The multiplicity of a Hermitian eigenvalue on graphs

For a graph $G$, let $\mathcal{S}(G)$ be the set consisting of Hermitian matrices whose graph is $G$. Denoted by $m_B(G,\lambda)$ the multiplicity of an eigenvalue $\lambda$ of $B(G)\in \mathcal{S}(G)$, we show that $m_B(G,\lambda)\le 2\theta(G)+p(G)$ where $\theta(G)$ and $p(G)$ are the cyclomatic number and the number of pendent vertices of $G$ respectively, and characterize the graphs attaining the equality. This is a generalization of a result on adjacency matrix by Wang et al.\cite{Wang1}. Moreover, they arose an open problem in \cite{Wang1}: \textit{characterize all graphs with $m_A(G,\lambda)=2\theta(G)+p(G)-1$ for any eigenvalue $\lambda$ of its adjacency matrix.} In this paper, we completely characterize the graphs with $m_B(G,\lambda)=2\theta(G)+p(G)-1$ for any eigenvalue $\lambda$ of an arbitrary Hermitian matrix $B(G)\in \mathcal{S}(G)$. This result provides a stronger answer to the above problem, and encompasses some previous known works considering $\lambda=-1$ or $0$ on the problem.