matrices - Properties of Graphs with an eigenvalue of -1 (adjacency matrix)?

There are a lot of graphs with this property. I just introduce two classes that are very famous:

The Friendship graph $F_n$ that is $K_1nabla nK_2$, where $nabla$ means the join of two graphs. These graphs has $n-1$ eigenvalue $-1$.

The second class is graph $K_n$ that is removed $1,2,3$ or $4$ edges from it. So, this class contain also five different classes!!!

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