We present a new approach to analysing finite graphs which admit a vertex intransitive group of automorphisms G and are either locally (G, s)-arc transitive for s > 2 or G-locally primitive. Such graphs are bipartite with the two parts of the bipartition being the orbits of G. Given a normal subgroup N which is intransitive on both parts of the bipartition, we show that taking quotients with respect to the orbits of N preserves both local primitivity and local s-arc transitivity and leads us to study graphs where G acts faithfully on both orbits and quasiprimitively on at least one. We determine the possible quasiprimitive types for G in these two cases and give new constructions of examples for each possible type. The analysis raises several open problems which are discussed in the final section.

/pdf/analysing-finite-locally-s-arc-transitive-graphs-3vyblmoapc.pdf

Analysing Finite Locally s-arc Transitive Graphs

/pdf/the-finite-vertex-primitive-and-vertex-biprimitive-1z54ah29m2.pdf

The finite vertex-primitive and vertex-biprimitive -transitive graphs for ≥4

An s-arc in a graph is a vertex sequence (α0,α1,…,αs) such that {αi−1,αi} ∈ EΓ for 1 [les ] i [les ] s and αi−1 ≠ αi+1 for 1 [les ] i [les ] s − 1. This paper gives a characterization of a class of s-transitive graphs; that is, graphs for which the automorphism group is transitive on s-arcs but not on (s + 1)-arcs. It is proved that if Γ is a finite connected s-transitive graph (where s [ges ] 2) of order a p-power with p prime, then s = 2 or 3; further, either s = 3 and Γ is a normal cover of the complete bipartite graph K2m,2m, or s = 2 and Γ is a normal cover of one of the following 2-transitive graphs: Kpm+1 (the complete graph of order pm+1), K2m,2m − 2mK2 (the complete bipartite graph of order 2m+1 minus a 1-factor), a primitive affine graph, or a biprimitive affine graph. (Finite primitive and biprimitive affine 2-arc transitive graphs were classified by Ivanov and Praeger in 1993.)

Finite s-Arc Transitive Graphs of Prime-Power Order

Let Γ be a finite connected undirected vertex transitive locally primitive graph of prime-power order. It is shown that either Γ is a normal Cayley graph of a 2-group, or Γ is a normal cover of a complete graph, a complete bipartite graph, or Σ×l, where Σ=Kpm with p prime or Σ is the Schlafli graph (of order 27). In particular, either Γ is a Cayley graph, or Γ is a normal cover of a complete bipartite graph.

/pdf/locally-primitive-graphs-of-prime-power-order-4jif768xkz.pdf

Locally primitive graphs of prime-power order

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.

/pdf/finite-edge-transitive-oriented-graphs-of-valency-four-a-g3qvv48mvr.pdf

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

A graph &#x1d6e4; is called G-locally-quasiprimitive if each normal subgroup of Gv acts on &#x1d6e4;(v) trivially or transitively for every vertex v. In this paper we analyse the global action and the structural information of such groups G, extending the previous results for locally-primitive graphs and vertex-transitive locally-quasiprimitive graphs.

Finite Locally-quasiprimitive Graphs

A characterization is given of the class of edge-transitive Cayley graphs of Frobenius groups $\mathbb{Z}_{p^{d}}{:}\mathbb{Z}_{q}$ with p,q odd prime, of valency coprime to p. This characterization is then used to study an isomorphism problem regarding Cayley graphs, and to construct new families of half-arc-transitive graphs.

/pdf/classifying-a-family-of-edge-transitive-metacirculant-graphs-4cxjebd42d.pdf

Classifying a family of edge-transitive metacirculant graphs

Reliability Analysis of the Cayley Graphs of Dihedral Groups

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Finite Locally-quasiprimitive Graphs

Classifying a family of edge-transitive metacirculant graphs

Reliability Analysis of the Cayley Graphs of Dihedral Groups