The degree-diameter problem asks for the largest possible vertex count 
of a simple graph with maximum
vertex degree at most 
and graph diameter at
most 
.
For 
,
counting vertices at successive distances from a
fixed graph vertex gives the Moore
bound
 |
(1)
|
Equality is attained by complete graphs when 
and by odd cycle graphs when 
. For 
and 
, equality is possible only for 
, 7, and possibly 57. For 
and 
, the Moore bound is not attained. The equality cases
are Moore graphs (Bermond et al. 1992).
Even when equality is impossible, graphs can approach the bound asymptotically. Cames van Batenburg and Korsky (2026) proved that for every
fixed positive integer 
,
 |
(2)
|
Their proof constructs graphs attaining the Moore bound asymptotically. The authors credit GPT-5.6 Sol with suggesting the decisive separation
of a construction into odd and even parts, which they developed into the proof. This
asymptotic result does not determine 
for every fixed pair of parameters.
-Graphs."
Disc. Appl. Math. 37/38, 575-577, 1992.Bozóki,
S.; Szádoczki, Z.; and Tekile, H. A. "Filling in Pattern Designs
for Incomplete Pairwise Comparison Matrices: (Quasi-)Regular Graphs with Minimal
Diameter." 31 May 2020.