Distance spectral radius of complete multipartite graphs and majorization.

Let G be a simple connected graph with vertices v 1 , ... , v n. The distance matrix of G , denoted by D (G) , is the n × n matrix whose (i , j) -entry is equal to d (v i , v j) (length of a shortest path between v i and v j). 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 = (m 1 , ... , m t) and Y = (n 1 , ... , n t) , where m 1 ≥ ⋯ ≥ m t ≥ 1 and n 1 ≥ ⋯ ≥ n t ≥ 1 are integer. We say X majorizes Y and let X ⪰ M Y , if for every j , 1 ≤ j ≤ t − 1 , ∑ i = 1 j m i ≥ ∑ i = 1 j n i 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 (m 1 , ... , m t) ⪰ M (n 1 , ... , n t) and (m 1 , ... , m t) ≠ (n 1 , ... , n t) then μ (K m 1 , ... , m t ) > μ (K n 1 , ... , n t ) , where K n 1 , ... , n t is the complete multipartite graph with t parts of size n 1 , ... , n t. 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 n 1 , ... , n t be some positive integers. We show that n + a + b − 4 + (n + a + b) 2 − 4 a b (t + 1) 2 ≤ μ (K n 1 , ... , n t ) ≤ 2 n − t − 2 + (2 n − 2 t + 1) 2 + t 2 − 1 2 , where n = n 1 + ⋯ + n t , a = ⌈ n t ⌉ and b = ⌊ n t ⌋. In addition we investigate the equality in both sides. Finally we obtain that μ (K n 1 , ... , n t ) ≤ 2 n − t − 1. [ABSTRACT FROM AUTHOR]