scholarly journals A Note on the Critical Group of a Line Graph

10.37236/611 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
David Perkinson ◽  
Nick Salter ◽  
Tianyuan Xu
Keyword(s):  
Directed Graph ◽  
Line Graph ◽  
Critical Group ◽  
Directed Line

This note answers a question posed by Levine. The main result shows that under certain circumstances a critical group of a directed graph is the quotient of a critical group of its directed line graph.

scholarly journals An Algebraic Analogue of a Formula of Knuth

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Lionel Levine
Keyword(s):  
Directed Graph ◽  
Spanning Trees ◽  
Line Graph ◽  
Torsion Subgroup ◽  
Directed Line ◽  
De Bruijn Graphs ◽  
International Audience ◽  
De Bruijn ◽  
Kautz Graphs ◽  
Sandpile Group

International audience We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph $G$ and its directed line graph $\mathcal{L} G$. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at a fixed vertex. In the case when $G$ is regular of degree $k$, we show that the sandpile group of $G$ is isomorphic to the quotient of the sandpile group of $\mathcal{L} G$ by its $k$-torsion subgroup. As a corollary we compute the sandpile groups of two families of graphs widely studied in computer science, the de Bruijn graphs and Kautz graphs. Nous généralisons un théorème de Knuth qui relie les arbres couvrants dirigés d'un graphe orienté $G$ au graphe adjoint orienté $\mathcal{L} G$. On peut associer à tout graphe orienté un groupe abélien appelé groupe du tas de sable, et dont l'ordre est le nombre d'arbres couvrants dirigés enracinés en un sommet fixé. Lorsque $G$ est régulier de degré $k$, nous montrons que le groupe du tas de sable de $G$ est isomorphe au quotient du groupe du tas de sable de $\mathcal{L} G$ par son sous-groupe de $k$-torsion. Comme corollaire, nous déterminons les groupes de tas de sable de deux familles de graphes étudiées en informatique: les graphes de de Bruijn et les graphes de Kautz.

scholarly journals SIFAT KEPRIMAAN MODUL SEDERHANA CHEN UNTUK GRAF 𝐴∞

2019 ◽  
Vol 12 (2) ◽  
Author(s):  
Risnawita Risnawita ◽  
Irawati Irawati ◽  
Intan Muchtadi Alamsyah
Keyword(s):  
Directed Graph ◽  
Infinite Sequence ◽  
Line Graph ◽  
Simple Module ◽  
Path Algebra ◽  
Leavitt Path Algebra ◽  
Directed Line ◽  
Simple Modules ◽  
Infinite Path ◽  
Prime Module

Let 𝐾𝐾 be a field, 𝐸𝐸 is a directed graph. Let 𝐴𝐴~ is a directed line graph. Suppose that 𝑉𝑉[𝑝𝑝] is a class of Chen simple module for the Leavitt path algebra (𝐿𝐿𝐾𝐾 (𝐸𝐸)), with [p] being equivalent classes containing an infinite path. An infinite path p is an infinite sequence from the sides of a graph. In this paper it will be shown that 𝑉𝑉[𝑝𝑝]is not a prime module of the Leavitt path algebra for graph 𝐴𝐴∞ .Keywords : Leavitt path algebra, Graph 𝐴𝐴~, Chen simple modules, Prime modules

scholarly journals The Critical Group of a Line Graph

2012 ◽  
Vol 16 (3) ◽  
pp. 449-488 ◽  
Author(s):  
Andrew Berget ◽  
Andrew Manion ◽  
Molly Maxwell ◽  
Aaron Potechin ◽  
Victor Reiner
Keyword(s):  
Line Graph ◽  
Critical Group

scholarly journals The Smith and Critical Groups of the Square Rook's Graph and its Complement

10.37236/5442 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Joshua E. Ducey ◽  
Jonathan Gerhard ◽  
Noah Watson
Keyword(s):  
Normal Form ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Line Graph ◽  
Cartesian Product ◽  
Laplacian Matrix ◽  
Critical Group ◽  
Critical Groups ◽  
Complete Graphs ◽  
Vertex Set

Let $R_{n}$ denote the graph with vertex set consisting of the squares of an $n \times n$ grid, with two squares of the grid adjacent when they lie in the same row or column.  This is the square rook's graph, and can also be thought of as the Cartesian product of two complete graphs of order $n$, or the line graph of the complete bipartite graph $K_{n,n}$.  In this paper we compute the Smith group and critical group of the graph $R_{n}$ and its complement.  This is equivalent to determining the Smith normal form of both the adjacency and Laplacian matrix of each of these graphs.  In doing so we verify a 1986 conjecture of Rushanan.

scholarly journals Superdiffusive quantum stochastic walk definable on arbitrary directed graph

2017 ◽  
Vol 17 (11&12) ◽  
pp. 973-986
Author(s):  
Krzysztof Domino ◽  
Adam Glos ◽  
Mateusz Ostaszewski
Keyword(s):  
Directed Graph ◽  
Line Graph ◽  
Global Environment ◽  
Environment Interaction ◽  
Graph Topology ◽  
New Model ◽  
Correction Scheme ◽  
Moral Graph

In this paper we define a quantum stochastic walk on arbitrary directed graph with super-diffusive propagation on a line graph. Our model is based on global environment interaction QSW, which is known to have ballistic propagation. However we discovered, that in this case additional amplitude transitions occur, hence graph topology is changed into moral graph. Because of that we call the effect a spontaneous moralization. We propose a general correction scheme, which is proved to remove unnecessary transition and thus to preserve the graph topology. In the end we numerically show, that superdiffusive propagation is preserved. Because of that our new model may be applied as effective evolution on arbitrary directed graph.

Sign in / Sign up

Export Citation Format

Share Document