Locally arc-transitive graphs of valence {3,4}\{3,4\} with trivial edge kernel

In this paper we consider connected locally $G$-arc-transitive graphs with vertices of valence 3 and 4, such that the kernel $G_{uv}^{[1]}$ of the action of an edge-stabiliser on the neighourhood $\Gamma(u) \cup \Gamma(v)$ is trivial. We find nineteen finitely presented groups with the property that any such group $G$ is a quotient of one of these groups. As an application, we enumerate all connected locally arc-transitive graphs of valence ${3,4}$ on at most 350 vertices whose automorphism group contains a locally arc-transitive subgroup $G$ with $G_{uv}^{[1]} = 1$

Resolution of a conjecture about linking ring structures

An LR-structure is a tetravalent vertex-transitive graph together with a special type of a decomposition of its edge-set into cycles. LR-structures were introduced in a paper by P. Poto\v{c}nik and S. Wilson, titled `Linking rings structures and tetravalent semisymmetric graphs', in Ars Math. Contemp. 7 (2014), as a tool to study tetravalent semisymmetric graphs of girth 4. In this paper, we use the methods of group amalgams to resolve some problems left open in the above-mentioned paper

Core-Free, Rank Two Coset Geometries from Edge-Transitive Bipartite Graphs

It is known that the Levi graph of any rank two coset geometry is an edge-transitive graph, and thus coset geometries can be used to construct many edge transitive graphs. In this paper, we consider the reverse direction. Starting from edge- transitive graphs, we construct all associated core-free, rank two coset geometries. In particular, we focus on 3-valent and 4-valent graphs, and are able to construct coset geometries arising from these graphs. We summarize many properties of these coset geometries in a sequence of tables; in the 4-valent case we restrict to graphs that have relatively small vertex-stabilizers

A note on pentavalent s-transitive graphs

AbstractA graph, with a group G of its automorphisms, is said to be (G,s)-transitive if G is transitive on s-arcs but not on (s+1)-arcs of the graph. Let X be a connected (G,s)-transitive graph for some s≥1, and let Gv be the stabilizer of a vertex v∈V(X) in G. In this paper, we determine the structure of Gv when X has valency 5 and Gv is non-solvable. Together with the results of Zhou and Feng [J.-X. Zhou, Y.-Q. Feng, On symmetric graphs of valency five, Discrete Math. 310 (2010) 1725–1732], the structure of Gv is completely determined when X has valency 5. For valency 3 or 4, the structure of Gv is known