scholarly journals Edge ideals of oriented graphs

2019 ◽  
Vol 29 (03) ◽  
pp. 535-559 ◽  
Author(s):  
Huy Tài Hà ◽  
Kuei-Nuan Lin ◽  
Susan Morey ◽  
Enrique Reyes ◽  
Rafael H. Villarreal
Keyword(s):  
Bipartite Graph ◽  
Natural Condition ◽  
Perfect Matching ◽  
Undirected Graph ◽  
Oriented Graph ◽  
Edge Ideal ◽  
Oriented Graphs ◽  
Edge Ideals ◽  
Underlying Graph ◽  
Equivalent Conditions

Let [Formula: see text] be a weighted oriented graph and let [Formula: see text] be its edge ideal. Under a natural condition that the underlying (undirected) graph of [Formula: see text] contains a perfect matching consisting of leaves, we provide several equivalent conditions for the Cohen–Macaulayness of [Formula: see text]. We also completely characterize the Cohen–Macaulayness of [Formula: see text] when the underlying graph of [Formula: see text] is a bipartite graph. When [Formula: see text] fails to be Cohen–Macaulay, we give an instance where [Formula: see text] is shown to be sequentially Cohen–Macaulay.

scholarly journals Skew Spectra of Oriented Graphs

10.37236/270 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Bryan Shader ◽  
Wasin So
Keyword(s):  
Directed Graph ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Oriented Graph ◽  
Oriented Graphs ◽  
Underlying Graph ◽  
Adjacency Spectrum ◽  
The Relationship ◽  
Skew Adjacency Matrix

An oriented graph $G^{\sigma}$ is a simple undirected graph $G$ with an orientation $\sigma$, which assigns to each edge a direction so that $G^{\sigma}$ becomes a directed graph. $G$ is called the underlying graph of $G^{\sigma}$, and we denote by $Sp(G)$ the adjacency spectrum of $G$. Skew-adjacency matrix $S( G^{\sigma} )$ of $G^{\sigma}$ is introduced, and its spectrum $Sp_S( G^{\sigma} )$ is called the skew-spectrum of $G^{\sigma}$. The relationship between $Sp_S( G^{\sigma} )$ and $Sp(G)$ is studied. In particular, we prove that (i) $Sp_S( G^{\sigma} ) = {\bf i} Sp(G)$ for some orientation $\sigma$ if and only if $G$ is bipartite, (ii) $Sp_S(G^{\sigma}) = {\bf i} Sp(G)$ for any orientation $\sigma$ if and only if $G$ is a forest, where ${\bf i}=\sqrt{-1}$.

scholarly journals Symbolic powers in weighted oriented graphs

2021 ◽  
pp. 1-17
Author(s):  
Mousumi Mandal ◽  
Dipak Kumar Pradhan
Keyword(s):  
Oriented Graph ◽  
Explicit Description ◽  
Oriented Graphs ◽  
Edge Ideals ◽  
Underlying Graph ◽  
Symbolic Powers ◽  
Odd Cycle ◽  
Waldschmidt Constant

Let [Formula: see text] be a weighted oriented graph with the underlying graph [Formula: see text] when vertices with non-trivial weights are sinks and [Formula: see text] be the edge ideals corresponding to [Formula: see text] and [Formula: see text] respectively. We give an explicit description of the symbolic powers of [Formula: see text] using the concept of strong vertex covers. We show that the ordinary and symbolic powers of [Formula: see text] and [Formula: see text] behave in a similar way. We provide a description for symbolic powers and Waldschmidt constant of [Formula: see text] for certain classes of weighted oriented graphs. When [Formula: see text] is a weighted oriented odd cycle, we compute [Formula: see text] and prove [Formula: see text] and show that equality holds when there is only one vertex with non-trivial weight.

scholarly journals Characteristic Polynomials of Skew-Adjacency Matrices of Oriented Graphs

10.37236/643 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Yaoping Hou ◽  
Tiangang Lei
Keyword(s):  
Characteristic Polynomial ◽  
Directed Graph ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Oriented Graph ◽  
Bipartite Graphs ◽  
Characteristic Polynomials ◽  
Oriented Graphs ◽  
Underlying Graph ◽  
Skew Adjacency Matrix

An oriented graph $\overleftarrow{G}$ is a simple undirected graph $G$ with an orientation, which assigns to each edge a direction so that $\overleftarrow{G}$ becomes a directed graph. $G$ is called the underlying graph of $\overleftarrow{G}$ and we denote by $S(\overleftarrow{G})$ the skew-adjacency matrix of $\overleftarrow{G}$ and its spectrum $Sp(\overleftarrow{G})$ is called the skew-spectrum of $\overleftarrow{G}$. In this paper, the coefficients of the characteristic polynomial of the skew-adjacency matrix $S(\overleftarrow{G}) $ are given in terms of $\overleftarrow{G}$ and as its applications, new combinatorial proofs of known results are obtained and new families of oriented bipartite graphs $\overleftarrow{G}$ with $Sp(\overleftarrow{G})={\bf i} Sp(G) $ are given.

scholarly journals Monomial Ideals of Weighted Oriented Graphs

10.37236/8479 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Yuriko Pitones ◽  
Enrique Reyes ◽  
Jonathan Toledo
Keyword(s):  
Oriented Graph ◽  
Monomial Ideals ◽  
Associated Primes ◽  
Edge Ideal ◽  
Combinatorial Characterization ◽  
Oriented Graphs ◽  
Irreducible Decomposition ◽  
Underlying Graph

Let $I=I(D)$ be the edge ideal of a weighted oriented graph $D$ whose underlying graph is $G$. 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 $G$ is bipartite, $G$ is a graph with whiskers or $G$ is a cycle. Finally, we study the Cohen–Macaulay property of $I$.

scholarly journals Betti Numbers of Weighted Oriented Graphs

10.37236/9887 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Beata Casiday ◽  
Selvi Kara
Keyword(s):  
Betti Number ◽  
Betti Numbers ◽  
Oriented Graph ◽  
Projective Dimension ◽  
Edge Ideal ◽  
Oriented Graphs ◽  
Edge Ideals ◽  
Mapping Cone ◽  
Recursive Formulas ◽  
Extremal Betti Number

Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ be its edge ideal. In this paper, we investigate the Betti numbers of $I(\mathcal{D})$ via upper-Koszul simplicial complexes, Betti splittings and the mapping cone construction. In particular, we provide recursive formulas for the Betti numbers of edge ideals of several classes of weighted oriented graphs. We also identify classes of weighted oriented graphs whose edge ideals have a unique extremal Betti number which allows us to compute the regularity and projective dimension for the identified classes. Furthermore, we characterize the structure of a weighted oriented graph $\mathcal{D}$ on $n$ vertices such that $\textrm{pdim } (R/I(\mathcal{D}))=n$ where $R=k[x_1,\ldots, x_n]$.

The powers of unmixed bipartite edge ideals

2019 ◽  
Vol 18 (11) ◽  
pp. 1950209 ◽  
Author(s):  
Arindam Banerjee ◽  
Vivek Mukundan
Keyword(s):  
Bipartite Graph ◽  
Future Research ◽  
Edge Ideal ◽  
Edge Ideals ◽  
Powers Of Ideals

In this paper, we study the depth and the Castelnuovo–Mumford regularity of the powers of edge ideals which are unmixed and whose underlying graphs are bipartite. In particular, we prove that the depth of the powers of the edge ideal stabilizes when the exponent is the same as half the number of vertices in the underlying connected bipartite graph. We also define the idea of “drop” in the sequence of depth of powers of ideals. Further, we show that the sequence of depth of the powers of such edge ideals may have any number of “drops”. In the process of proving these results we put forward some interesting examples and some questions for future research. As for regularity, we establish a formula for the regularity of the powers of such edge ideals in terms of the regularity of the edge ideal itself.

scholarly journals Aspherical word labeled oriented graphs and cyclically presented groups

2015 ◽  
Vol 24 (05) ◽  
pp. 1550025 ◽  
Author(s):  
Jens Harlander ◽  
Stephan Rosebrock
Keyword(s):  
Oriented Graph ◽  
Central Importance ◽  
Oriented Edge ◽  
Oriented Graphs ◽  
Underlying Graph

A word labeled oriented graph (WLOG) is an oriented graph [Formula: see text] on vertices X = {x1,…,xn}, where each oriented edge is labeled by a word in X±1. WLOGs give rise to presentations which generalize Wirtinger presentations of knots. WLOG presentations, where the underlying graph is a tree, are of central importance in view of Whitehead's Asphericity Conjecture. We present a class of aspherical word labeled oriented graphs. This class can be used to produce highly non-injective aspherical labeled oriented trees and also aspherical cyclically presented groups.

scholarly journals On the Skew Spectra of Cartesian Products of Graphs

10.37236/2864 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Cui Denglan ◽  
Hou Yaoping
Keyword(s):  
Directed Graph ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Oriented Graph ◽  
Bipartite Graphs ◽  
Cartesian Products ◽  
Product Graph ◽  
Underlying Graph ◽  
Skew Energy ◽  
Skew Adjacency Matrix

An oriented graph ${G^{\sigma}}$ is a simple undirected graph $G$ with an orientation, which assigns to each edge of $G$ a direction so that ${G^{\sigma}}$ becomes a directed graph. $G$ is called the underlying graph of ${G^{\sigma}}$ and we denote by $S({G^{\sigma}})$ the skew-adjacency matrix of ${G^{\sigma}}$ and its spectrum $Sp({G^{\sigma}})$ is called the skew-spectrum of ${G^{\sigma}}$. In this paper, the skew spectra of two orientations of the Cartesian products are discussed, as applications, new families of oriented bipartite graphs ${G^{\sigma}}$ with $Sp({G^{\sigma}})={\bf i} Sp(G)$ are given and the orientation of a product graph with maximum skew energy is obtained.

scholarly journals Unmixed and Cohen–Macaulay Weighted Oriented Kőnig Graphs

2021 ◽  
Vol 58 (3) ◽  
pp. 276-292
Author(s):  
Yuriko Pitones ◽  
Enrique Reyes ◽  
Rafael H. Villarreal
Keyword(s):  
Oriented Graph ◽  
Edge Ideal ◽  
Underlying Graph

Let D be a weighted oriented graph, whose underlying graph is G, and let I (D) be its edge ideal. If G has no 3-, 5-, or 7-cycles, or G is Kőnig, we characterize when I (D) is unmixed. If G has no 3- or 5-cycles, or G is Kőnig, we characterize when I (D) is Cohen–Macaulay. We prove that I (D) is unmixed if and only if I (D) is Cohen–Macaulay when G has girth greater than 7 or G is Kőnig and has no 4-cycles.

scholarly journals Stanley depth of edge ideals

2012 ◽  
Vol 49 (4) ◽  
pp. 501-508 ◽  
Author(s):  
Muhammad Ishaq ◽  
Muhammad Qureshi
Keyword(s):  
Complete Graph ◽  
Upper Bound ◽  
Edge Ideal ◽  
Stanley Depth ◽  
Edge Ideals

We give an upper bound for the Stanley depth of the edge ideal I of a k-partite complete graph and show that Stanley’s conjecture holds for I. Also we give an upper bound for the Stanley depth of the edge ideal of a s-uniform complete bipartite hypergraph.

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