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.

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 Cubic Vertex-Transitive Non-Cayley Graphs of Order $8p$

10.37236/2087 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Jin-Xin Zhou ◽  
Yan-Quan Feng
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Infinite Family ◽  
The Other ◽  
Transitive Graph ◽  
Vertex Transitive ◽  
Unique Graph ◽  
Vertex Transitive Graph

A graph is vertex-transitive if its automorphism group acts transitively on its vertices. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper, the cubic vertex-transitive non-Cayley graphs of order $8p$ are classified for each prime $p$. It follows from this classification that there are two sporadic and two infinite families of such graphs, of which the sporadic ones have order $56$,  one infinite family exists for every prime $p>3$ and the other family exists if and only if $p\equiv 1\mod 4$. For each family there is a unique graph for a given order.

scholarly journals Cayley Graphs of Order 27p Are Hamiltonian

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Ebrahim Ghaderpour ◽  
Dave Witte Morris
Keyword(s):  
Finite Group ◽  
Cayley Graph ◽  
Hamiltonian Cycle ◽  
Cayley Graphs ◽  
Generating Set ◽  
A Finite Group

Suppose that G is a finite group, such that |G|=27p, where p is prime. We show that if S is any generating set of G, then there is a Hamiltonian cycle in the corresponding Cayley graph Cay (G;S).

A Cayley graph for F2 × F2 which is not minimally almost convex

2021 ◽  
pp. 1-12
Author(s):  
Andrew Elvey Price
Keyword(s):  
Cayley Graph ◽  
The Other ◽  
Fellow Traveler ◽  
Generating Set ◽  
Other Hand ◽  
Almost Convex ◽  
Graph Property ◽  
Almost Convexity

We give an example of a Cayley graph [Formula: see text] for the group [Formula: see text] which is not minimally almost convex (MAC). On the other hand, the standard Cayley graph for [Formula: see text] does satisfy the falsification by fellow traveler property (FFTP), which is strictly stronger. As a result, any Cayley graph property [Formula: see text] lying between FFTP and MAC (i.e., [Formula: see text]) is dependent on the generating set. This includes the well-known properties FFTP and almost convexity, which were already known to depend on the generating set as well as Poénaru’s condition [Formula: see text] and the basepoint loop shortening property (LSP) for which dependence on the generating set was previously unknown. We also show that the Cayley graph [Formula: see text] does not have the LSP, so this property also depends on the generating set.

Hamiltonicity of a class of toroidal graphs

2020 ◽  
Vol 70 (2) ◽  
pp. 497-503
Author(s):  
Dipendu Maity ◽  
Ashish Kumar Upadhyay
Keyword(s):  
Connected Graph ◽  
Hamiltonian Cycle ◽  
Partial Solution ◽  
The Other ◽  
Hamiltonian Cycles ◽  
The Face ◽  
Toroidal Graphs

Abstract If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. There are eleven types of semi-equivelar maps on the torus. In 1972 Altshuler has presented a study of Hamiltonian cycles in semi-equivelar maps of three types {36}, {44} and {63} on the torus. In this article we study Hamiltonicity of semi-equivelar maps of the other eight types {33, 42}, {32, 41, 31, 41}, {31, 61, 31, 61}, {34, 61}, {41, 82}, {31, 122}, {41, 61, 121} and {31, 41, 61, 41} on the torus. This gives a partial solution to the well known Conjecture that every 4-connected graph on the torus has a Hamiltonian cycle.

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.

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 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.

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.

Sign in / Sign up

Export Citation Format

Share Document