Time: 9h00, Wednesday, January 28, 2015

Location: Room 6, Building A14, Institute of Mathematics, 18 Hoang Quoc Viet, Cau Giay, Hanoi

Abstract: Let  $R=k[x_1,\\ldots,x_n]$ be a polynomial ring in $n$ variables   over field $k$. Let $\G$ be a simple graph with vertex set   $\\{x_1,\\ldots,x_n\\}$ and edge set $E(\\G)$. The squarefree monomial ideal  $$I(\G) = (x_ix_j|x_ix_j\\in E(\\G))\\subseteq R$$  is called the {\\it edge ideal} of  $\G$. We say that  $\G$ is {\it Cohen-Macaulay} (resp. {\it Gorenstein})  over field  $k$  if  $R/I(\\G)$ is   Cohen-Macaulay (resp.   Gorenstein) over field $k$. In this thesis, we classify all Cohen-Macaulay graphs of girth at least 5  and all triangle-free Gorenstein graphs. By this classification, we give  a purely combinatorial characterization for  Cohen-Macaulay property of the second power of edge ideals and      of saturation of the second power of edge ideals.