A relation between the signless Laplacian spectral radius of complete multipartite graphs and majorization

Abstract

Let G be a simple graph with vertices v1,…,vn. Let A(G) be the adjacency matrix of G and D(G) be the diagonal matrix (d1,…,dn), where di is the degree of vertex vi, for i=1,…,n. The matrix Q(G)=D(G)+A(G) is called the signless Laplacian matrix of G. By the signless Laplacian spectral radius of G, denoted by q(G), we mean the largest eigenvalue of Q(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 signless Laplacian spectral radius of complete multipartite graphs. We show that if (m1,…,mt)⪰M(n1,…,nt) and (m1,…,mt)≠(n1,…,nt) then q(Kn1,…,nt)>q(Km1,…,mt), where Kn1,…,nt is the complete multipartite graph with t parts of size n1,…,nt. Using the above relation we find that among all complete multipartite graphs with n vertices and t≥3 parts, the split graphs have the minimum signless Laplacian spectral radius and the Turán graphs have the maximum signless Laplacian spectral radius. Finally we obtain that for every positive integers t≥2 and n1,…,nt,n+2t−4+n2+(n−t)(4t−8)2≤q(Kn1,…,nt)≤3n−2(a+b)+n2−4n(a+b)+4(a−b)2+8abt2, where n=n1+⋯+nt, a=⌈nt⌉ and b=⌊nt⌋. In addition we investigate the equality in both sides.