Vertex-transitive graphs which are not Cayley graphs, I

Journal of Combinatorial Theory, Series B, 2020

A graph Γ admitting a group H of automorphisms acting semi-regularly on the vertices with exactly two orbits is called a bi-Cayley graph over H. Such a graph Γ is called normal if H is normal in the full automorphism group of Γ, and normal edge-transitive if the normaliser of H in the full automorphism group of Γ is transitive on the edges of Γ. In this paper, we give a characterisation of normal edgetransitive bi-Cayley graphs, and in particular, we give a detailed description of 2-arc-transitive normal bi-Cayley graphs. Using this, we investigate three classes of bi-Cayley graphs, namely those over abelian groups, dihedral groups and metacyclic p-groups. We find that under certain conditions, 'normal edgetransitive' is the same as 'normal' for graphs in these three classes. As a by-product, we obtain a complete classification of all connected trivalent edge-transitive graphs of girth at most 6, and answer some open questions from the literature about 2-arc-transitive, half-arc-transitive and semisymmetric graphs.

2010

For any d ≥ 5 and k ≥ 3 we construct a family of Cayley graphs of degree d, diameter k, and order at least k((d−3)/3) k . By comparison with other available results in this area we show that our family gives the

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.

Journal of Symbolic Computation, 1989

This paper describes the construction of all the vertex-transitive graphs on 24 vertices, thus extending the currently available catalogues. This construction differs significantly from previous constructions of the vertex-transitive graphs of order up to 23 in that we are forced to use far more sophisticated group-theoretic techniques. We include an analysis of all the symmetric graphs on 24 vertices.

2018

The Cayley graph X = Cay(G,S) on group G with respect to connection set S is normal edge-transitive, if NAut(X)(R(G)) acts transitively on edge set. In this paper we determine the structure of automorphism group of all tetravalent normal edge-transitive Cayley graphs of order pqr.

Journal of Graph Theory, 2011

We examine the existing constructions of the smallest known vertex-transitive graphs of a given degree and girth 6. It turns out that most of these graphs can be described in terms of regular lifts of suitable quotient graphs. A further outcome of our analysis is a precise identification of which of these graphs are Cayley. We also investigate higher level of transitivity of the smallest known vertex-transitive graphs of a given degree and girth 6 and relate their constructions to near-difference sets. ᭧

Canadian Journal of Mathematics, 1982

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.

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.

2021

A graph Γ is called (G, s)-arc-transitive if G ≤Aut(Γ) is transitive on the set of vertices of Γ and the set of s-arcs of Γ, where for an integer s ≥ 1 an s-arc of Γ is a sequence of s+1 vertices (v_0,v_1,…,v_s) of Γ such that v_i-1 and v_i are adjacent for 1 ≤ i ≤ s and v_i-1 v_i+1 for 1 ≤ i ≤ s-1. Γ is called 2-transitive if it is (Aut(Γ), 2)-arc-transitive but not (Aut(Γ), 3)-arc-transitive. A Cayley graph Γ of a group G is called normal if G is normal in Aut(Γ) and non-normal otherwise. It was proved by X. G. Fang, C. H. Li and M. Y. Xu that if Γ is a tetravalent 2-transitive Cayley graph of a finite simple group G, then either Γ is normal or G is one of the groups PSL_2(11), M_11, M_23 and A_11. However, it was unknown whether Γ is normal when G is one of these four groups. In the present paper we answer this question by proving that among these four groups only M_11 produces connected tetravalent 2-transitive non-normal Cayley graphs. We prove further that there are exactly tw...