Many mathematicians have noted this fact and have expressed surprise that it does not appear to have a ready explanation.
Note that the maximization ranges over subsets A of G, subject to A being over half the size of G In words, this means that one can take a subset of size at least half of G, and blow it up by only
Long "stringy" (i.e. not "compact") families of graphs such as the cycle graph of order n clearly don't have such a property: one could consider a subset comprising the n/2 neighborhood of a point (midnight to six o'clock, say).
A Levy family would have this neighborhood covering almost all the set.
It should be clear that a Levy family must have a very special type of compact structure.