Eigen value of principal submatrix. I was studying "interlacing property" and trying to find out the below fact- $A$ is an adjacency matrix of a $r$ regular graph $G$. $u,v \in G $;$u,v$ are not similar vertices. $B$ is the adjacency matrix of graph$(G-v)$( deleting v) and $C$ is the adjacency matrix of graph $(G-u)$(deleting u). **Is there always exists different eigenvalues(at least one ) in $B,C$? i.e. there always exists $\lambda$ , an eigen value of $B$, which is not an eigen value of $C$. ** I have come to know about theorems of Perron,Friedland,Cauchy regarding the "interlacing property" of eigen value of principal sub-matrix, but not much relation to above problem. I doubt there is no guarantee of the above statement, since co-spectral graph can be non-isomorphic. In that case, **when it is guaranteed??** Any comment/answer/reference would help.

Any strongly regular graph that is not vertex transitive provides examples. If $G$ is strongly regular, all subgraphs obtained by deleting one vertex have the same spectrum. The smallest such strongly regular graphs are on 25 vertices, and can be constructed from Latin squares of order five.

Why are these subgraphs cospectral. If $a\in V(G)$, then the rational function $$ \phi(G\setminus a,t)\phi(G,t)$$ is a form of the generating function for the closed walks in $G$ that start at $a$. But for strongly regular graphs, this generating function is determined by the parameters.

I am practically certain that there will be smaller examples of regular graphs with vertices that are "cospectral" but not similar.

I am not aware of any characterization of cases where the vertex-deleted subgraphs of a graph are determined by their spectrum. I would be very surprised if there was one.