Two-Distance-Transitive Graphs of Valency $7$

Abstract

A graph $\mathrm{\Gamma }$ is said to be $(G,2)$-distance-transitive if, for $i=1,2$ and for any two vertex pairs $(u_1,v_1)$ and $(u_2,v_2)$ with $d_{\mathrm{\Gamma }}(u_1,v_1)=d_{\mathrm{\Gamma }}(u_2,v_2)=i$, there exists $g\in G$ such that $(u_1,v_1{)}^g=(u_2,v_2)$. This paper classifies the family of $(G,2)$-distance-transitive graphs of valency $7$.