On the relationship between the skew-rank of an oriented graph and the rank of its underlying graph

2018 ◽  
Vol 554 ◽  
pp. 205-223 ◽  
Author(s):  
Wenjun Luo ◽  
Jing Huang ◽  
Shuchao Li
Keyword(s):  
Oriented Graph ◽  
Underlying Graph ◽  
The Relationship

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 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 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 The Square of a Directed Graph

2019 ◽  
Vol 8 (4) ◽  
pp. 8331-8335
Keyword(s):  
Directed Graph ◽  
Oriented Graph ◽  
Directed Cycle ◽  
Directed Path ◽  
The Relationship ◽  
Special Case ◽  
Cycle Graph

The square of an oriented graph is an oriented graph such that if and only if for some , both and exist. According to the square of oriented graph conjecture (SOGC), there exists a vertex such that . It is a special case of a more general Seymour’s second neighborhood conjecture (SSNC) which states for every oriented graph , there exists a vertex such that . In this study, the methods to square a directed graph and verify its correctness were introduced. Moreover, some lemmas were introduced to prove some classes of oriented graph including regular oriented graph, directed cycle graph and directed path graphs are satisfied the SOGC. Besides that, the relationship between SOGC and SSNC are also proved in this study. As a result, the verification of the SOGC in turn implies partial results for SSNC.

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 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.

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