Smallest tetravalent half-arc-transitive graphs with the vertex-stabiliser isomorphic to the dihedral group of order 8 (original) (raw)

Tetravalent -transitive graphs of order

Discrete Mathematics, 2009

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, and 1 2-arc-transitive if its automorphism group is transitive on vertices, edges but not on arcs. Let p be a prime. Feng et al. [Y.-Q. Feng, K.S. Wang, C.X. Zhou, Tetravalent half-trasnitive graphs of order 4p, European J. Combin. 28 (2007) 726-733] classified tetravalent 1 2-arc-transitive graphs of order 4p. In this article a complete classification of tetravalent s-transitive graphs of order 4p is given. It follows from this classification that with the exception of two graphs of orders 8 or 28, all such graphs are 1-transitive. As a result, all tetravalent vertex-and edge-transitive graphs of order 4p are known.

Classification of half-arc-transitive graphs of order

European Journal of Combinatorics, 2013

A vertex-transitive graph X is said to be half-arc-transitive if its automorphism group acts transitively on the set of edges of X but does not act transitively on the set of arcs of X. A classification of half-arc-transitive graphs on 4p vertices, where p is a prime, is given. Apart from an obvious infinite family of metacirculants, which exist for p ≡ 1(mod 4) and have been known before, there is an additional somewhat unique family of half-arc-transitive graphs of order 4p and valency 12; the latter exists only when p ≡ 1 (mod 6) is of the form 2 2k + 2 k + 1, k > 1.

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.

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.

Finite graphs of valency 4 and girth 4 admitting half-transitive group actions

Journal of the Australian Mathematical Society, 2002

Finite graphs of valency 4 and girth 4 admitting 1/2-transitive group actions, that is, vertex-and edge-but not arc-transitive group actions, are investigated. A graph is said to be 1/2-transitive if its automorphism group acts 1/2-transitively. There is a natural orientation of the edge set of a 1/2-transitive graph induced and preserved by its automorphism group. It is proved that in a finite 1/2-transitive graph of valency 4 and girth 4 the set of 4-cycles decomposes the edge set in such a way that either every 4-cycle is alternating or every 4-cycle is directed relative to this orientation. In the latter case vertex stabilizers are isomorphic to 2 .

A Family of Tetravalent Half-transitive Graphs

2020

In this paper, we introduce a new family of graphs, Gamma(n,a)\Gamma(n,a)Gamma(n,a). We show that it is an infinite family of tetravalent half-transitive Cayley graphs. Apart from that, we determine some structural properties of Gamma(n,a)\Gamma(n,a)Gamma(n,a).

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.

Tetravalent Graphs Admitting Half-Transitive Group Actions: Alternating Cycles

Journal of Combinatorial Theory, Series B, 1999

In this paper we study finite, connected, 4-valent graphs X which admit an action of a group G which is transitive on vertices and edges, but not transitive on the arcs of X. Such a graph X is said to be (G, 1Â2)-transitive. The group G induces an orientation of the edges of X, and a certain class of cycles of X (called alternating cycles) determined by the group G is identified as having an important influence on the structure of X. The alternating cycles are those in which consecutive edges have opposite orientations. It is shown that X is a cover of a finite, connected, 4-valent, (G, 1Â2)-transitive graph for which the alternating cycles have one of three additional special properties, namely they are tightly attached, loosely attached, or antipodally attached. We give examples with each of these special attachment properties, and moreover we complete the classification (begun in a separate paper by the first author) of the tightly attached examples.

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