Combinatorial characterizations of the Cohen–Macaulayness of the second power of edge ideals

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). application of our results, we will classify all bipartite graphs in which the second (symbolic) powers are Cohen–Macaulay (resp. Buchsbaum, generalized Cohen–Macaulay).