Minimal graphs with eigenvalue multiplicity of $n-d$

Abstract: For a connected graph $G$ with order $n$, let $e(G)$ be the number of its distinct eigenvalues and $d$ be the diameter. We denote by $m_G(\mu)$ the eigenvalue multiplicity of $\mu$ in $G$. It is well known that $e(G)\geq d+1$, which shows $m_G(\mu)\leq n-d$ for any real number $\mu$. A graph is called $minimal$ if $e(G)= d+1$. In 2013, Wang (\cite{WD}, Linear Algebra Appl.) characterize all minimal graphs with $m_G(0)=n-d$. In 2023, Du et al. (\cite{Du}, Linear Algebra Appl.) characterize all the trees for which there is a real symmetric matrix with nullity $n-d$ and $n-d-1$. In this paper, by applying the star complement theory, we prove that if $G$ is not a path and $m_G(\mu)= n-d$, then $\mu \in {0,-1}$. Furthermore, we completely characterize all minimal graphs with $m_G(-1)=n-d$.