On graphs with $m({\partial }_1^L)=n-3$

Abstract

Let ${\partial }_1^L\ge {\partial }_2^L\ge \cdots \ge {\partial }_n^L$ be the distance Laplacian eigenvalues of a connected graph $G$ and $m({\partial }_i^L)$ the multiplicity of ${\partial }_i^L$. It is well known that the graphs with $m({\partial }_1^L)=n-1$ are complete graphs. Recently, the graphs with $m({\partial }_1^L)=n-2$ have been characterized by Celso et al. In this paper, we completely determine the graphs with $m({\partial }_1^L)=n-3$.

13 pages, 3 figures