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).

(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 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.

Buckley, F. and Harary, F. "On the Euclidean Dimension of a Wheel." Graphs and Combin.4, 23-30, 1988.Erdős,
P.; Harary, F.; and Tutte, W. T. "On the Dimension of a Graph." Mathematika12,
118-122, 1965.Frankl, N.; Kupavskii, A.; Swanepoel, K. J. "Embedding
Graphs in Euclidean Space." 12 Feb 2018. https://arxiv.org/abs/1802.03092.