On Unique Neighborhoods in Bipartite and Expander Graphs

An undirected graph is said to have \emph{unique neighborhoods} if any two distinct nodes have also distinct sets of neighbors. In this way, the connections of a node to other nodes can characterize a node like an "identity", irrespectively of how nodes are named, as long as two nodes are distinguishable. We study the uniqueness of neighborhoods in (random) bipartite graphs, and expander graphs.