Complete multipartite graphs are determined by their distance spectra

Abstract

It is well known that the complete multipartite graphs can not be determined by their adjacency spectra. But in this paper, we prove that they can be determined by their distance spectra, which confirms the conjecture proposed

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

“…In the same paper, they conjectured that the complete multipartite graphs are determined by their distance spectra. Recently, Jin and Zhang [6] confirmed the conjecture. Let a be the smallest root of a] if and only if G ∼ = G(s, n 1 , .…”

“…If s = 0, then G ∼ = K n 1 ,...,n k and the result follows from Jin and Zhang [6]. If s ≥ 1, then by Lemmas 2.5 and 3.3, the graph cospectral to G must be G ′…”

“…In [8], Lin et al proved that the complete bipartite graph K n 1 ,n 2 and the complete split graph K a ∨ K c b are determined by D-spectra, and conjectured that the complete k-partite graph K n 1 ,n 2 ,...,n k is determined by its D-spectra. Recently, Jin and Zhang [6] confirmed the conjecture. Lin, Zhai and Gong [9] showed that the graph K r s,t is determined by its distance spectra.…”

Self‐Assembly of Aggregation‐Induced‐Emission Molecules

“…There are many similar AIE‐based polymers, such as TPE‐mPEG, synthesized by Liang et al., and AIE PCL‐b‐PEG‐based AIE Pdots, reported by Fu et al . All these particles display strong fluorescence and excellent biocompatibility, which makes them promising for cell imaging– and imaging‐guided photodynamic cancer therapy applications …”

Hypercubes are determined by their distance spectra

“…The study of question whether a graph is determined by its distance spectrum is just in its beginning. See McKay [15], Lin et al [14], and Jin & Zhang [12] for recent papers on this subject. This paper is organized as follows.…”