scholarly journals Cayley Graphs over La-Groups and La-Polygroups

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Nabilah Abughazalah ◽  
Naveed Yaqoob ◽  
Asif Bashir
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Simple Graph ◽  
Simple Graphs ◽  
Cycle Graph

The purpose of this paper is the study of simple graphs that are generalized Cayley graphs over LA-polygroups GCLAP − graphs . In this regard, we construct two new extensions for building LA-polygroups. Then, we define Cayley graph over LA-group and GCLAP-graph. Further, we investigate a few properties of them to show that each simple graph of order three, four, and five (except cycle graph of order five which may or may not be a GCLAP-graph) is a GCLAP-graph and then we prove this result.

On the Diameter of Unitary Cayley Graphs of Rings

2016 ◽  
Vol 59 (3) ◽  
pp. 652-660
Author(s):  
Huadong Su
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Simple Graph ◽  
Unitary Cayley Graph

AbstractThe unitary Cayley graph of a ringR, denoted Γ(R), is the simple graph defined on all elements ofR, and where two verticesxandyare adjacent if and only ifx−yis a unit inR. The largest distance between all pairs of vertices of a graphGis called the diameter ofGand is denoted by diam(G). It is proved that for each integern≥ 1, there exists a ringRsuch that diam(Γ(R)) =n. We also show that diam(Γ(R)) ∊ {1, 2, 3,∞} for a ringRwithR/J(R) self-injective and classify all those rings with diam(Γ(R)) = 1, 2, 3, and ∞, respectively.

A characterization of rings whose unitary Cayley graphs are planar

2018 ◽  
Vol 17 (09) ◽  
pp. 1850178 ◽  
Author(s):  
Huadong Su ◽  
Yiqiang Zhou
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Simple Graph ◽  
Vertex Set ◽  
Unitary Cayley Graph

Let [Formula: see text] be a ring with identity. The unitary Cayley graph of [Formula: see text] is the simple graph with vertex set [Formula: see text], where two distinct vertices [Formula: see text] and [Formula: see text] are linked by an edge if and only if [Formula: see text] is a unit of [Formula: see text]. A graph is said to be planar if it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In this paper, we completely characterize the rings whose unitary Cayley graphs are planar.

scholarly journals Cayley Graphs Without a Bounded Eigenbasis

2020 ◽  
Author(s):  
Ashwin Sah ◽  
Mehtaab Sawhney ◽  
Yufei Zhao
Keyword(s):  
Cayley Graph ◽  
Orthonormal Basis ◽  
Cayley Graphs ◽  
The Other ◽  
Transitive Graph ◽  
Classic Result ◽  
Other Hand ◽  
Small Set ◽  
Vertex Transitive ◽  
Vertex Transitive Graph

Abstract Does every $n$-vertex Cayley graph have an orthonormal eigenbasis all of whose coordinates are $O(1/\sqrt{n})$? While the answer is yes for abelian groups, we show that it is no in general. On the other hand, we show that every $n$-vertex Cayley graph (and more generally, vertex-transitive graph) has an orthonormal basis whose coordinates are all $O(\sqrt{\log n / n})$, and that this bound is nearly best possible. Our investigation is motivated by a question of Assaf Naor, who proved that random abelian Cayley graphs are small-set expanders, extending a classic result of Alon–Roichman. His proof relies on the existence of a bounded eigenbasis for abelian Cayley graphs, which we now know cannot hold for general groups. On the other hand, we navigate around this obstruction and extend Naor’s result to nonabelian groups.

scholarly journals Normal Edge-Transitive Cayley Graphs of the Group U6n

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
A. Assari ◽  
F. Sheikhmiri
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
The Right ◽  
Edge Transitive

A Cayley graph of a group G is called normal edge-transitive if the normalizer of the right representation of the group in the automorphism of the Cayley graph acts transitively on the set of edges of the graph. In this paper, we determine all connected normal edge-transitive Cayley graphs of the group U6n.

scholarly journals Approximating Cayley Diagrams Versus Cayley Graphs

2012 ◽  
Vol 21 (4) ◽  
pp. 635-641
Author(s):  
ÁDÁM TIMÁR
Keyword(s):  
Cayley Graph ◽  
Hamiltonian Cycle ◽  
Cayley Graphs ◽  
The Other ◽  
Finite Graphs ◽  
Similar Construction ◽  
Graph Property

We construct a sequence of finite graphs that weakly converge to a Cayley graph, but there is no labelling of the edges that would converge to the corresponding Cayley diagram. A similar construction is used to give graph sequences that converge to the same limit, and such that a Hamiltonian cycle in one of them has a limit that is not approximable by any subgraph of the other. We give an example where this holds, but convergence is meant in a stronger sense. This is related to whether having a Hamiltonian cycle is a testable graph property.

Some properties of subgroup complementary addition Cayley graphs on abelian groups

2021 ◽  
Author(s):  
Naveen Palanivel ◽  
Chithra A. Velu
Keyword(s):  
Cayley Graph ◽  
Abelian Groups ◽  
Cayley Graphs ◽  
Graph Invariants

In this paper, we introduce subgroup complementary addition Cayley graph [Formula: see text] and compute its graph invariants. Also, we prove that [Formula: see text] if and only if [Formula: see text] for all [Formula: see text] where [Formula: see text].

scholarly journals Cubic arc-transitive Cayley graphs on Frobenius groups

2018 ◽  
Vol 17 (07) ◽  
pp. 1850126 ◽  
Author(s):  
Hailin Liu ◽  
Lei Wang
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Frobenius Groups

A Cayley graph [Formula: see text] is called arc-transitive if its automorphism group [Formula: see text] is transitive on the set of arcs in [Formula: see text]. In this paper, we give a characterization of cubic arc-transitive Cayley graphs on a class of Frobenius groups.

On tetravalent s-regular Cayley graphs

2017 ◽  
Vol 16 (10) ◽  
pp. 1750195 ◽  
Author(s):  
Jing Jian Li ◽  
Bo Ling ◽  
Jicheng Ma
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Normal Cover

A Cayley graph [Formula: see text] is said to be core-free if [Formula: see text] is core-free in some [Formula: see text] for [Formula: see text]. A graph [Formula: see text] is called [Formula: see text]-regular if [Formula: see text] acts regularly on its [Formula: see text]-arcs. It is shown in this paper that if [Formula: see text], then there exist no core-free tetravalent [Formula: see text]-regular Cayley graphs; and for [Formula: see text], every tetravalent [Formula: see text]-regular Cayley graph is a normal cover of one of the three known core-free graphs. In particular, a characterization of tetravalent [Formula: see text]-regular Cayley graphs is given.

scholarly journals Cayley Graphs with an Infinite Heesch Number

10.37236/5114 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Azer Akhmedov
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs

We construct a 2-generated group $\Gamma $ such that its Cayley graph possesses finite connected subsets with arbitrarily large finite Heesch number. Thus we obtain an example of a Cayley graph with an infinite Heesch number.

scholarly journals Integral Cayley Graphs over Abelian Groups

10.37236/353 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Walter Klotz ◽  
Torsten Sander
Keyword(s):  
Finite Group ◽  
Abelian Group ◽  
Boolean Algebra ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Additive Group ◽  
Cyclic Groups ◽  
Vertex Set ◽  
A Finite Group ◽  
Gamma S

Let $\Gamma$ be a finite, additive group, $S \subseteq \Gamma, 0\notin S, -S=\{-s: s\in S\}=S$. The undirected Cayley graph Cay$(\Gamma,S)$ has vertex set $\Gamma$ and edge set $\{\{a,b\}: a,b\in \Gamma$, $a-b \in S\}$. A graph is called integral, if all of its eigenvalues are integers. For an abelian group $\Gamma$ we show that Cay$(\Gamma,S)$ is integral, if $S$ belongs to the Boolean algebra $B(\Gamma)$ generated by the subgroups of $\Gamma$. The converse is proven for cyclic groups. A finite group $\Gamma$ is called Cayley integral, if every undirected Cayley graph over $\Gamma$ is integral. We determine all abelian Cayley integral groups.

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