a b s t r a c tA graph is called very well-covered if it is unmixed without isolated vertices such that the cardinality of each minimal vertex cover is half the number of vertices. We first prove that a very well-covered graph is Cohen-Macaulay if and only if it is vertex decomposable. Next, we show that the Castelnuovo-Mumford regularity of the quotient ring of the edge ideal of a very well-covered graph is equal to the maximum number of pairwise 3-disjoint edges.
Abstract. Let G be a finite simple graph on a vertex set V (G) = {x 11 , . . . , x n1 }. Also let m 1 , . . . , , m n ≥ 2 be integers and G 1 , . . . , G n be connected simple graphs on the vertex sets V (G i ) = {x i1 , . . . , x imi }. In this paper, we provide necessary and sufficient conditions on G 1 , . . . , G n for which the graph obtained by attaching G i to G is unmixed or vertex decomposable. Then we characterize Cohen-Macaulay and sequentially Cohen-Macaulay graphs obtained by attaching the cycle graphs or connected chordal graphs to an arbitrary graphs.
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Abstract. Let G and H be two simple graphs and let G * H denotes the graph theoretical product of G by H. In this paper we provide some results on graded Betti numbers, Castelnuovo-Mumford regularity, projective dimension, h-vector, and Hilbert series of G * H in terms of that information of G and H. To do this, we will provide explicit formulae to compute graded Betti numbers, h-vector, and Hilbert series of disjoint union of complexes. Also we will prove that the family of graphs whose regularity equal the maximum number of pairwise 3-disjoint edges, is closed under product of graphs.
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