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I'm searching for regular graphs with the least eigenvalue -2. Is there any characterization or something that represent all these graphs?also number of vertices is power of n?

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Connected regular graphs with smallest eigenvalue at least $−2$ are either a line graph, a cocktail party graph, or the number of vertices is at most 28.

P. J. Cameron, J. M. Goethals, J. J. Seidel and E. E. Shult, Line graphs, root systems, and elliptic geometry, J. Algebra 43, 305–327 (1976).

This is Theorem 3.12.2 in Distance-Regular Graphs (2012).

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