The 120-cell graph is the skeleton of the 120-cell. It has 600 vertices and 1200 edges.
Several drawings are illustrated above. It is a quartic
graph of girth 5 and graph
diameter 15. The numbers of vertices at graph
distance 
,
1, 2, ... from a given graph vertex on the 120-cell
graph are 1, 4, 12, 24, 36, 52, 68, 76, 78, 72, 64, 56, 40, 12, 4, and 1 (OEIS A108997). The 120-cell graph has graph
spectrum
![4^1(alpha,beta,gamma)^(25)(mu,eta,sigma)^(36)[1/2(3+/-sqrt(13))]^(16)[1/2(-1+/-sqrt(21))]^(16)(-1+/-sqrt(2))^(48)[1/2(5+/-sqrt(5))]^91^(40)0^(18)(-1)^8(-2)^8(-3phi+2)^4(3phi-1)^4(+/-sqrt(5))^(24)phi^(24)(1-phi)^(24)(phi-2)^(30)(-1-phi)^(30),](/images/equations/120-CellGraph/NumberedEquation1.svg) |
where 
,
, and 
are the real roots of 
, 
, 
,
and 
are the roots of 
, and 
is the golden
ratio. The 120-cell graph is implemented in the Wolfram
Language as GraphData["HundredTwentyCellGraph"].
The independence number of the 120-cell graph is 220 (Debroni et al. 2013) and its chromatic number is 3 (S. Wagon and R. Pratt, pers. comm., Dec. 2, 2011). R. Pratt has also found a balanced vertex coloring with three colors having 200 vertices of each color.
Its graph automorphism group has order 
(Buekenhout and Parker 1998).
." Disc. Math. 186, 69-94, 1998.Debroni,
S.; Delisle, E.; Myrvold, W.; Sethi, A.; Whitney, J.; Woodcock, J.; Fowler, P. W.;
de La Vaissière, B.; and Deza, M. "Maximum Independent Sets of the 120-Cell
and Other Regular Polytopes." Ars Math. Contemp. 6, 197-210, 2013.