A new family of locally 5-arc transitive graphs (original) (raw)

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

Journal of Algebraic Combinatorics, 2013

In this paper, we consider connected locally G-arc-transitive graphs with vertices of valence 3 and 4, such that the kernel G [1] uv of the action of an edge-stabiliser on the neighbourhood Γ (u) ∪ Γ (v) is trivial. We find 19 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 [1] uv = 1. Keywords Edge-transitive • Locally arc-transitive • Graph • Symmetry • Amalgam 1 Introduction An arc in a simple graph Γ is an ordered pair of adjacent vertices of Γ. Let Γ be graph and G a group of automorphisms of Γ. Then Γ is said to be G-arc-transitive provided that G acts transitively on the set of arcs of Γ. Similarly, Γ is said to be locally G-arc-transitive if for every vertex v the stabiliser G v of v acts transitively on the set of all arcs of Γ with the initial vertex being v. A graph Γ is arc-transitive if it is Aut(Γ)-arc-transitive. In this paper, we shall be particularly interested in the structure of the vertex-stabilisers (and thus of the group G itself) in certain locally G-arc-transitive graphs. All the graphs in this paper are assumed to be connected. If Γ is a connected locally G-arc-transitive graph, then it is well known that G is transitive on the edges of Γ and that it has at most two orbits on the vertex set V (Γ). If G is transitive on V (Γ), then it is, in fact, arc-transitive. On the other hand, if G has two orbits on V (Γ), then we say that Γ is genuinely locally G-arc-transitive. In this

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.

A classification of tetravalent arc-transitive graphs of order 5p2

Indian Journal of Pure and Applied Mathematics, 2020

Let s be a positive integer. A graph is s-transitive if its automorphism group is transitive on s-arcs but not on (s + 1)-arcs. Let p be a prime. In this article a complete classification of tetravalent s-transitive graphs of order 5p 2 is given.

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.

Locally s‐distance transitive graphs

2012

Abstract We give a unified approach to analyzing, for each positive integer s, a class of finite connected graphs that contains all the distance transitive graphs as well as the locally s-arc transitive graphs of diameter at least s. A graph is in the class if it is connected and if, for each vertex v, the subgroup of automorphisms fixing v acts transitively on the set of vertices at distance i from v, for each i from 1 to s. We prove that this class is closed under forming normal quotients.

On the vertex-stabiliser in arc-transitive digraphs

Journal of Combinatorial Theory, Series B, 2010

We discuss a possible approach to the study of finite arc-transitive digraphs and prove an upper bound on the order of a vertexstabiliser in locally cyclic arc-transitive digraphs of prime outvalence.

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.

Tetravalent vertex-transitive graphs of order 6p6p6p

2022

A graph is vertex-transitive if its automorphism group acts transitively on vertices of the graph. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper, the tetravalent vertex-transitive non-Cayley graphs of order 6p are classified for each prime p. 1. Introduction In this paper we consider undirected finite connected graphs without loops or multiple edges. For a graph X we use V (X), E(X), A(X) and Aut(X) to denote its vertex set, edge set, arc set and its full automorphism group, respectively. For u, v ∈ V (X), u ∼ v represents that u is adjacent to v, and is denoted by {u, v} the edge incident to u and v in X, and N X (u) is the neighborhood of u in X, that is, the set of vertices adjacent to u in X. A graph X is said to be G-vertex-transitive, G-edge-transitive and G-arctransitive (or G-symmetric) if G ≤ Aut(X) acts transitively on V (X), E(X) and A(X), respectively. In the special case, if G = Aut(X) then X is said to be vertex-transitive, edge-transitive and arc-transitive (or symmetric). An s-arc in a graph X is an ordered (s + 1)-tuple (v 0 , v 1 , • • • , v s) of vertices of X such that v i−1 is adjacent to v i for 1 ≤ i ≤ s, and v i−1 = v i+1 for 1 ≤ i ≤ s; in other words, a directed walk of length s which never includes a backtracking. A graph X is said to be s-arc-transitive if Aut(X) is transitive on the set of s-arcs in X. A subgroup of the automorphism group of a graph X is said to be s-regular if it acts regularly on the set of s-arcs of X. Recall that a permutation group G acting on a set Ω is called semiregular if the stabilizer of α ∈ G, G α = 1 for all α ∈ G and is called regular if it is semiregular and transitive.

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.