The finite vertex-primitive and vertex-biprimitive 𝑠-transitive graphs for 𝑠≥4

Li, Cai

doi:10.1090/S0002-9947-01-02768-4

The finite vertex-primitive and vertex-biprimitive $s$-transitive graphs for $s\ge 4$
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Trans. Amer. Math. Soc. 353 (2001), 3511-3529 Request permission

Abstract:

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

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