  We find strong relationships between the zero-divisor graphs of apparently
disparate kinds of nilpotent-free semigroups by introducing the notion of an
\emph{Armendariz map} between such semigroups, which preserves many
graph-theoretic invariants. We use it to give relationships between the
zero-divisor graph of a ring, a polynomial ring, and the annihilating-ideal
graph. Then we give relationships between the zero-divisor graphs of certain
topological spaces (so-called pearled spaces), prime spectra, maximal spectra,
tensor-product semigroups, and the semigroup of ideals under addition,
obtaining surprisingly strong structure theorems relating ring-theoretic and
topological properties to graph-theoretic invariants of the corresponding
graphs.
