I search about known graphs have spectrum with the most zero eigenvalues respect to their adjacency matrix. I know null, complete, bipartite and cocktel party graphs. Any kind of suggestion is appreciated. Thanks to everyone for the help.
2026-03-25 12:45:49.1774442749
Graphs with zero eigenvalues
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Take the bipartite graph $K_{m,n}$, then its spectrum is $$ \{ \sqrt{mn}, 0^{m+n-2}, - \sqrt{mn}\}$$
Surely one of the most non-trivial graph with the desired property. Otherwise yes as stated in the comments, take the null matrix.
Edit you should take a look at this article, talking about cospectrality of complete bipartite graph. You should be able to construct graph with spectrum close to the complete bipartite's one, hence with some null values
Edit 2 Looking at this, you can generalize on complete multi partite graphs, with $k$ sets and eigenvalue $0^{n-k}$. After that i don't know if you can do much better.