  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 $m_G(-1)=n-d$.
