Locally arc-transitive graphs of valence {3,4} with trivial edge kernel (original) (raw)

Analysing finite locally s-arc transitive graphs

Transactions of the American Mathematical Society, 2004

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.

The locally 2-arc transitive graphs admitting a Ree simple group

Journal of Algebra, 2004

In this paper, three infinite families of locally 2-arc transitive graphs are constructed, which are vertex-intransitive, regular and all vertex stabilizers are conjugate. To the best of our knowledge these are the first infinite families of graphs with these properties. In particular, they are semi-symmetric. It is then shown that the only locally 2-arc transitive graphs admitting a Ree simple group are (i) the graphs in these three families, (ii) (vertex-transitive) 2-arc transitive graphs admitting a Ree simple group, previously classified by the first and third authors, and (iii) standard double covers of the graphs in (ii). This is the first complete classification of locally 2-arc transitive graphs for an infinite family of simple groups.

A new family of locally 5-arc transitive graphs

2007

Let Γ be a graph with vertex set . An s-arc is an (s+1)-tuple (v 0 ,v 1 ,…,v s ) of vertices in Γ such that each v i is adjacent to v i+1 while v i ≠v i+2 . We say that Γ is locallys-arc transitive if for each , the stabiliser in of v acts transitively on the set of s-arcs whose initial vertex is v. Locally s-arc transitive graphs have been the subject of much investigation see for example the results in [8], [9], [10] and [11]; and examples of locally s-arc transitive graphs with large values of s are of particular interest.

Finite 2-arc-transitive strongly regular graphs and 3-geodesic-transitive graphs

2019

We classify all the 222-arc-transitive strongly regular graphs, and use this classification to study the family of finite (G,3)(G,3)(G,3)-geodesic-transitive graphs of girth 444 or 555 for some group GGG of automorphisms. For this application we first give a reduction result on the latter family of graphs: let NNN be a normal subgroup of GGG which has at least 333 orbits on vertices. We show that Gamma\GammaGamma is a cover of its quotient GammaN\Gamma_NGammaN modulo the NNN-orbits, and that either GammaN\Gamma_NGammaN is (G/N,3)(G/N,3)(G/N,3)-geodesic-transitive of the same girth as Gamma\GammaGamma, or GammaN\Gamma_NGammaN is a (G/N,2)(G/N,2)(G/N,2)-arc-transitive strongly regular graph, or GammaN\Gamma_NGammaN is a complete graph with G/NG/NG/N acting 3-transitively on vertices. The classification of 222-arc-transitive strongly regular graphs allows us to characterise the (G,3)(G,3)(G,3)-geodesic-transitive covers Gamma\GammaGamma when GammaN\Gamma_NGammaN is complete or strongly regular.

On 2-arc transitive graphs of girth 4

Journal of Combinatorial Theory, Series B, 1983

It is shown that a 2-arc-transitive graph must be the incidence graph of a (known) symmetric design if (i) the stabilizer of some vertex acts faithfully on the set of neighbours of that vertex as a known doubly transitive group with no abelian normal subgroup and (ii) some pair of vertices at distance 2 is joined by more than six paths of length 2.

Symmetric Graphs with 2-ARC Transitive Quotients

Journal of the Australian Mathematical Society, 2014

A graph Γ is G-symmetric if Γ admits G as a group of automorphisms acting transitively on the set of vertices and the set of arcs of Γ, where an arc is an ordered pair of adjacent vertices. In the case when G is imprimitive on V (Γ), namely when V (Γ) admits a nontrivial G-invariant partition B, the quotient graph Γ B of Γ with respect to B is always G-symmetric and sometimes even (G, 2)-arc transitive. (A G-symmetric graph is (G, 2)-arc transitive if G is transitive on the set of oriented paths of length two.) In this paper we obtain necessary conditions for Γ B to be (G, 2)-arc transitive (regardless of whether Γ is (G, 2)-arc transitive) in the case when v − k is an odd prime p, where v is the block size of B and k is the number of vertices in a block having neighbours in a fixed adjacent block. These conditions are given in terms of v, k and two other parameters with respect to (Γ, B) together with a certain 2-point transitive block design induced by (Γ, B). We prove further that if p = 3 or 5 then these necessary conditions are essentially sufficient for Γ B to be (G, 2)-arc transitive.

Finite symmetric graphs with two-arc transitive quotients

2005

This paper forms part of a study of 2-arc transitivity for finite imprimitive symmetric graphs, namely for graphs Γ admitting an automorphism group G that is transitive on ordered pairs of adjacent vertices, and leaves invariant a nontrivial vertex partition B. Such a group G is also transitive on the ordered pairs of adjacent vertices of the quotient graph ΓB corresponding to B.

Characterizing finite locally s-arc transitive graphs with a star normal quotient

2006

Abstract Let Γ be a finite locally (G, s)-arc transitive graph with s≥ 2 such that G is intransitive on vertices. Then Γ is bipartite and the two parts of the bipartition are G-orbits. In previous work the authors showed that if G has a non-trivial normal subgroup intransitive on both of the vertex orbits of G, then Γ is a cover of a smaller locally s-arc transitive graph. Thus the 'basic'graphs to study are those for which G acts quasiprimitively on at least one of the two orbits.

Lobe, edge, and arc transitivity of graphs of connectivity 1

Ars Mathematica Contemporanea

We give necessary and sufficient conditions for lobe-transitivity of locally finite and locally countable graphs whose connectivity equals 1. We show further that, given any biconnected graph Λ and a "code" assigned to each orbit of Aut(Λ), there exists a unique lobe-transitive graph Γ of connectivity 1 whose lobes are copies of Λ and is consistent with the given code at every vertex of Γ. These results lead to necessary and sufficient conditions for a graph of connectivity 1 to be edge-transitive and to be arc-transitive. Countable graphs of connectivity 1 the action of whose automorphism groups is, respectively, vertex-transitive, primitive, regular, Cayley, and Frobenius had been previously characterized in the literature.