give an example to show it is possible to remove one vertex and the multiplicity of one of eigenvalue rise.

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.
• $C_3$ is 2-regular,and its biggest eigenvalue is 2,so you mean that the multiplicity of $\lambda=2$ increase,am I right? – kpax Dec 23 '14 at 6:37