A complete characterization of graphs for which $m_G(-1)=n-d-1$

Let $G$ be a simple connected graph of order $n$ with diameter $d$. Let $m_G(-1)$ denote the multiplicity of the eigenvalue $-1$ of the adjacency matrix of $G$, and let $P=P_{d+1}$ be the diameter path of $G$. If $-1$ is not an eigenvalue of $P$, then by the interlacing theorem, we have $m_G(-1)\le n-d-1$. In this article, we characterize the extremal graphs where equality holds. Moreover, for the completeness of the results, we also characterize the graphs $G$ that achieve $m_G(-1)=n-d-1$ when $-1$ is an eigenvalue of $P$. Thus, we provide a complete characterization of the graphs $G$ for which $m_G(-1)=n-d-1$.