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Let G be a finite simple graph, and J G denote the binomial edge ideal of G. In this paper, we first compute the v -number of binomial edge ideals corresponding to Cohen–Macaulay closed graphs. As a consequence, we obtain the v -number for paths. For cycle and binary tree graphs, we obtain a sharp upper bound for v (J G) using the number of vertices of the graph. We characterize all connected graphs G with v (J G) = 2. We show that for a given pair (k , m) , k ≤ m , there exists a graph G with an associated monomial edge ideal I having v -number equal to k and regularity m. We also show that if 2 k ≤ m , then there exists a binomial edge ideal with v -number k and regularity m. Finally, we compute v -number of powers of binomial edge ideals with linear resolution, thus proving a conjecture on the v -number of powers of a graded ideal having linear powers, for the class of binomial edge ideals. [ABSTRACT FROM AUTHOR]

The study of the edge ideal I(DG) of a weighted oriented graph DG with underlying graph G started in the context of Reed–Muller type codes. We generalize some Cohen–Macaulay constructions for I(DG), which Villarreal gave for edge ideals of simple graphs. Our constructions can be used to produce large classes of Cohen–Macaulay weighted oriented edge ideals. We use these constructions to classify all the Cohen–Macaulay weighted oriented edge ideals, whose underlying graph is a cycle. We also show that I(DCn) is Cohen–Macaulay if and only if I(DCn) is unmixed and I(Cn) is Cohen–Macaulay, where Cn denotes the cycle of length n. Miller generalized the concept of Alexander dual ideals of square-free monomial ideals to arbitrary monomial ideals, and in that direction, we study the Alexander dual of I(DG) and its conditions to be Cohen–Macaulay. [ABSTRACT FROM AUTHOR]

For a graph G and the binomial edge ideal JG of G, Bolognini et al. have proved the following: JG is strongly unmixed ⇒JG is Cohen–Macaulay ⇒G is accessible. Moreover, they have conjectured that the converse of these implications is true. Accessible and strongly unmixed properties are purely combinatorial. We give some motivations to focus only on blocks with whiskers for the characterization of all G with Cohen–Macaulay JG. We show that accessible and strongly unmixed properties of G depend only on the corresponding properties of its blocks with whiskers and vice versa. We give a new family of graphs whose binomial edge ideals are Cohen–Macaulay, and from that family, we classify all r-regular r-connected graphs, with the property that, after attaching some special whiskers to it, the binomial edge ideals become Cohen–Macaulay. To prove the Cohen–Macaulay conjecture, it is enough to show that every non-complete accessible graph G has a cut vertex v such that G ∖{v} is accessible. We show that any non-complete accessible graph G having at most three cut vertices has a cut vertex v for which G ∖{v} is accessible. [ABSTRACT FROM AUTHOR]