Existence of directed strongly regular graphs for remaining open parameter sets of order at most 31

Determine whether directed strongly regular graphs with the remaining open parameter sets of order at most 31 exist, without imposing vertex-transitivity on their automorphism groups.

Background

The paper constructs and classifies quasi-strongly regular digraphs arising from transitive permutation groups of degrees 3 through 30 and from primitive permutation groups of degrees 31 through 110 with rank at most 30. The authors note that their construction does not realize the remaining open directed strongly regular graph parameter sets of order at most 31, consistently with an earlier enumeration showing that those parameter sets admit no vertex-transitive realizations.

Consequently, the unresolved issue is whether directed strongly regular graphs with those parameter sets exist at all. The paper establishes only that, if such graphs exist, they cannot have a vertex-transitive automorphism group; it does not settle their general existence.

References

Gy"urki in extended the enumeration of vertex-transitive directed strongly regular graphs to order 31, obtaining 140 equivalence classes and proving that the remaining open parameter sets of order at most 31 admit no vertex-transitive realizations. Consistently with this result, our construction does not produce any of these open parameter sets. Hence, if directed strongly regular graphs with such parameters exist, none of them can admit a vertex-transitive automorphism group.

— Quasi-strongly regular digraphs constructed from transitive groups of degree $n\leq 110$  (2609.10075 - Crnković et al., 9 Sep 2026) in Remark following the classification theorems, Section 4