Let 
denote the graph with vertex set 
, where 
is the 
-hypercube and two vertices are
adjacent in 
iff they are at distance 
in 
. 
is therefore the graph square
of the hypercube graph 
.
is not connected, but it contains
two isomorphic components on 
vertices, each of which is called a halved 
-cube graph, half cube graph, the halved 
-cube, or sometimes the 
-demi-cube graph (Steinerberger 2023). The most common notation
is 
(Godsil 2004), but Steinerberger (2023) uses 
.
The halved 
-cube
graph can also be defined as the graph on the binary vectors of length 
with even weight, where two such vectors are adjacent if and
only if their sum has weight two (Godsil 2004), or as the 2nd graph
power (i.e., the graph square) of 
, where 
denotes the 
-hypercube graph.
Embeddings for small-order halved graphs are illustrated above and special cases are summarized in the table below.
 | name |
| 1 | singleton graph  |
| 2 | path graph  |
| 3 | tetrahedral graph  |
| 4 | 16-cell graph |
| 5 | graph complement of the Clebsch graph |
The 5-halved cube graph is the graph complement of the Clebsch graph. It has Lovász
number 
(Fung 2011, p. 34). Note that Brouwer et al. (1989, pp. 104 and
224) confusingly use the term "Clebsch graph" to refer to the halved 5-cube
graph instead of the folded 5-cube graph meant
by other authors.
The 6-halved cube graph is a distance-regular graph with intersection array 
, and therefore also a Taylor
graph.
The chromatic numbers of the 
-halved cube graphs for 
, 5, ... are 4, 8, 8, 8, 8, 13 or 14, [13, 15], 
, 
, ... (Godsil 2004, p. 67; typos corrected). Brouwer
agrees that 
has chromatic number 4 and gives its independence
number as 5.
The independence numbers of the 
-halved cube graphs for 
, 2, ... are 1, 1, 1, 2, 2, 4, 8, 16, 20, 40, 72, 144, ...,
where values for 
to 12 are from Godsil (2004, p. 67). This sequence appears identical to the
error-correcting coding function 
(OEIS A005864;
E. W. Weisstein, Dec. 31, 2015).
The domination numbers of 
-halved cube graphs for 
, 2, ... are 1, 1, 1, 2, 2, 2, 4, 7, 12, ..., which agrees
with OEIS A029866 for known terms (E. Weisstein,
Aug. 31, 2016).
