scholarly journals The talented monoid of a directed graph with applications to graph algebras

2021 ◽  
Author(s):  
Luiz Gustavo Cordeiro ◽  
Daniel Gonçalves ◽  
Roozbeh Hazrat
Keyword(s):  
Directed Graph ◽  
Graph Algebras

scholarly journals Faithful Representations of Graph Algebras via Branching Systems

2016 ◽  
Vol 59 (01) ◽  
pp. 95-103 ◽  
Author(s):  
Daniel Gonçalves ◽  
Hui Li ◽  
Danilo Royer
Keyword(s):  
Large Class ◽  
Uniqueness Theorem ◽  
Directed Graph ◽  
Directed Graphs ◽  
Graph Algebras

Abstract We continue to investigate branching systems of directed graphs and their connections with graph algebras. We give a suõcient condition under which the representation induced from a branching systemof a directed graph is faithful and construct a large class of branching systems that satisfy this condition. We ûnish the paper by providing a proof of the converse of the Cuntz–Krieger uniqueness theorem for graph algebras by means of branching systems.

scholarly journals A class of corners of a Leavitt path algebra

2019 ◽  
Vol 2 (4) ◽  
pp. 75-81
Author(s):  
Deo Thanh Trinh
Keyword(s):  
Directed Graph ◽  
Finite Subset ◽  
Path Algebra ◽  
Graph Algebras ◽  
Leavitt Path Algebra ◽  
C Algebras ◽  
Path Algebras ◽  
Leavitt Path Algebras

Let E be a directed graph, K a field and LK(E) the Leavitt path algebra of E over K. The goal of this paper is to describe the structure of a class of corners of Leavitt path algebras LK(E). The motivation of this work comes from the paper “Corners of Graph Algebras” of Tyrone Crisp in which such corners of graph C*-algebras were investigated completely. Using the same ideas of Tyrone Crisp, we will show that for any finite subset X of vertices in a directed graph E such that the hereditary subset HE(X) generated by X is finite, the corner ( ) ( )( )     K v X v X v L E v is isomorphic to the Leavitt path algebra LK(EX) of some graph EX. We also provide a way how to construct this graph EX.

scholarly journals NEST REPRESENTATIONS OF DIRECTED GRAPH ALGEBRAS

2006 ◽  
Vol 92 (3) ◽  
pp. 762-790 ◽  
Author(s):  
KENNETH R. DAVIDSON ◽  
ELIAS KATSOULIS
Keyword(s):  
Irreducible Representation ◽  
Directed Graph ◽  
Tensor Algebra ◽  
Graph Algebras ◽  
Finite Dimensional ◽  
Comprehensive Study

This paper is a comprehensive study of the nest representations for the free semigroupoid algebra ${\mathfrak{L}}_G$ of a countable directed graph $G$ as well as its norm-closed counterpart, the tensor algebra ${\mathcal{T}}^{+}(G)$.We prove that the finite-dimensional nest representations separate the points in ${\mathfrak{L}}_G$, and a fortiori, in ${\mathcal{T}}^{+}(G)$. The irreducible finite-dimensional representations separate the points in ${\mathfrak{L}}_G$ if and only if $G$ is transitive in components (which is equivalent to being semisimple). Also the upper triangular nest representations separate points if and only if for every vertex $x \in {\mathcal{T}}(G)$ supporting a cycle, $x$ also supports at least one loop edge.We also study faithful nest representations. We prove that ${\mathfrak{L}}_G$ (or ${\mathcal{T}}^{+}(G)$) admits a faithful irreducible representation if and only if $G$ is strongly transitive as a directed graph. More generally, we obtain a condition on $G$ which is equivalent to the existence of a faithful nest representation. We also give a condition that determines the existence of a faithful nest representation for a maximal type ${\mathbb{N}}$ nest.

A Graph-Theoretic Approach to Comparing and Integrating Genetic, Physical and Sequence-Based Maps

Genetics ◽  
2003 ◽  
Vol 165 (4) ◽  
pp. 2235-2247
Author(s):  
Immanuel V Yap ◽  
David Schneider ◽  
Jon Kleinberg ◽  
David Matthews ◽  
Samuel Cartinhour ◽  
...  
Keyword(s):  
Directed Graph ◽  
Source Material ◽  
Complete Picture ◽  
Theoretic Approach ◽  
Research Groups ◽  
Genome Coverage ◽  
Graph Theoretic ◽  
Novel Approach ◽  
Graph Theoretic Approach ◽  
Locus Order

AbstractFor many species, multiple maps are available, often constructed independently by different research groups using different sets of markers and different source material. Integration of these maps provides a higher density of markers and greater genome coverage than is possible using a single study. In this article, we describe a novel approach to comparing and integrating maps by using abstract graphs. A map is modeled as a directed graph in which nodes represent mapped markers and edges define the order of adjacent markers. Independently constructed graphs representing corresponding maps from different studies are merged on the basis of their common loci. Absence of a path between two nodes indicates that their order is undetermined. A cycle indicates inconsistency among the mapping studies with regard to the order of the loci involved. The integrated graph thus produced represents a complete picture of all of the mapping studies that comprise it, including all of the ambiguities and inconsistencies among them. The objective of this representation is to guide additional research aimed at interpreting these ambiguities and inconsistencies in locus order rather than presenting a “consensus order” that ignores these problems.

scholarly journals A Fault Identification Method in Distillation Process Based on Dynamic Mechanism Analysis and Signed Directed Graph

Processes ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 229
Author(s):  
Wende Tian ◽  
Shifa Zhang ◽  
Zhe Cui ◽  
Zijian Liu ◽  
Shaochen Wang ◽  
...  
Keyword(s):  
Directed Graph ◽  
Process Simulation ◽  
Expert Knowledge ◽  
Dynamic Mechanism ◽  
Fault Identification ◽  
Mechanism Analysis ◽  
Distillation Process ◽  
Complex Network Theory ◽  
Identification Method ◽  
Signed Directed Graph

Due to the complexity of materials and energy cycles, the distillation system has numerous working conditions difficult to troubleshoot in time. To address the problem, a novel DMA-SDG fault identification method that combines dynamic mechanism analysis based on process simulation and signed directed graph is proposed for the distillation process. Firstly, dynamic simulation is employed to build a mechanism model to provide the potential relationships between variables. Secondly, sensitivity analysis and dynamic mechanism analysis in process simulation are introduced to the SDG model to improve the completeness of this model based on expert knowledge. Finally, a quantitative analysis based on complex network theory is used to select the most important nodes in SDG model for identifying the severe malfunctions. The application of DMA-SDG method in a benzene-toluene-xylene (BTX) hydrogenation prefractionation system shows sound fault identification performance.

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