Graph algebras and graph varieties

1990 ◽  
Vol 27 (4) ◽  
pp. 559-577 ◽  
Author(s):  
Reinhard P�schel
Keyword(s):  
Graph Algebras ◽  
Graph Varieties

scholarly journals Graph varieties containing Murskii's groupoid

1989 ◽  
Vol 39 (2) ◽  
pp. 265-276
Author(s):  
R. Pöschel
Keyword(s):  
Finitely Based ◽  
Graph Algebras ◽  
Undirected Graphs ◽  
Graph Algebra ◽  
Subvariety Lattice ◽  
As Graph ◽  
Graph Varieties ◽  
Nonfinitely Based

In this paper varieties are investigated which are generated by graph algebras of undirected graphs and—in most cases—contain Murskii's groupoid (that is the graph algebra of the graph with two adjacent vertices and one loop). Though these varieties are inherently nonfinitely based, they can be finitely based as graph varieties (finitely graph based) like, for example, the varitey generated by Murskii's groupoid. Many examples of nonfinitely based graph varities containing Murskii's groupoid are given, too. Moreover, the coatoms in the subvariety lattice of the graph variety of all undirected graphs are described. There are two coatoms and they are finitely graph based.

IDENTITIES IN BIREGULAR LEFTMOST GRAPH VARIETIES OF TYPE (2,0)

2009 ◽  
Vol 02 (01) ◽  
pp. 1-17
Author(s):  
Apinant Anantpiniwatna ◽  
Tiang Poomsa-Ard
Keyword(s):  
Universal Algebra ◽  
Directed Graphs ◽  
Graph Algebras ◽  
Universal Algebras ◽  
Graph Algebra ◽  
Basic Concepts ◽  
Multiple Edges ◽  
Graph Varieties

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type (2,0). We say that a graph G satisfies a term equation s ≈ t if the corresponding graph algebra A(G) satisfies s ≈ t. A class of graph algebras V is called a graph variety if V = ModgΣ where Σ is a subset of T(X) × T(X). A graph variety V' = ModgΣ' is called a biregular leftmost graph variety if Σ' is a set of biregular leftmost term equations. A term equation s ≈ t is called an identity in a variety V if G satisfies s ≈ t for all G ∈ V. In this paper we characterize identities in each biregular leftmost graph variety. For identities, varieties and other basic concepts of universal algebra see e.g. [1].

Graph variety generated by linear terms

2019 ◽  
Vol 12 (05) ◽  
pp. 1950074
Author(s):  
C. Manyuen ◽  
P. Jampachon ◽  
T. Poomsa-ard
Keyword(s):  
Directed Graphs ◽  
Linear Term ◽  
Graph Algebras ◽  
Universal Algebras ◽  
Type Formula ◽  
Graph Algebra ◽  
Multiple Edges ◽  
Graph Varieties

Graph algebras establish a connection between directed graphs without multiple edges and special universal algebras of type [Formula: see text]. We say that a graph [Formula: see text] satisfies a term equation [Formula: see text] if the corresponding graph algebra [Formula: see text] satisfies [Formula: see text]. The set of all term equations [Formula: see text], which the graph [Formula: see text] satisfies, is denoted by [Formula: see text]. The class of all graph algebras satisfy all term equations in [Formula: see text] is called the graph variety generated by [Formula: see text] denoted by [Formula: see text]. A term is called a linear term if each variable which occurs in the term, occurs only once. A term equation [Formula: see text] is called a linear term equation if [Formula: see text] and [Formula: see text] are linear terms. This paper is devoted to a thorough investigation of graph varieties defined by linear term equations. In particular, we give a complete description of rooted graphs generating a graph variety described by linear term equations.

scholarly journals Associative spectra of graph algebras I

2021 ◽  
Author(s):  
Erkko Lehtonen ◽  
Tamás Waldhauser
Keyword(s):  
Catalan Numbers ◽  
Graph Algebras ◽  
Undirected Graphs ◽  
Powers Of 2

AbstractAssociative spectra of graph algebras are examined with the help of homomorphisms of DFS trees. Undirected graphs are classified according to the associative spectra of their graph algebras; there are only three distinct possibilities: constant 1, powers of 2, and Catalan numbers. Associative and antiassociative digraphs are described, and associative spectra are determined for certain families of digraphs, such as paths, cycles, and graphs on two vertices.

scholarly journals Lie bialgebra structures on 2-step nilpotent graph algebras

2018 ◽  
Vol 505 ◽  
pp. 70-91 ◽  
Author(s):  
Marco A. Farinati ◽  
Alejandra Patricia Jancsa
Keyword(s):  
Graph Algebras ◽  
Lie Bialgebra

Graph algebras and widths of graphs

2012 ◽  
pp. 80-187
Author(s):  
Bruno Courcelle ◽  
Joost Engelfriet
Keyword(s):  
Graph Algebras

scholarly journals GROUP ACTIONS AND COVERINGS OF BRAUER GRAPH ALGEBRAS

2013 ◽  
Vol 56 (2) ◽  
pp. 439-464 ◽  
Author(s):  
EDWARD L. GREEN ◽  
SIBYLLE SCHROLL ◽  
NICOLE SNASHALL
Keyword(s):  
Group Actions ◽  
Multiplicity Function ◽  
Graph Algebras ◽  
Multiple Edges

AbstractWe develop a theory of group actions and coverings on Brauer graphs that parallels the theory of group actions and coverings of algebras. In particular, we show that any Brauer graph can be covered by a tower of coverings of Brauer graphs such that the topmost covering has multiplicity function identically one, no loops, and no multiple edges. Furthermore, we classify the coverings of Brauer graph algebras that are again Brauer graph algebras.

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