Minimal graphs with eigenvalue multiplicity of n − d

Abstract

For a connected graph G with order n, let $e\left(G\right)$ be the number of its distinct eigenvalues and d be the diameter. We denote by $m_G\left(\mu \right)$ the eigenvalue multiplicity of μ in G. It is well known that $e\left(G\right)\ge d+1$, which shows $m_G\left(\mu \right)\le n-d$ for any real number μ. A graph is called $minimal$ if $e\left(G\right)=d+1$. In 2013, Wong et al. characterize all minimal graphs with $m_G\left(0\right)=n-d$. In this paper, by applying the star complement theory, we prove that if G is not a path and $m_G\left(\mu \right)=n-d$, then $\mu \in \{0,-1\}$. Furthermore, we completely characterize all minimal graphs with $m_G(-1)=n-d$.