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ALGEBRAIC STUDY ON CAMERON–WALKER GRAPHS
, 2014
"... Let G be a finite simple graph on [n] and I(G) ⊂ S the edge ideal of G, where S = K[x1,..., xn] is the polynomial ring over a field K. Let m(G) denote the maximum size of matchings of G and im(G) that of induced matchings of G. It is known that im(G) ≤ reg(S/I(G)) ≤ m(G), where reg(S/I(G)) is th ..."
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Let G be a finite simple graph on [n] and I(G) ⊂ S the edge ideal of G, where S = K[x1,..., xn] is the polynomial ring over a field K. Let m(G) denote the maximum size of matchings of G and im(G) that of induced matchings of G. It is known that im(G) ≤ reg(S/I(G)) ≤ m(G), where reg(S/I(G)) is the Castelnuovo–Mumford regularity of S/I(G). Cameron and Walker succeeded in classifying the finite connected simple graphs G with im(G) = m(G). We say that a finite connected simple graph G is a Cameron–Walker graph if im(G) = m(G) and if G is neither a star nor a star triangle. In the present paper, we study Cameron–Walker graphs from a viewpoint of commutative algebra. First, we prove that a Cameron–Walker graph G is unmixed if and only if G is Cohen– Macaulay and classify all Cohen–Macaulay Cameron–Walker graphs. Second, we prove that there is no Gorenstein Cameron–Walker graph. Finally, we prove that every Cameron–Walker graph is sequentially Cohen–Macaulay.
THE MACAULAY 2 PACKAGE EDGEIDEALS
"... Abstract. We introduce the Macaulay 2 package EdgeIdeals as a tool to study edge and cover ideals. These tutorials complement the lectures given at MONICA: MONo ..."
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Abstract. We introduce the Macaulay 2 package EdgeIdeals as a tool to study edge and cover ideals. These tutorials complement the lectures given at MONICA: MONo
A BEGINNER’S GUIDE TO EDGE AND COVER IDEALS
"... Abstract. Our goal is to introduce the basics properties of edge and cover ideals, and to introduce some current research themes. We also include an introduction to the Macaulay 2 computer package EdgeIdeals. These notes are an expanded version of my ..."
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Abstract. Our goal is to introduce the basics properties of edge and cover ideals, and to introduce some current research themes. We also include an introduction to the Macaulay 2 computer package EdgeIdeals. These notes are an expanded version of my
SEQUENTIALLY COHENMACAULAY BIPARTITE GRAPHS: VERTEX DECOMPOSABILITY AND REGULARITY
, 2009
"... Let G be a bipartite graph with edge ideal I(G) whose quotient ring R/I(G) is sequentially CohenMacaulay. We prove: (1) the independence complex of G must be vertex decomposable, and (2) the CastelnuovoMumford regularity of R/I(G) can be determined from the invariants of G. ..."
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Let G be a bipartite graph with edge ideal I(G) whose quotient ring R/I(G) is sequentially CohenMacaulay. We prove: (1) the independence complex of G must be vertex decomposable, and (2) the CastelnuovoMumford regularity of R/I(G) can be determined from the invariants of G.
SOME ALGEBRAIC PROPERTIES OF HYPERGRAPHS
, 2008
"... We consider Stanley–Reisner rings k[x1,..., xn]/I(H) where I(H) is the edge ideal associated to some particular classes of hypergraphs. For instance, we consider hypergraphs that are natural generalizations of graphs that are lines and cycles, and for these we compute the Betti numbers. We also ge ..."
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We consider Stanley–Reisner rings k[x1,..., xn]/I(H) where I(H) is the edge ideal associated to some particular classes of hypergraphs. For instance, we consider hypergraphs that are natural generalizations of graphs that are lines and cycles, and for these we compute the Betti numbers. We also generalize upon some known results about chordal graphs and study a weak form of shellability.