−1.