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Generalized Cayley graphs

Dragan Marušič, Raffaele Scapellato, Norma Zagaglia Salvi
1992 Discrete Mathematics  
Zf B(X) is connected, then B(X) is a Cayley graph if and only if X is a generalized Cayley graph, Proof.  ...  We shall now introduce the concept of generalized Cayley graphs.  ... 
doi:10.1016/0012-365x(92)90121-u fatcat:ic7xdg4qrra23jdmzkf23pmkye

Isomorphisms of generalized Cayley graphs

Xu Yang, Weijun Liu, Lihua Feng
2018 Ars Mathematica Contemporanea  
In this paper, we investigate the isomorphism problems of the generalized Cayley graphs, which are generalizations of the traditional Cayley graphs.  ...  We find that there are two types of natural isomorphisms for the generalized Cayley graphs.  ...  Thus GC(G, {1}, α) is a generalized Cayley graph of G. Also, GC(G, {2}, 1) is a generalized Cayley graph of G.  ... 
doi:10.26493/1855-3974.1345.ae6 fatcat:dajpugqm45cqtlt4imyylm4rxq

Cellular automata over generalized Cayley graphs

PABLO ARRIGHI, SIMON MARTIEL, VINCENT NESME
2017 Mathematical Structures in Computer Science  
We propose a notion of graphs, which preserves or generalizes these features. Whereas Cayley graphs are very regular, generalized Cayley graphs are arbitrary, although of a bounded degree.  ...  Cayley graphs have a number of useful features: the ability to graphically represent finitely generated group elements and their relations; to name all vertices relative to an origin; and the fact that  ...  Graphs as languages Generalized Cayley graphs.  ... 
doi:10.1017/s0960129517000044 fatcat:rrsg6zukbverfjnxyqan2ekbdu

Graph product of generalized Cayley graphs over polygroups

Dariush Heidari, Bijan Davvaz
2019 Algebraic structures and their applications  
In this paper, we introduce a suitable generalization of Cayley graphs that is defined over polygroups (GCP-graph) and give some examples and properties.  ...  Then, we mention a generalization of NEPS that contains some known graph operations and apply to GCP-graphs. Finally, we prove that the product of GCP-graphs is again a GCP-graph.  ...  Then we define the generalized Cayley graph GCP(P; S) as a simple graph with vertex set P and edge set Example 2. 2 . 2 The generalized Cayley graph of the polygroup P 2 = ⟨{1, 2, 3, 4}, •, 1, −1 ⟩ and  ... 
doi:10.29252/asta.6.1.49 fatcat:24xcpktffjeqjnhkfrkbmdfi5u

Asymptotic pressure on Cayley graphs of finitely generated semigroups [article]

Karl Petersen, Ibrahim Salama
2021 arXiv   pre-print
The vertices of the Cayley graph of a finitely generated semigroup form a set of sites which can be labeled by elements of a finite alphabet in a manner governed by a nonnegative real interaction matrix  ...  We prove that for shifts of finite type on generalized Fibonacci trees, under an added condition, the limit exists and is given by an infinite series.  ...  (If r = 0, so that R ij = 1 for all i, j, S is the free semigroup on k generators and its Cayley graph is the full k-tree.)  ... 
arXiv:2106.16246v4 fatcat:rzv4sp5xj5fa7pxegusdgwac34

Cayley graphs of finitely generated groups

Simon Thomas
2005 Proceedings of the American Mathematical Society  
There does not exist a Borel choice of generators for each finitely generated group which has the property that isomorphic groups are assigned isomorphic Cayley graphs.  ...  Of course, if G is an arbitrary finitely generated group, then G does not have a "canonical" finite generating set S; and if S, S ⊆ G 1 are different finite generating sets, then the Cayley graphs Cay(  ...  Our proof is based upon the observation that the classification problem for finitely generated groups is much more difficult than that for Cayley graphs.  ... 
doi:10.1090/s0002-9939-05-07987-6 fatcat:ulqefovl3bgrjcz3ozyltfai3e

CAYLEY GRAPHS OF GENERALIZED DIHEDRAL GROUPS [chapter]

H.S.M. COXETER, ROBERTO FRUCHT, DAVID L. POWERS
1981 Zero-Symmetric Graphs  
Also the Estrada index of Γ, which is defined in 2000 by Ernesto Estrada, is defined as In this paper, we compute the eigenvalues, energy and Estrada index of Cayley graphs on generalized dihedral groups  ...  As an application, we compute these items for honeycomb toroidal graphs and Cayley graphs on dihedral groups.  ...  Semi-Cayley graphs which are a generalization of Cayley graphs defined by Resmini and Jungnickel [12] , see also [3] .  ... 
doi:10.1016/b978-0-12-194580-0.50028-8 fatcat:vzpwb2sjyndzxk4g3jegv52uj4

Construction of Directed Strongly Regular Graphs as Generalized Cayley Graphs [article]

Rongquan Feng, Liwei Zeng
2014 arXiv   pre-print
In this paper, an infinite family of directed strongly regular graphs is constructed, as generalized Cayley graphs of cyclic groups.  ...  Directed strongly regular graphs were introduced by Duval in 1998 as one of the possible generalization of classical strongly regular graphs to the directed case.  ...  In section 2, we give the definition of generalized Cayley graphs and a necessary and sufficient condition that makes it easy to check whether a generalized Cayley graph is a DSRG.  ... 
arXiv:1410.1161v2 fatcat:jesqb5fb2zfo7lf7mwzcpigrqu

Automorphisms of Cayley graphs generated by transposition sets [article]

Ashwin Ganesan
2013 arXiv   pre-print
It is shown that the automorphism group of the Cayley graph of the permutation group H generated by S is the semidirect product R(H) (H,S), where R(H) is the right regular representation of H and (H,S)  ...  This result is a common generalization of previous results by Feng, Ganesan, Harary, Mirafzal, and Zhang and Huang.  ...  Thus, this Cayley graph Cay(S 4 , S) is not normal. In all of the papers mentioned above on Cayley graphs generated by transpositions, the set of transpositions S is assumed to generate S n .  ... 
arXiv:1303.5974v2 fatcat:v4fxosaknjctbby3zam4w7dkbq

Which finitely generated Abelian groups admit isomorphic Cayley graphs?

Clara Löh
2012 Geometriae Dedicata  
We show that Cayley graphs of finitely generated Abelian groups are rather rigid.  ...  As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion parts have the same cardinality.  ...  Cayley graphs.  ... 
doi:10.1007/s10711-012-9761-x fatcat:phelagp3brfaxkipvoakrnlyk4

A note on vertex-transitive non-Cayley graphs from Cayley graphs generated by involutions

Jana Tomanová
2010 Discrete Mathematics  
graphs from Cayley graphs generated by involutions.  ...  We show that the result of Watkins (1990) [19] on constructing vertex-transitive non-Cayley graphs from line graphs yields a simple method that produces infinite families of vertex-transitive non-Cayley  ...  Indirect methods try to identify structural properties of Cayley graphs which they do not share with vertex-transitive graphs in general.  ... 
doi:10.1016/j.disc.2009.07.022 fatcat:mdt5crix3ffy5hbtzigygzu7xu

Automorphism groups of Cayley graphs generated by connected transposition sets

Ashwin Ganesan
2013 Discrete Mathematics  
Let S be a set of transpositions that generates the symmetric group S_n, where n > 3. The transposition graph T(S) is defined to be the graph with vertex set {1,...  ...  We thus identify the existence of 4-cycles in the transposition graph as being an important factor in causing a potentially larger automorphism group of the Cayley graph.  ...  A special case of this Corollary whereby T (S) is a tree gives the result of Feng [3] that the automorphism group of a Cayley graph generated by a transposition tree is isomorphic to R(S n ) ⋊ Aut(S  ... 
doi:10.1016/j.disc.2013.07.013 fatcat:rkmrl5rsrvenpgfzb2npd5khte

Neighbourhood Graphs of Cayley Graphs for Finitely-generated Groups

Markus Neuhauser
2002 European journal of combinatorics (Print)  
In this short note the neighbourhood graph of a Cayley graph is considered. It has as nodes a symmetric generating set of a finitely-generated group Γ.  ...  Two nodes are connected by an edge if one is obtained from the other by multiplication on the right by one of the generators. Two necessary conditions on the graphs are shown.  ...  Let now S be a finite symmetric generating set of a finitely-generated group Γ. The neighbourhood graph (S, K) of 1 ∈ Γ of the Cayley graph of Γ with respect to S with edges K will be investigated.  ... 
doi:10.1006/eujc.2002.0596 fatcat:r4ojenmhnjhj5fsr37tfht26vi

Diameter of Cayley graphs of permutation groups generated by transposition trees [article]

Ashwin Ganesan
2011 arXiv   pre-print
Let Γ be a Cayley graph of the permutation group generated by a transposition tree T on n vertices.  ...  In an oft-cited paper Akers:Krishnamurthy:1989 (see also Hahn:Sabidussi:1997), it is shown that the diameter of the Cayley graph Γ is bounded as (Γ) <_π∈ S_nc(π)-n+∑_i=1^n _T(i,π(i)), where the maximization  ...  Let Γ be the Cayley graph generated by transposition tree T .  ... 
arXiv:1111.3114v3 fatcat:7bbu7j7cw5enbkgrewyc464cly

On a diameter bound for Cayley graphs generated by transposition trees [article]

Ashwin Ganesan
2011 arXiv   pre-print
Let Γ be a Cayley graph generated by a transposition tree. A natural problem is to understand how the properties of the Cayley graph depend on those of the underlying transposition tree.  ...  However, it can be verified that the diameter of the Cayley graph generated by T is 14.  ...  Thus, when we study the sharpness (or lack thereof) of the upper bounds, we assume throughout that T is a tree and Γ is the Cayley graph generated by a tree.  ... 
arXiv:1106.5353v3 fatcat:mjvvuqnr5vbexn5kvi2ayn26pe
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