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For a connected graph $G$ with order $n$, let $e(G)$ be the number of itsdistinct eigenvalues and $d$ be the diameter. We denote by $m_G(\mu)$ theeigenvalue 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 isa real symmetric matrix with nullity $n-d$ and $n-d-1$. In this paper, byapplying 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 completelycharacterize all minimal graphs with $m_G(-1)=n-d$.

Minimal graphs with eigenvalue multiplicity of $n-d$