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I am looking at comparing multiple graph topologies based on their spectra. From the set of all $N\times N$ adjacency matrices, is there any result which points to the adjacency matrix with the highest max eigenvalue and the one with the lowest min eigenvalue?
I know several results which provide closed form expressions for specific well-defined topologies, but have not come across anything which compares spectra across graphs.
The maximum eigenvalue of a connected graph is bounded between the average degree and the maximum degree of the vertices, so it follows that its maximum value is $N-1$, attained for the complete graph on $N$ vertices. See, e.g., E2 in section 2.1 of http://www.ams.org/bookstore/pspdf/cbms-115-prev.pdf
