Existence and Classification of 3‐Regular Symmetric Graphs of Order 6pq With Distinct Primes p and q

A graph Σ is said symmetric if its automorphism group acts transitively on the set of its arc. Let p < q be two distinct prime integers. This paper demonstrates that connected 3‐regular symmetric graphs of order 6 p q exist if and only if the pair ( p , q ) belongs to the set (5, 19), (19, 37), (37, 73), which up to isomorphism there are nine sporadic ones, or Σ is a Cayley graph on dihedral group D 6 p q , where and .