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 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 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 $I$.

Tournaments With a Given Automorphism Group

1964 ◽  
Vol 16 ◽  
pp. 485-489 ◽  
Author(s):  
J. W. Moon
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Oriented Graph ◽  
Not Given ◽  
Oriented Graphs ◽  
Vertex Set ◽  
A Finite Group ◽  
Graph Form

The set of all adjacency-preserving automorphisms of the vertex set of a graph form a group which is called the (automorphism) group of the graph. In 1938 Frucht (2) showed that every finite group is isomorphic to the group of some graph. Since then Frucht, Izbicki, and Sabidussi have considered various other properties that a graph having a given group may possess. (For pertinent references and definitions not given here see Ore (4).) The object in this paper is to treat by similar methods a corresponding problem for a class of oriented graphs. It will be shown that a finite group is isomorphic to the group of some complete oriented graph if and only if it has an odd number of elements.

Random collision models in oriented graphs

1979 ◽  
Vol 16 (01) ◽  
pp. 36-44 ◽  
Author(s):  
Yoshiaki Itoh
Keyword(s):  
Oriented Graph ◽  
Dominant Type ◽  
Oriented Graphs ◽  
Collision Model ◽  
Dominance Relations ◽  
Different Types ◽  
The Stability ◽  
Collision Models

We investigate a random collision model for competition between types of individuals in a population. There are dominance relations defined for each pair of types such that if two individuals of different types collide then after the collision both are of the dominant type. These dominance relations are represented by an oriented graph, called a tournament. It is shown that tournaments having a particular form are relatively stable, while other tournaments are relatively unstable. A measure of the stability of the stable tournaments is given in the main theorem.

scholarly journals Ford-Fulkerson algorithm refinement for the cooperation effectiveness increase of intensive and low-density lines

2021 ◽  
Vol 2131 (3) ◽  
pp. 032008
Author(s):  
K E Kovalev ◽  
A V Novichikhin
Keyword(s):  
Oriented Graph ◽  
Maximum Flow ◽  
Minimum Cut ◽  
Low Density ◽  
Railway Network ◽  
Oriented Graphs ◽  
Clear View ◽  
Railway Line ◽  
As Graph

Abstract The article describes tools of the railway control on intensive and low-density lines which is directed on the effectiveness increase of low-density line functioning, for the solution of perspective tasks of the railway network functioning and development. For the too realization the oriented graph with the Ford-Fulkerson algorithm which allows determining the maximum flow and the minimum cut for non-oriented graphs. Firstly as values of graph tops inverse values of the station rating and as graph edges inverse values of the railway line class are accepted. The use of this approach allows determining the maximum flow in the system and provides the clear view of relations of transportation capacities of railway lines and stations.

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