Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userEdge ideals of oriented graphs
156
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let $mathcal{D}$ be a weighted oriented graph and let $I(mathcal{D})$ be its edge ideal. Under a natural condition that the underlying (undirected) graph of $mathcal{D}$ contains a perfect matching consisting of leaves, we provide several equivalent conditions for the Cohen-Macaulayness of $I(mathcal{D})$. We also completely characterize the Cohen-Macaulayness of $I(mathcal{D})$ when the underlying graph of $mathcal{D}$ is a bipartite graph. When $I(mathcal{D})$ fails to be Cohen-Macaulay, we give an instance where $I(mathcal{D})$ is shown to be sequentially Cohen-Macaulay.
rate research
Read More
Let $mathcal{D}$ be a weighted oriented graph and $I(mathcal{D})$ be its edge ideal. In this paper, we show that all the symbolic and ordinary powers of $I(mathcal{D})$ coincide when $mathcal{D}$ is a weighted oriented certain class of tree. Finally, we give necessary and sufficient conditions for the equality of ordinary and symbolic powers of naturally oriented lines.
Let I=I(D) be the edge ideal of a weighted oriented graph D. We determine the irredundant irreducible decomposition of I. Also, we characterize the associated primes and the unmixed property of I. Furthermore, we give a combinatorial characterization for the unmixed property of I, when D is bipartite, D is a whisker or D is a cycle. Finally, we study the Cohen-Macaulay property of I.
Let $G$ be a connected simple graph on the vertex set $[n]$. Banerjee-Betancourt proved that $depth(S/J_G)leq n+1$. In this article, we prove that if $G$ is a unicyclic graph, then the depth of $S/J_G$ is bounded below by $n$. Also, we characterize $G$ with $depth(S/J_G)=n$ and $depth(S/J_G)=n+1$. We then compute one of the distinguished extremal Betti numbers of $S/J_G$. If $G$ is obtained by attaching whiskers at some vertices of the cycle of length $k$, then we show that $k-1leq reg(S/J_G)leq k+1$. Furthermore, we characterize $G$ with $reg(S/J_G)=k-1$, $reg(S/J_G)=k$ and $reg(S/J_G)=k+1$. In each of these cases, we classify the uniqueness of extremal Betti number of these graphs.
In this paper we provide some exact formulas for the projective dimension and the regularity of edge ideals associated to three special types of vertex-weighted oriented $m$-partite graphs. These formulas are functions of the weight and number of vertices. We also give some examples to show that these formulas are related to direction selection and the weight of vertices.
We classify generalized block graphs whose binomial edge ideals admit a unique extremal Betti number. We prove that the Castelnuovo-Mumford regularity of binomial edge ideals of generalized block graphs is bounded below by $m(G)+1$, where $m(G)$ is the number of minimal cut sets of the graph $G$ and obtain an improved upper bound for the regularity in terms of the number of maximal cliques and pendant vertices of $G$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
