On isomorphism of simplicial complexes and their related algebras

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.