On extremal properties of perfect 2-colorings

A coloring of vertices of a graph is called perfect if, for every vertex, the collection of colors of its neighbors depends only on its own color. The correspondent color partition of vertices is called equitable. We note that a number of bounds (Hoffman bound, Cheeger bound, Bierbrauer--Friedman bound and other) is only reached on perfect $2$-colorings. We show that the Expander Mixing Lemma is another example of an inequality that generates a perfect $2$-coloring. We prove a new upper bound for the size of $S\subset V(G)$ with the fixed average internal degree for an amply regular graph $G$. This bound is reached on the set $S$ if and only if $\{S, V(G)\setminus S\}$ is an equitable partition.