  In this paper, we provide a simple proof for the fact that two simplicial
complexes are isomorphic if and only if their associated Stanley-Reisner rings,
or their associated facet rings are isomorphic as $K$-algebras. As a
consequence, we show that two graphs are isomorphic if and only if their
associated edge rings are isomorphic as $K$-algebras. Based on an explicit
$K$-algebra isomorphism of two Stanley-Reisner rings, or facet rings or edge
rings, we present a fast algorithm to find explicitly the isomorphism of the
associated simplicial complexes, or graphs.
