The multiplicity of eigenvalue -2 of the distance matrix of graphs

Abstract

Let $D(G)$ be the
distance matrix and $\lambda_1(D(G))\geq\cdots\geq\lambda_n(D(G))$ be the distance spectra of a connected graph $G$. Let $m_\lambda(D)$ denote the multiplicity of the eigenvalue $\lambda$ of the distance matrix $D$ of $G$. In this paper, we characterized graphs with $m_{-2}(D(G))=n-i$, where $i=1,2,3,4$. Furthermore, we show that both $S_n^+$ and $S_{a,b}$ $(a+b=n-2)$ are determined by their $D$-spectrum.