The 16-cell graph is the skeleton of the 16-cell. It is the 4-cocktail party graph 
, which is a complete
multipartite graph, and is isomorphic to the circulant
graph 
.
It is a 6-regular graph of girth
3 and graph diameter 2. Its graph
spectrum is 
,
so it is an integral graph. Its cycle
polynomial is
 |
(OEIS A167982).
The 16-cell graph has five LCF notations: one of order 8, one of order 4, two of order 3, and one of order 2. The corresponding drawings are illustrated above.
The 16-cell graph has rectilinear crossing number 8, which is 2 greater than its graph crossing number of 6, making it one of exactly two minimally curvy graphs of graph order 8. The corresponding drawings are illustrated above.
Its projective plane crossing number is 4.
The 16-cell graph is the graph square of the cubical graph 
and the graph cube of the cycle
graph 
.
When drawn in three-space, it is a cube with an "X"
connecting diagonally opposite vertices on each face, and therefore can be viewed as a "crossed cube"
graph.
It is a toroidal graph, as illustrated above. The left-hand drawing shows an graph embedding in a fundamental region whose paired boundary sides are identified to form a torus. The right-hand drawing shows a finite patch of the corresponding periodic lift, illustrating how edges continue across these boundaries and where corresponding vertices in different regions represent the same vertex on the torus.
The 16-cell graph is implemented in the Wolfram Language as GraphData["SixteenCellGraph"].
."