Question about the relationship between the eigenvalues of graph and its subgraph

Dear all,

I have a question based on the following theorem,


Given an n*n matrix M, we can compute n eigenvalues after
normalization, denoted lamda_1 ... lamda_n. The maximum and minimum
eigenvalues are denoted lamda(max) and lamda(min), respectively.
Then, let G and H be two undirected unlabeled graphs and M_G, M_H be
their adjacency matrices. If H is an induced subgraph of G, then
lamda(min)(M_G) <= lamda(min)(M_H) <= lamda(max)(M_H) <=
lamda(max)(M_G).


Above is a nice property, however, it is for the induced subgraph. My
question is that, H is a subgraph of G, instead of an induced subgraph,

can we deduce similar relationship among the eigenvalues?


Regards,
Joe

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