A $D,\epsilon$ lossless bipartite vertex expander is a $D$-left-regular bipartite graph such that for each subset $S$ of the left vertices which is not too large, the neighborhood of $S$ has size at least $(1-\epsilon)|S|$.
Given a graph, we are often interested in deciding if it satisfies a certain property. Some examples of graph properties include connectivity, bipartiteness, and planarity.