  Let $I(G)$ be the edge ideal of a simple graph $G$. In this paper, we will
give sufficient and necessary combinatorial conditions of $G$ in which the
second symbolic and ordinary power of its edge ideal are Cohen-Macaulay (resp.
Buchsbaum, generalized Cohen-Macaulay).
  As an application of our results, we will classify all bipartite graphs in
which the second (symbolic) powers are Cohen-Macaulay (resp. Buchsbaum,
generalized Cohen-Macaulay).
