January 1, 2025Open Access

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

Key Points

Connected 3-regular symmetric graphs of order 6pq exist for specific distinct prime pairs (5, 19), (19, 37), or (37, 73), showing unique graph structures.
Nine sporadic 3-regular symmetric graphs exist up to isomorphism, illustrating diversity in graph forms within given parameters.
The automorphism group of a symmetric graph acts transitively on its arcs, enhancing understanding of graph symmetry.
Cayley graphs on the dihedral group D6pq provide another classification avenue for these graph types.

Abstract

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 .

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

Key Points

Abstract

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