A maximally dense unit-distance graph is a unit-distance graph having the maximum possible number of edges (i.e., maximum graph density) for a given number of vertices. Finding a maximally dense unit-distance graph is equivalent to solving the Erdős unit distance problem.
Special cases are summarized in the following table, where 
is the vertex count.
 | maximally dense unit-distance graphs |
| 1 | singleton
graph  |
| 2 | path graph  |
| 3 | triangle graph  |
| 4 | diamond graph  |
| 5 | gem graph  |
| 6 | prism graph  , fan graph  ,
8-Beineke graph, 2-triangular
graph |
| 7 | wheel
graph  |
| 8 | (8,10)-self-complementary graph, Johnson
skeleton graph  , (unnamed) |
| 9 | generalized quadrangle  |
Alexeev et al. (2025) determined all nonisomorphic maximally dense unit-distance graph up to 21 vertices. Their counts for 
, 2, ... are 1, 1, 1, 1, 1, 4, 1, 3, 1, 1, 2, 1, 1, 2, 1,
1, 7, 16, 3, 1, 5, ... (OEIS A385657). All
but one of these graphs were previously discovered by Engel et al. (2025),
with the exception being a graph on 17 vertices whose unit-distance embedding does
not reside within the Moser ring they searched (Alexeev et al. 2025). The
maximally dense unit-distance graphs for 
to 15 are illustrated above.
The 
th
maximally dense unit-distance graph on 
vertices will be implemented in a future version of
the Wolfram Language as GraphData[
"MaximallyDenseUnitDistance",
n,
k
].
Points." Amer. Math. Monthly 53, 248-250,
1946.Schade, C. "Exakte Maximale Anzahlen Gleicher Abstände."
Thesis. Braunschweig, Germany: TU Braunschweig, 1993.Sloane, N. J. A.
Sequences