Classification of vertex-transitive digraphs via automorphism group (original) (raw)

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.

VERTEX-TRANSITIVE GRAPHS OF ORDER 2p

Annals of the New York Academy of Sciences, 1979

In the past ten years there has been a considerable amount of activity in the area of circulant graphs and digraphs. Most of this has consisted of investigation of basic properties of circulants along with some applications. We shall now summarize some of this activity.

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.

Non-Cayley Vertex-Transitive Graphs of Order Twice the Product of 2 Odd Primes

Journal of Algebraic Combinatorics, 1994

For a positive integer n, does there exist a vertex-transitive graph r on n vertices which is not a Cayley graph, or, equivalently, a graph r on n vertices such that Aut F is transitive on vertices but none of its subgroups are regular on vertices? Previous work (by Alspach and Parsons, Frucht, Graver and Watkins, MaruSic and Scapellato, and McKay and the second author) has produced answers to this question if n is prime, or divisible by the square of some prime, or if n is the product of two distinct primes. In this paper we consider the simplest unresolved case for even integers, namely for integers of the form n = 2pq, where 2 < q < p, and p and q are primes. We give a new construction of an infinite family of vertex-transitive graphs on 2pq vertices which are not Cayley graphs in the case where p = 1 (mod q). Further, if p = 1 (mod q), p = q = 3(mod 4), and if every vertex-transitive graph of order pq is a Cayley graph, then it is shown that, either 2pq = 66, or every vertex-transitive graph of order 2pq admitting a transitive imprimitive group of automorphisms is a Cayley graph.

Vertex-transitive digraphs with extra automorphisms that preserve the natural arc-colouring

Australas. J Comb., 2017

In a Cayley digraph on a group G, if a distinct colour is assigned to each arc-orbit under the left-regular action of G, it is not hard to show that the elements of the left-regular action of G are the only digraph automorphisms that preserve this colouring. In this paper, we show that the equivalent statement is not true in the most straightforward generalisation to G-vertex-transitive digraphs, even if we restrict the situation to avoid some obvious potential problems. Specifically, we display an infinite family of 2-closed groups G, and a G-arc-transitive digraph on each (without any digons) for which there exists an automorphism of the digraph that is not an element of G (it is an automorphism of G). Since the digraph is G-arc-transitive, the arcs would all be assigned the same colour under the colouring by arc-orbits, so this digraph automorphism is colour-preserving.

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.

On the automorphism groups of vertex-transitive Cayley digraphs of monoids

Journal of Algebraic Combinatorics, 2020

In 1982, Babai and Godsil conjectured that almost all Cayley digraphs are digraphical regular representations. In 1998, Xu conjectured that almost all Cayley digraphs are normal [i.e., G L is a normal subgroup of the automorphism group of Cay(G, C)]. Finally, in 1994, Praeger and Mckay conjectured that almost all undirected vertextransitive graphs are Cayley graphs of groups. In this paper, first we present the variants of these conjectures for Cayley digraphs of monoids and we determine when the variant of Babai and Godsil's conjecture is equivalent to the variant of Xu's conjecture. Then, as a special consequence of our results, we conclude that Xu's conjecture is equivalent to Babai and Godsil's conjecture. On the other hand, we give affirmative answer to the variant of Praeger and Mckay's conjecture and we prove that a Cayley digraph of a monoid is vertex-transitive if and only if it is isomorphic to a Cayley digraph of a group. Finally, we use this characterization to give an affirmative answer to a question raised by Kelarev and Praeger about vertex-transitivity of Cayley digraphs of monoids. Also using this characterization, we explicitly determine the automorphism groups of vertex-transitive Cayley digraphs of monoids.

On the automorphism groups of almost all circulant graphs and digraphs

Ars Mathematica Contemporanea

We attempt to determine the structure of the automorphism group of a generic circulant graph. We first show that almost all circulant graphs have automorphism groups as small as possible. The second author has conjectured that almost all of the remaining circulant (di)graphs (those whose automorphism group is not as small as possible) are normal circulant (di)graphs. We show this conjecture is not true in general, but is true if we consider only those circulant (di)graphs whose order is in a "large" subset of integers. We note that all non-normal circulant (di)graphs can be classified into two natural classes (generalized wreath products, and deleted wreath type), and show that neither of these classes contains almost every non-normal circulant digraph.