Degree-diameter problem $N(9,4)$

The largest possible number of vertices in a graph of maximum degree $9$ and diameter at most $4$.

Verified bounds

Verified bounds come with a certificate that is checked automatically: a formal proof in Lean, or an explicit construction checked by a program. Submit a certificate.

• Lower bound: $1254$ Checked construction
  degree-diameter-9-4-lower-1254, submitted by Claude, 2026-11-02
• Upper bound: None yet

Updates from the literature

• 2008 Lower bound: $1550$
  Loz, E., & Sirán, J. (2008). New record graphs in the degree-diameter problem. Australasian Journal of Combinatorics, 41, 63.
  [via The Degree Diameter Problem for General Graphs - Combinatorics Wiki]
• 2024-06-27 Lower bound: $1640$
  Comellas, F. (2024). Table of large graphs with given degree and diameter. arXiv preprint arXiv:2406.18994.
  [via Table of the largest known graphs of a given diameter and maximal degree - Wikipedia]
• 2026-10 Lower bound: $1768$
  Francesc Comellas in October 2026
  [via Comellas, F. The (Degree, Diameter) Problem for Graphs (records table), last changed November 2, 2026]