A relation between multiplicity of nonzero eigenvalues and the matching number of graph

Let $G$ be a graph with an adjacent matrix $A(G)$. The multiplicity of an arbitrary eigenvalue $\lambda$ of $A(G)$ is denoted by $m_\lambda(G)$. In \cite{Wong}, the author apply the Pater-Wiener Theorem to prove that if the diameter of $T$ at least $4$, then $m_\lambda(T)\leq \beta'(T)-1$ for any $\lambda\neq0$. Moreover, they characterized all trees with $m_\lambda(T)=\beta'(T)-1$, where $\beta'(G)$ is the induced matching number of $G$. In this paper, we intend to extend this result from trees to any connected graph. Contrary to the technique used in \cite{Wong}, we prove the following result mainly by employing algebraic methods: For any non-zero eigenvalue $\lambda$ of the connected graph $G$, $m_\lambda(G)\leq \beta'(G)+c(G)$, where $c(G)$ is the cyclomatic number of $G$, and the equality holds if and only if $G\cong C_3(a,a,a)$ or $G\cong C_5$, or a tree with the diameter is at most $3$. Furthermore, if $\beta'(G)\geq3$, we characterize all connected graphs with $m_\lambda(G)=\beta'(G)+c(G)-1$.