Explicit expression of eigenvalue and eigenvector of a graph

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Could any one tell me what kind of graph has the explicit expression of its eigenvalue and eigenvector? Thanks!

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It's not really clear what you want. If it's just a list of graphs then: (a) Paths. (b) Cayley graphs for abelian groups. (c) Strongly regular graphs.

There are other classes, but these would be the simplest.