The dimension 
,
also called the Euclidean dimension (e.g., Buckley and Harary 1988) of a graph, is
the smallest dimension 
of Euclidean 
-space
in which 
can be embedded with every edge length equal to 1 and every vertex position distinct
(but where edges may cross or overlap and points may lie on edges that are not incident
on them; Erdős et al. 1965).
Any connected graph with maximum vertex degree 
has graph dimension at most 
, with the exception of the utility
graph 
(Frankl et al. 2018). Furthermore, any graph with chromatic
number 
has graph dimension at most 
. This can be seen by partitioning the space into 
orthogonal two-dimensional planes, then in each plane placing
the vertices with one color on a circle with radius 
centred on the plane's origin (so all points have
a squared norm of 1/2) (J. Tan, pers. comm., Oct. 26, 2021).
For any nonempty graph 
,
the graph Cartesian product satisfies
 |
(1)
|
(Erdős et al. 1965, Buckley and Harary 1988). While the theorem is stated as holding for "any" graph by both references, if 
is taken as the empty graph 
, then 
is isomorphic to the ladder
rung graph 
.
Yet 
for 
(since vertices may not overlap by the definition of graph dimension) and 
since each of the 
paths can be placed on a 1-dimensional line.
The singleton graph 
has graph dimension 
, the path graphs 
for 
have graph dimension 
, and in general, any graph with dimension 2 or less
is said to be a unit-distance graph.
The dimension of the complete graph 
is 
(Erdős et al. 1965, Buckley and Harary
1988). For the complete bipartite graph 
with 
,
 |
(2)
|
(Erdős et al. 1965, Buckley and Harary 1988).
The dimension of 
is given by 
for 
(Erdős et al. 1965).
The hypercube graph 
has dimension 
for 
(Erdős et al. 1965).
The wheel graph 
has graph dimension 2 for 
(and hence is unit-distance)
and dimension 3 otherwise (and hence is not unit-distance) (Erdős et al.
1965, Buckley and Harary 1988).
The following table summarizes the graph dimensions for various families of parametrized graphs.
