On the least distance eigenvalue and its applications on the distance spread

• Published in: Discrete mathematics Vol. 338; no. 6; pp. 868 - 874
• Main Author: Lin, Huiqiu
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 06.06.2015
• Subjects: Distance spectra; Distance spectral radius; Distance spread; The least distance eigenvalue; Distance spectra; Distance spread; The least distance eigenvalue; Distance spectral radius
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2015.01.006

Let G be a connected graph with order n and D(G) be its distance matrix. Suppose that λ1(D)≥⋯≥λn(D) are the distance eigenvalues of G. In this paper, we give an upper bound on the least distance eigenvalue and characterize all the connected graphs with −1−2≤λn(D)≤a where a is the smallest root of x3−x2−11x−7=0 and a∈(−1−2,−2). Furthermore, we show that connected graphs with λn(D)≥−1−2 are determined by their distance spectra. As applications, we give some lower bounds on the distance spread of graphs with given some parameters. In the end, we characterize connected graphs with the (k+1)th smallest distance spread.