  We analyze graphs attaining the extreme values of various spectral indices in
the class of all simple connected graphs, as well as in the class of graphs
which are not complete multipartite graphs. We also present results on density
of spectral gap indices and its nonpersistency with respect to small
perturbations of the underlying graph. We show that a small change in the set
set of edges may result in a significant change of the spectral index like,
e.g., the spectral gap or spectral index. We also present a statistical and
numerical analysis of spectral indices of graphs of the order $m\le 10$. We
analyze the extreme values for spectral indices for graphs and their small
perturbations. Finally, we present the statistical and extreme properties of
graphs on $m\le 10$ vertices.
