The sum of the k largest distance eigenvalues of graphs

Motivated by progresses on study of eigenvalue sum in adjacency matrix and Laplacian matrix, in this paper, we focus on the sum of the k largest distance eigenvalues of graphs. Denote by Sk(D(G)) = angstrom 1(D(G)) + center dot center dot center dot + angstrom k(D(G)) the sum of the k largest distance eigenvalues of G. We initially determine the extremal graph attaining the minimum Sk(D(G)) among all threshold graphs except for complete graphs. Besides, we prove that the complete r-partite graph attains the minimum Sk(D(G)) among all graphs with chromatic number chi(G) = r and we also give the lower bounds on Sk(D(G)) when G is Kr+1-free, which are extensions of previous results from Lin (2019) and Lu & Lu (2022), respectively. (c) 2023 Elsevier B.V. All rights reserved.