Let G be a graph with an adjacent matrix A(G). The multiplicity of an
arbitrary eigenvalue λ of A(G) is denoted by mλ​(G). In
\cite{Wong}, the author apply the Pater-Wiener Theorem to prove that if the
diameter of T at least 4, then mλ​(T)≤β′(T)−1 for any
Î»î€ =0. Moreover, they characterized all trees with
mλ​(T)=β′(T)−1, where β′(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
λ of the connected graph G, mλ​(G)≤β′(G)+c(G), where
c(G) is the cyclomatic number of G, and the equality holds if and only if
G≅C3​(a,a,a) or G≅C5​, or a tree with the diameter is at most
3. Furthermore, if β′(G)≥3, we characterize all connected graphs
with mλ​(G)=β′(G)+c(G)−1