I know that if we consider a graph $G$ with $\lambda$ as one of its eigenvalue of adjacency matrix with multiplicity $n$ ,there is a vertex of $G$ that by removing it ,the multiplicity of $\lambda$ will be $n-1$.
now I want to say an example to show that it is possible to remove one vertex and the multiplicity of one of eigenvalue rise,but I couldn't,it will be great if you help me with this,thanks.
Consider $C_{3} \cup C_{3} \cup K_{4}$. We have an eigenvalue $\lambda = 2$ with multiplicity two. Removing a vertex from $K_{4}$ yields $C_{3}$, thus increasing the multiplicity of $\lambda = 2$ to three.