Related papers: Tetravalent Vertex- and Edge-Transitive Graphs Ove…
A graph $\G$ is {\em symmetric} or {\em arc-transitive} if its automorphism group $\Aut(\G)$ is transitive on the arc set of the graph, and $\G$ is {\em basic} if $\Aut(\G)$ has no non-trivial normal subgroup $N$ such that the quotient…
These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are…
A graph is said to be symmetric if its automorphism group is transitive on its arcs. Guo et al. (Electronic J. Combin. 18, \#P233, 2011) and Pan et al. (Electronic J. Combin. 20, \#P36, 2013) determined all pentavalent symmetric graphs of…
A graph is said to be edge-transitive if its automorphism group acts transitively on its edges. It is known that edge-transitive graphs are either vertex-transitive or bipartite. In this paper we present a complete classification of all…
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…
A graph is called {\em half-arc-transitive} if its full automorphism group acts transitively on vertices and edges, but not on arcs. It is well known that for any prime $p$ there is no tetravalent half-arc-transitive graph of order $p$ or…
We classify non-complete prime valency graphs satisfying the property that their automorphism group is transitive on both the set of arcs and the set of $2$-geodesics. We prove that either $\Gamma$ is 2-arc transitive or the valency $p$…
In this paper, we examine the structure of vertex- and edge-transitive strongly regular graphs, using normal quotient reduction. We show that the irreducible graphs in this family have quasiprimitive automorphism groups, and prove (using…
This paper is to accompany the Census of Edge-Transitive Tetravalent Graphs, available at jan.ucc.nau.edu/~swilson/C4FullSite/index.html, which is a collection of all known edge-transitive graphs of valence 4 up to 512 vertices. The Census…
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, we introduce a family of tetravalent graphs called propeller graphs, denoted by $Pr_{n}(b,c,d)$. We then produce three infinite subfamilies and one finite subfamily of arc-transitive propeller graphs, and show that all such…
A graph or hypergraph is said to be vertex-transitive if its automorphism group acts transitively upon its vertices. A classic theorem of Mader asserts that every connected vertex-transitive graph is maximally edge-connected. We generalise…
A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not its arc set. In this paper, we study all tetravalent half-arc-transitive graphs of order $12p$.
A graph is edge-transitive if the natural action of its automorphism group on its edge set is transitive. An automorphism of a graph is semiregular if all of the orbits of the subgroup generated by this automorphism have the same length.…
A graph $\Gamma$ is said to be symmetric if its automorphism group $\rm Aut(\Gamma)$ acts transitively on the arc set of $\Gamma$. In this paper, we show that if $\Gamma$ is a finite connected heptavalent symmetric graph with solvable…
Every finite, self-dual, regular (or chiral) 4-polytope of type {3,q,3} has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by dropping self-duality, we obtain a construction for semisymmetric trivalent graphs (which…
The theory of voltage graphs has become a standard tool in the study graphs admitting a semiregular group of automorphisms. We introduce the notion of a cyclic generalised voltage graph to extend the scope of this theory to graphs admitting…
Given a finite group $G$, its prime graph $\Gamma(G)$ (also known as its Gruenberg-Kegel graph) is the graph whose vertices are the prime divisors of $|G|$ and where edges $\{p, q\}$ exist whenever $G$ contains an element of order $pq$. We…
In this paper a classification of tetravalent edge-transitive metacirculants is given. It is shown that a tetravalent edge-transitive metacirculant $\Gamma$ is a normal graph except for four known graphs. If further, $\Gamma$ is a Cayley…
A graph $\Gamma$ of even order is a bicirculant if it admits an automorphism with two orbits of equal length. Symmetry properties of bicirculants, for which at least one of the induced subgraphs on the two orbits of the corresponding…