Big subsets with small boundaries in a graph with a vertex-transitive group of automorphisms

2017 ◽  
Vol 81 (1) ◽  
pp. 137-155
Author(s):  
N Seifter ◽  
T I Trofimov
Keyword(s):  
Transitive Group ◽  
Group Of Automorphisms ◽  
Vertex Transitive

scholarly journals Semiregular Automorphisms of Cubic Vertex-Transitive Graphs and the Abelian Normal Quotient Method

10.37236/4842 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Joy Morris ◽  
Pablo Spiga ◽  
Gabriel Verret
Keyword(s):  
Transitive Group ◽  
Cubic Graphs ◽  
Cubic Graph ◽  
Maximum Order ◽  
Abelian Normal Subgroup ◽  
Group Of Automorphisms ◽  
Normal Quotient ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

We characterise connected cubic graphs admitting a vertex-transitive group of automorphisms with an abelian normal subgroup that is not semiregular. We illustrate the utility of this result by using it to prove that the order of a semiregular subgroup of maximum order in a vertex-transitive group of automorphisms of a connected cubic graph grows with the order of the graph.

scholarly journals Finite Edge-Transitive Oriented Graphs of Valency Four: a Global Approach

10.37236/4779 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Jehan A. Al-bar ◽  
Ahmad N. Al-kenani ◽  
Najat M. Muthana ◽  
Cheryl E. Praeger ◽  
Pablo Spiga
Keyword(s):  
Transitive Group ◽  
Open Problems ◽  
Oriented Graphs ◽  
Group Of Automorphisms ◽  
Normal Quotient ◽  
Normal Cover ◽  
Vertex Transitive ◽  
Edge Transitive ◽  
Basic Graph ◽  
New Framework

We develop a new framework for analysing finite connected, oriented graphs of valency four, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a sub-family of `basic' graphs such that each graph of this type is a normal cover of at least one basic graph. The basic graphs either admit an edge-transitive group of automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit an (oriented or unoriented) cycle as a normal quotient. We anticipate that each of these additional properties will facilitate effective further analysis, and we demonstrate that this is so for the quasiprimitive basic graphs. Here we obtain strong restrictions on the group involved, and construct several infinite families of such graphs which, to our knowledge, are different from any recorded in the literature so far. Several open problems are posed in the paper.

scholarly journals Cubic Non-Cayley Vertex-Transitive Bi-Cayley Graphs over a Regular $p$-Group

10.37236/3915 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Jin-Xin Zhou ◽  
Yan-Quan Feng
Keyword(s):  
Cayley Graph ◽  
Cayley Graphs ◽  
Equal Size ◽  
Group Of Automorphisms ◽  
Vertex Transitive ◽  
Vertex Transitive Graphs ◽  
Transitive Graphs

A bi-Cayley graph is a graph which admits a semiregular group of automorphisms with two orbits of equal size. In this paper, we give a characterization of cubic non-Cayley vertex-transitive bi-Cayley graphs over a regular $p$-group, where $p>5$ is an odd prime. As an application, a classification of cubic non-Cayley vertex-transitive graphs of order $2p^3$ is given for each prime $p$.

scholarly journals INTRANSITIVE GEOMETRIES

2006 ◽  
Vol 93 (3) ◽  
pp. 666-692 ◽  
Author(s):  
RALF GRAMLICH ◽  
HENDRIK VAN MALDEGHEM
Keyword(s):  
Transitive Group ◽  
Incidence Geometry ◽  
Orthogonal Groups ◽  
Group Of Automorphisms ◽  
Simple Connectivity ◽  
Universal Completion

A lemma of Tits establishes a connection between the simple connectivity of an incidence geometry and the universal completion of an amalgam induced by a sufficiently transitive group of automorphisms of that geometry. In the present paper, we generalize this lemma to intransitive geometries, thus opening the door for numerous applications. We treat ourselves some amalgams related to intransitive actions of finite orthogonal groups, as a first class of examples.

scholarly journals Distance-transitive strongly regular graphs

2021 ◽  
Vol 38 (1) ◽  
pp. 21-34
Author(s):  
MONTHER RASHED ALFRUIDAN ◽  
Keyword(s):  
Open Problem ◽  
Transitive Group ◽  
Complete List ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Group Of Automorphisms ◽  
Transitive Groups ◽  
Strongly Regular ◽  
First Time

We present a complete description of strongly regular graphs admitting a distance-transitive group of automorphisms. Parts of the list have already appeared in the literature; however, this is the first time that the complete list appears in one place. The description is complemented, where possible, with the discussion of the corresponding distance-transitive groups and some further properties of the graphs. We also point out an open problem.

ON ISOMORPHISMS OF VERTEX-TRANSITIVE CUBIC GRAPHS

2015 ◽  
Vol 99 (3) ◽  
pp. 341-349
Author(s):  
JING CHEN ◽  
CAI HENG LI ◽  
WEI JUN LIU
Keyword(s):  
Simple Group ◽  
Isomorphism Problem ◽  
Cubic Graphs ◽  
Group Of Automorphisms ◽  
Vertex Transitive

We study the isomorphism problem of vertex-transitive cubic graphs which have a transitive simple group of automorphisms.

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