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Catalan graph determinant conjecture

About 8 years old · traced to

Let CGnCG_n be the Catalan graph, and let

Γ={11,13,15,23,33,51,61,63}.\Gamma=\{11,13,15,23,33,51,61,63\}.

Catalan graph determinant conjecture. The determinant satisfies

det⁡(CGn)=0\det(CG_n)=0

if and only if n≥6n\ge6 is even or n∈Γn\in\Gamma. The paper proves the even-order case det⁡(CG2n)=0\det(CG_{2n})=0 for n≥3n\ge3 and reports computational evidence for the exceptional set Γ\Gamma; the stated if-and-only-if classification remains conjectural in the supplied text.

References

Primary source

Gi-Sang Cheon, Ji-Hwan Jung, Sergey Kitaev and Seyed Ahmad Mojallal, “Riordan graphs II: Spectral properties”, arXiv:1801.07021 (2018).

catalan-numbersriordan-graphsspectral-graph-theory
Posted · edited

Progress summary

Refreshed
Open

The even-dimensional cases are proved, but the full classification—including the listed exceptional cases—remains unproved.

The conjecture predicts exactly when the determinant of the Catalan graph CGnCG_n vanishes: for even n≥6n\ge6 or for nn in the exceptional set Γ\Gamma. The January 2018 paper proves the even-order vanishing result and presents computational evidence for Γ\Gamma, but does not prove the converse.

Known results

  • det⁡(CG2n)=0\det(CG_{2n})=0 for n≥3n\ge3.
  • The exceptional set Γ={11,13,15,23,33,51,61,63}\Gamma=\{11,13,15,23,33,51,61,63\} is supported by computation, not established theoretically.
  • No subsequently retrieved source reports a proof, counterexample, or verification of the full if-and-only-if statement.

Current status (as of September 2026): The even-order case n≥6n\ge6 is settled, while the exceptional cases and the full classification remain open.

Sources

Solutions 0

No solutions have been posted yet.