Speaker: Do Trong Hoang
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.
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