  Let f be a function mapping an n dimensional vector space over GF(p) to
GF(p). When p is 2, Bernasconi et al. have shown that there is a correspondence
between certain properties of f (e.g., if it is bent) and properties of its
associated Cayley graph. Analogously, but much earlier, Dillon showed that f is
bent if and only if the "level curves" of f had certain combinatorial
properties (again, only when p is 2). The attempt is to investigate an
analogous theory when p is greater than 2 using the (apparently new)
combinatorial concept of a weighted partial difference set. More precisely, we
try to investigate which properties of the Cayley graph of f can be
characterized in terms of function-theoretic properties of f, and which
function-theoretic properties of f correspond to combinatorial properties of
the set of "level curves", i.e., the inverse map of f. While the natural
generalizations of the Bernasconi correspondence and Dillon correspondence are
not true in general, using extensive computations, we are able to determine a
classification in some small cases. Our main conjecture is Conjecture 67.
