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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.

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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.

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  • $\begingroup$ thanks,I didn't think this way,if we want to have a connected example,can we have one? $\endgroup$ – kpax Dec 23 '14 at 6:31
  • $\begingroup$ $C_3$ is 2-regular,and its biggest eigenvalue is 2,so you mean that the multiplicity of $\lambda=2$ increase,am I right? $\endgroup$ – kpax Dec 23 '14 at 6:37
  • $\begingroup$ That's correct. Serves me right for posting late at night! :-) I can't think of a connected example, but I'm no expert with spectral graph theory. Sorry I can't be of more help! $\endgroup$ – ml0105 Dec 23 '14 at 6:39
  • $\begingroup$ thanks a lot,it was so helpful for me,thank you for answering my question. $\endgroup$ – kpax Dec 23 '14 at 6:43
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    $\begingroup$ The graph on five vertices got by joining a new vertex to one vertex on the the 4-cycle is a connected example. Its eigenvalues are simple and there are three vertices you can delete. $\endgroup$ – Chris Godsil Dec 23 '14 at 13:03

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