The pentakis dodecahedral graph is Archimedean dual graph which is the skeleton of the disdyakis triacontahedron. It is implemented in the Wolfram Language as GraphData["PentakisDodecahedralGraph"].

The plots above show the adjacency, incidence, and graph distance matrices for the deltoidal hexecontahedral graph.

The following table summarizes some properties of the graph.

property	value
automorphism group order	120
characteristic polynomial
chromatic number	4
chromatic polynomial	?
claw-free	no
clique number	3
determined by spectrum	?
diameter	5
distance-regular graph	no
dual graph name	truncated icosahedral graph
edge chromatic number	6
edge connectivity	5
edge count	90
Eulerian	no
girth	3
Hamiltonian	yes
Hamiltonian cycle count	?
Hamiltonian path count	?
integral graph	no
independence number	12
line graph	?
perfect matching graph	no
planar	yes
polyhedral graph	yes
polyhedron embedding names	pentakis dodecahedron, small stellated dodecahedron
radius	5
regular	no
square-free	no
traceable	yes
triangle-free	no
vertex connectivity	5
vertex count	32