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 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 Semiregular elements in cubic vertex-transitive graphs and the restricted Burnside problem

2014 ◽  
Vol 157 (1) ◽  
pp. 45-61
Author(s):  
PABLO SPIGA
Keyword(s):  
Automorphism Group ◽  
Positive Solution ◽  
Cayley Graph ◽  
Burnside Problem ◽  
Transitive Graph ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Restricted Burnside Problem ◽  
Vertex Transitive Graph ◽  
Transitive Graphs

AbstractIn this paper, we prove that the maximal order of a semiregular element in the automorphism group of a cubic vertex-transitive graph Γ does not tend to infinity as the number of vertices of Γ tends to infinity. This gives a solution (in the negative) to a conjecture of Peter Cameron, John Sheehan and the author [4, conjecture 2].However, with an application of the positive solution of the restricted Burnside problem, we show that this conjecture holds true when Γ is either a Cayley graph or an arc-transitive graph.

scholarly journals Vertex-transitive graphs which are not Cayley graphs, I

1994 ◽  
Vol 56 (1) ◽  
pp. 53-63 ◽  
Author(s):  
Brendan D. McKay ◽  
Cheryl E. Praeger
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Petersen Graph ◽  
Fourth Power ◽  
Transitive Graph ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Vertex Transitive Graph ◽  
Transitive Graphs

AbstractThe Petersen graph on 10 vertices is the smallest example of a vertex-transitive graph which is not a Cayley graph. We consider the problem of determining the orders of such graphs. In this, the first of a series of papers, we present a sequence of constructions which solve the problem for many orders. In particular, such graphs exist for all orders divisible by a fourth power, and all even orders which are divisible by a square.

scholarly journals Cubic Edge-Transitive Bi-$p$-Metacirculants

10.37236/6417 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
Yan-Li Qin ◽  
Jin-Xin Zhou
Keyword(s):  
Automorphism Group ◽  
Cayley Graph ◽  
Cayley Graphs ◽  
Infinite Family ◽  
Group Of Automorphisms ◽  
Group A ◽  
Semisymmetric Graphs ◽  
Edge Transitive

A graph is said to be a bi-Cayley graph over a group $H$ if it admits $H$ as a group of automorphisms acting semiregularly on its vertices with two orbits. For a prime $p$, we call a bi-Cayley graph over a metacyclic $p$-group a bi-$p$-metacirculant. In this paper, the automorphism group of a connected cubic edge-transitive bi-$p$-metacirculant is characterized for an odd prime $p$, and the result reveals that a connected cubic edge-transitive bi-$p$-metacirculant exists only when $p=3$. Using this, a classification is given of connected cubic edge-transitive bi-Cayley graphs over an inner-abelian metacyclic $3$-group. As a result, we construct the first known infinite family of cubic semisymmetric graphs of order twice a $3$-power.

A survey on Hamiltonicity in Cayley graphs and digraphs on different groups

2019 ◽  
Vol 11 (05) ◽  
pp. 1930002
Author(s):  
G. H. J. Lanel ◽  
H. K. Pallage ◽  
J. K. Ratnayake ◽  
S. Thevasha ◽  
B. A. K. Welihinda
Keyword(s):  
Cayley Graphs ◽  
Current Status ◽  
Transitive Graph ◽  
Famous Conjecture ◽  
Hamilton Path ◽  
Vertex Transitive ◽  
Future Direction ◽  
Lovász Conjecture ◽  
Vertex Transitive Graph ◽  
Transitive Graphs

Lovász had posed a question stating whether every connected, vertex-transitive graph has a Hamilton path in 1969. There is a growing interest in solving this longstanding problem and still it remains widely open. In fact, it was known that only five vertex-transitive graphs exist without a Hamiltonian cycle which do not belong to Cayley graphs. A Cayley graph is the subclass of vertex-transitive graph, and in view of the Lovász conjecture, the attention has focused more toward the Hamiltonicity of Cayley graphs. This survey will describe the current status of the search for Hamiltonian cycles and paths in Cayley graphs and digraphs on different groups, and discuss the future direction regarding famous conjecture.

Local Expansion of Symmetrical Graphs

1992 ◽  
Vol 1 (1) ◽  
pp. 1-11 ◽  
Author(s):  
László Babai ◽  
Mario Szegedy
Keyword(s):  
Automorphism Group ◽  
Finite Subset ◽  
Harmonic Mean ◽  
Expansion Rate ◽  
Transitive Graph ◽  
Local Expansion ◽  
Vertex Set ◽  
Vertex Transitive ◽  
Edge Transitive ◽  
Vertex Transitive Graph

A graph is vertex-transitive (edge-transitive) if its automorphism group acts transitively on the vertices (edges, resp.). The expansion rate of a subset S of the vertex set is the quotient e(S):= |∂(S)|/|S|, where ∂(S) denotes the set of vertices not in S but adjacent to some vertex in S. Improving and extending previous results of Aldous and Babai, we give very simple proofs of the following results. Let X be a (finite or infinite) vertex-transitive graph and let S be a finite subset of the vertices. If X is finite, we also assume |S| ≤|V(X)/2. Let d be the diameter of S in the metric induced by X. Then e(S) ≥1/(d + 1); and e(S) ≥ 2/(d +2) if X is finite and d is less than the diameter of X. If X is edge-transitive then |δ(S)|/|S| ≥ r/(2d), where ∂(S) denotes the set of edges joining S to its complement and r is the harmonic mean of the minimum and maximum degrees of X. – Diverse applications of the results are mentioned.

scholarly journals Presentations for vertex-transitive graphs

2021 ◽  
Author(s):  
Agelos Georgakopoulos ◽  
Alex Wendland
Keyword(s):  
Cayley Graph ◽  
Group Presentation ◽  
Transitive Graph ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Vertex Transitive Graph ◽  
Transitive Graphs

AbstractWe generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. The resulting notion of presentation allows us to represent every vertex-transitive graph.

scholarly journals Regular graphs and stability

1975 ◽  
Vol 20 (3) ◽  
pp. 377-384 ◽  
Author(s):  
D. A. Holton ◽  
Douglas D. Grant
Keyword(s):  
Automorphism Group ◽  
Regular Graphs ◽  
Transitive Graph ◽  
Trivial Vertex ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Vertex Transitive Graph ◽  
Stable Vertex ◽  
Transitive Graphs

AbstractWe show that a graph G is semi-stable at vertex v if and only if the set of vertices of G adjacent to v is fixed by the automorphism group of Gv, the subgraph of G obtained by deleting v and its incident edges. This result leads to a neat proof that regular graphs are semi-stable at each vertex. We then investigate stable regular graphs, concentrating mainly on stable vertex-transitive graphs. We conjecture that if G is a non-trivial vertex-transitive graph, then G is stable if and only if γ(G) contains a transposition, offering some evidence for its truth.

scholarly journals Cores of Vertex Transitive Graphs

10.37236/3144 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
David E. Roberson
Keyword(s):  
Cayley Graphs ◽  
Transitive Graph ◽  
The Core ◽  
Vertex Set ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Vertex Transitive Graph ◽  
Transitive Graphs ◽  
Vertex Sets ◽  
Normal Cayley Graphs

A core of a graph X is a vertex minimal subgraph to which X admits a homomorphism. Hahn and Tardif have shown that for vertex transitive graphs, the size of the core must divide the size of the graph. This motivates the following question: when can the vertex set of a vertex transitive graph be partitioned into sets each of which induce a copy of its core? We show that normal Cayley graphs and vertex transitive graphs with cores half their size always admit such partitions. We also show that the vertex sets of vertex transitive graphs with cores less than half their size do not, in general, have such partitions.

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