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In Gephi I visualized a graph to calculate the eigenvalues,then I choose a portion of graph (e.g 6 vertex with their edges) and delete all others. I calculate the eigenvalues again and noticed that I get the same result.By same result I mean same ranking but with different values. The question is if this is happens by chance(I tried three times) or there is a mathematical fact back of it? If yes, what theorem should I search for? or if anybody can explain here and give a proof of it I'll be appreciate it.

Thanks alot, Behzad

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Yes. The adjacency matrix for the subgraph is a principle submatrix of the adjacency matrix of the entire graph, so the eigenvalues will be related by interlacing inequalities. Here is one paper that discusses these inequalities and some graph-theoretic results derived from them.

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