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 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 CORES OF SYMMETRIC GRAPHS

2008 ◽  
Vol 85 (2) ◽  
pp. 145-154 ◽  
Author(s):  
PETER J. CAMERON ◽  
PRISCILA A. KAZANIDIS
Keyword(s):  
Semigroup Theory ◽  
Finite Geometry ◽  
Transitive Graph ◽  
Induced Subgraph ◽  
The Core ◽  
Spanning Subgraph ◽  
Vertex Transitive ◽  
Edge Transitive ◽  
Vertex Transitive Graph ◽  
Two Alternatives

AbstractThe core of a graph Γ is the smallest graph Δ that is homomorphically equivalent to Γ (that is, there exist homomorphisms in both directions). The core of Γ is unique up to isomorphism and is an induced subgraph of Γ. We give a construction in some sense dual to the core. The hull of a graph Γ is a graph containing Γ as a spanning subgraph, admitting all the endomorphisms of Γ, and having as core a complete graph of the same order as the core of Γ. This construction is related to the notion of a synchronizing permutation group, which arises in semigroup theory; we provide some more insight by characterizing these permutation groups in terms of graphs. It is known that the core of a vertex-transitive graph is vertex-transitive. In some cases we can make stronger statements: for example, if Γ is a non-edge-transitive graph, we show that either the core of Γ is complete, or Γ is its own core. Rank-three graphs are non-edge-transitive. We examine some families of these to decide which of the two alternatives for the core actually holds. We will see that this question is very difficult, being equivalent in some cases to unsolved questions in finite geometry (for example, about spreads, ovoids and partitions into ovoids in polar spaces).

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 Consistent Cycles in $1\over2$-Arc-Transitive Graphs

10.37236/94 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Marko Boben ◽  
Štefko Miklavič ◽  
Primož Potočnik
Keyword(s):  
Directed Cycle ◽  
Vertex Transitive ◽  
Edge Transitive ◽  
Transitive Graphs

A directed cycle $C$ of a graph is called $1\over k$-consistent if there exists an automorphism of the graph which acts as a $k$-step rotation of $C$. These cycles have previously been considered by several authors in the context of arc-transitive graphs. In this paper we extend these results to the case of graphs which are vertex-transitive, edge-transitive but not arc-transitive.

An Edge but not Vertex Transitive Cubic Graph*

1968 ◽  
Vol 11 (4) ◽  
pp. 533-535 ◽  
Author(s):  
I. Z. Bouwer
Keyword(s):  
Undirected Graph ◽  
The Other ◽  
Cubic Graph ◽  
Vertex Transitive ◽  
Multiple Edges ◽  
Edge Transitive

Let G be an undirected graph, without loops or multiple edges. An automorphism of G is a permutation of the vertices of G that preserves adjacency. G is vertex transitive if, given any two vertices of G, there is an automorphism of the graph that maps one to the other. Similarly, G is edge transitive if for any two edges (a, b) and (c, d) of G there exists an automorphism f of G such that {c, d} = {f(a), f(b)}. A graph is regular of degree d if each vertex belongs to exactly d edges.

Vertex and Edge Transitive, but not 1-Transitive, Graphs

1970 ◽  
Vol 13 (2) ◽  
pp. 231-237 ◽  
Author(s):  
I. Z. Bouwer
Keyword(s):  
Undirected Graph ◽  
The Other ◽  
Vertex Transitive ◽  
Edge Transitive ◽  
Transitive Graphs

A (simple, undirected) graphGisvertex transitiveif for any two vertices ofGthere is an automorphism ofGthat maps one to the other. Similarly,Gisedge transitiveif for any two edges [a,b] and [c,d] ofGthere is an automorphism ofGsuch that {c,d} = {f(a),f(b)}. A 1-pathofGis an ordered pair (a,b) of (distinct) verticesaandbofG, such thataandbare joined by an edge.Gis 1-transitiveif for any two 1-paths (a,b) and (c,d) ofGthere is an automorphismfofGsuch thatc=f(a) andd=f(b). A graph isregular of valency dif each of its vertices is incident with exactlydof its edges.

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