Minimal transitive graphs of finite simple groups (Q1317643)

The minimal transitive graph of a finite simple group \(G\) is a graph \(\Gamma\) such that 1) \(\Aut \Gamma\) acts transitively on the vertices of \(\Gamma\); 2) \(G\leq\Aut \Gamma\) and \(G\) is the only nonsolvable composition factor in \(\Aut \Gamma\); 3) of the graphs that satisfy the conditions 1) and 2), \(\Gamma\) has the minimal number of vertices. The author constructs minimal transitive graphs for the classical groups \(L_ 3(4)\), \(U_ 4(2)\), \(U_ 3(5)\), \(\text{Sp}_ 4(4)\), \(\text{Sp}_ 6(2)\) of permutational rank 3 and for the sporadic group \(Co_ 3\) of rank 5.