Algebraic Graph Theory: Automorphism Groups and Cayley graphs

Journal of Algebraic Combinatorics, 2020

In 1982, Babai and Godsil conjectured that almost all Cayley digraphs are digraphical regular representations. In 1998, Xu conjectured that almost all Cayley digraphs are normal [i.e., G L is a normal subgroup of the automorphism group of Cay(G, C)]. Finally, in 1994, Praeger and Mckay conjectured that almost all undirected vertextransitive graphs are Cayley graphs of groups. In this paper, first we present the variants of these conjectures for Cayley digraphs of monoids and we determine when the variant of Babai and Godsil's conjecture is equivalent to the variant of Xu's conjecture. Then, as a special consequence of our results, we conclude that Xu's conjecture is equivalent to Babai and Godsil's conjecture. On the other hand, we give affirmative answer to the variant of Praeger and Mckay's conjecture and we prove that a Cayley digraph of a monoid is vertex-transitive if and only if it is isomorphic to a Cayley digraph of a group. Finally, we use this characterization to give an affirmative answer to a question raised by Kelarev and Praeger about vertex-transitivity of Cayley digraphs of monoids. Also using this characterization, we explicitly determine the automorphism groups of vertex-transitive Cayley digraphs of monoids.

arXiv: Combinatorics, 2020

In the mid-1990s, two groups of authors independently obtained classifications of vertex-transitive graphs whose order is a product of two distinct primes. In the intervening years it has become clear that there is additional information concerning these graphs that would be useful, as well as making explicit the extensions of these results to digraphs. Additionally, there are several small errors in some of the papers that were involved in this classification. The purpose of this paper is to fill in the missing information as well as correct all known errors.

2018

In this supplement, we will assume that all graphs are undirected graphs with no loops or multiple edges. In graph theory, we talk about graph isomorphisms. As a reminder, an isomorphism between graphs G and H is a bijection φ : V (G) → V (H) such that uv ∈ E(G) if and only if φ(u)φ(v) ∈ E(H). A graph automorphism is simply an isomorphism from a graph to itself. In other words, an automorphism on a graph G is a bijection φ : V (G) → V (G) such that uv ∈ E(G) if and only if φ(u)φ(v) ∈ E(G). This definition generalizes to digraphs, multigraphs, and graph with loops. Let Aut(G) denote the set of all automorphisms on a graph G. Note that this forms a group under function composition. In other words,

2009

… ASI Series C …, 1997

Facta Universitatis, Series: Mathematics and Informatics, 2021

The power graph of a group $G$ is the graph with vertex set $G$,having an edge joining $x$ and $y$ whenever one is a power of theother. The purpose of this paper is to study the automorphismgroups of the power graphs of infinite groups.

Bulletin of the Australian Mathematical Society, 1999

An approach to analysing the family of Cayley graphs for a finite group G is given which identifies normal edge-transitive Cayley graphs as a sub-family of central importance. These are the Cayley graphs for G for which a subgroup of automorphisms exists which both normalises G and acts transitively on edges. It is shown that, for a nontrivial group G, each normal edge-transitive Cayley graph for G has at least one homomorphic image which is a normal edge-transitive Cayley graph for a characteristically simple quotient group of G. Moreover, given a normal edge-transitive Cayley graph ΓH for a quotient group G/H, necessary and sufficient conditions are obtained for the existence of a normal edge-transitive Cayley graph Γ for G which has ΓH as a homomorphic image, and a method for obtaining all such graphs Γ is given.

Discrete Mathematics, 2010

In this paper, we first give a characterization of Cayley graphs of rectangular groups. Then, vertex transitivity of Cayley graphs of rectangular groups is considered. Further, it is shown that Cayley graphs Cay(S, C) which are automorphism-vertex transitive, are in fact Cayley graphs of rectangular groups, if the subsemigroup generated by C is an orthodox semigroup. Finally, a characterization of vertex transitive graphs which are Cayley graphs of finite semigroups is concluded.

2019

There have been many investigations on the combinatorial structures and invariants over the group actions on the subsets of its elements. Automorphism groups from Graphs containing the cyclic and dihedral groups, <em>C<sub>n</sub></em> and <em>D<sub>n </sub></em>respectively have been constructed using Schur's Algorithm. In this paper, we seek to extend the work to graphs whose automorphism groups contain the Alternating group <em>A<sub>n</sub></em>. We plan to construct the graphs whose automorphism groups contain the Alternating group <em>A<sub>n</sub></em>. To construct these graphs, we will engage Schur's algorithm. We will first consider cases for , which will in turn help us make generalizations for any value of, the degree of the group in study. The process will involve, establishing the transitivity of the Alternating group <em>A<sub>n</sub></em> fi...

Journal of Combinatorial Theory, Series B, 2020

A graph Γ admitting a group H of automorphisms acting semi-regularly on the vertices with exactly two orbits is called a bi-Cayley graph over H. Such a graph Γ is called normal if H is normal in the full automorphism group of Γ, and normal edge-transitive if the normaliser of H in the full automorphism group of Γ is transitive on the edges of Γ. In this paper, we give a characterisation of normal edgetransitive bi-Cayley graphs, and in particular, we give a detailed description of 2-arc-transitive normal bi-Cayley graphs. Using this, we investigate three classes of bi-Cayley graphs, namely those over abelian groups, dihedral groups and metacyclic p-groups. We find that under certain conditions, 'normal edgetransitive' is the same as 'normal' for graphs in these three classes. As a by-product, we obtain a complete classification of all connected trivalent edge-transitive graphs of girth at most 6, and answer some open questions from the literature about 2-arc-transitive, half-arc-transitive and semisymmetric graphs.