The finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\ge 4$

Abstract

Abstract. A complete classification is given for finite vertex-primitive and vertex-biprimitive s-transitive graphs for s ≥ 4. The classification involves the construction of new 4-transitive graphs, namely a graph of valency 14 admitting the Monster simple group M, and an infinite family of graphs of valency 5 admitting projective symplectic groups PSp(4, p) with p prime and p ≡ ±1 (mod 8). As a corollary of this classification, a conjecture of Biggs and Hoare (1983) is proved.

Tetravalent edge-transitive Cayley graphs of Frobenius groups

“…Furthermore, AutΓ = PGL (3,3).Z 2 , and Γ is a Cayley graph of G = D 26 . Refer to [14,15], for example.…”

Finite Vertex Bi-Primitive 2-Arc Transitive Graphs Admitting a Two-Dimensional Linear Group

“…Li et al [21] and Praeger and Xu [26] classified vertex primitive symmetric graphs of order kp with k < p, and Ivanov and Praeger [17] classified affine primitive 2-arc transitive graphs. Li [18] classified vertex primitive and vertex bi-primitive s-transitive…”

Pentavalent Symmetric Graphs of Order $12p$

“…Following this, by using deep group theory, all symmetric graphs of order 2p, 3p or qp were classified in [5,34,29,30], where p, q are distinct primes. Recently, Li [22] classified vertex-primitive and vertex-biprimitive s-transitive graphs for s ≥ 4, and Fang et al [8] classified vertex-primitive 2-regular graphs. For more results on symmetric graphs with general valencies, see, for example, [10,20,21,22].…”

“…Recently, Li [22] classified vertex-primitive and vertex-biprimitive s-transitive graphs for s ≥ 4, and Fang et al [8] classified vertex-primitive 2-regular graphs. For more results on symmetric graphs with general valencies, see, for example, [10,20,21,22]. Despite all of these efforts, however, further classifications of symmetric graphs with general valencies seem to be very difficult.…”