A short note on the sum of [formula omitted] largest distance eigenvalues of bipartite graphs

Abstract

For a connected graph of order n and an integer k≥1, denoted by Sk(D(G))=λ1(D(G))+⋯+λk(D(G)) the sum of the k largest distance eigenvalues of G. In this note, by using a basic technique, we show that λ1(D(G))+λ2(D(G))≥2n−4 when G is bipartite. It leads to that Sk(D(G))≥2n−2k when n is large enough with respect to k, which completely settles a problem posed by Lin (2019).