In this paper we show that if the Stanley-Reisner ring of the simplicial
complex of independent sets of a bipartite graph $G$ satisfies Serre's
condition $S_2$, then $G$ is Cohen-Macaulay. As a consequence, the
characterization of Cohen-Macaulay bipartite graphs due to Herzog and Hibi
carries over this family of bipartite graphs. We check that the equivalence of
Cohen-Macaulay property and the condition $S_2$ is also true for chordal graphs
and we classify cyclic graphs with respect to the condition $S_2$.