On distance Laplacian spectral determination of complete multipartite graphs

Abstract

The (distance) Laplacian spectrum is said to determine a graph Γ if there is no non-isomorphic graph whose (distance) Laplacian spectra is same as that of Γ. Aouchiche et al. [3] proved that “complete k-partite graph is determined by its distance Laplacian spectrum”. This result is not true. In this paper, we determine the correct result on this. Further motivated by this, the distance Laplacian spectral determination of complete k-partite graph with edge addition is studied and at last it is shown that the graphs whose complements are disconnected and determined by their Laplacian spectra are also determined by their distance Laplacian spectra.