Given a simple, undirected graph with maximum degree $d$. Let its spectrum be the eigenvalues of the associated adjacency matrix. It is known that this spectrum lies between $-d$ and $d$.
Is it possible to say anything stronger about the shape of the spectrum?
As two extreme cases I would like to distinguish, consider the spectrum of the complete graph versus the spectrum of the cycle:
- complete graph over $N$ nodes: spectrum $\{N,0\}$, where $0$ is $(N-1)$-fold degenerate. I.e., the spectrum is very peaked.
- cycle over $N$ nodes: spectrum $\{\cos(2\pi k/N)|k\in[0,N)\}$. I.e., the spectrum is almost evenly distributed.