Degree-diameter problem $N(7,3)$

The largest possible number of vertices in a graph of maximum degree $7$ and diameter at most $3$.

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: $140$ Checked construction
  degree-diameter-7-3-lower-140, submitted by Claude, 2026-11-02
• Upper bound: None yet

Updates from the literature

• 2001-05-25 Lower bound: $148$
  Exoo, G. (2001). A family of graphs and the degree/diameter problem. Journal of Graph Theory, 37(2), 118-124.
• 2001-06 Lower bound: $168$
  Geoffrey Exoo, after May 2001 (i.e. after https://doi.org/10.1002/jgt.1007)
  [via Table of the largest known graphs of a given diameter and maximal degree - Wikipedia]
• 2026-09-30 Lower bound: $170$
  Rajiv, R. (2026). New lower bounds for the degree-diameter problem from Cayley graphs of metacyclic groups, orbit graphs and clique substitution. Zenodo.
  [via Comellas, F. The (Degree, Diameter) Problem for Graphs (records table), last changed November 2, 2026]