Extremal problems on distance spectra of graphs

Abstract

Motivated by recent progresses on spectral extremal graph theory, in this paper, we aim to investigate the existence of cycles with given length in a graph in terms of its distance spectral radius. First of all, we show that if G is a connected bipartite graph with λ1(D(G))≤λ1(D(K1,n−1)), then G contains a C4 unless G≅K1,n−1. When n is sufficiently large with respect to k, as a corollary, we show that Sk(D(G))≥2n−2k if G is a C4-free bipartite graph. Besides, we prove that Sk(D(G))≥2n−2k if G is a bipartite distance regular graph. These two results partially solve a problem proposed by Lin (2019). Secondly, we give sufficient conditions for the existence of a Hamilton cycle or Hamilton path in a balanced bipartite graph in terms of the distance spectral radius.