An icosidodecahedral graph is a quartic symmetric graph on 30 vertices and 60 edges corresponding to the skeleton of the icosidodecahedron (an Archimedean solid), great dodecahemidodecahedron, great icosidodecahedron, great icosihemidodecahedron, small dodecahemidodecahedron, and small icosihemidodecahedron. It is illustrated above in a number of symmetrical drawings.
The icosidodecahedral graph is implemented in the Wolfram Language as GraphData["IcosidodecahedralGraph"].
It has a total of 
distinct (directed) Hamiltonian
cycles, which yield two inequivalent order-6 generalized LCF
notations, one of order 3, two of order 2, and 649 of order 1. The LCF
drawings of order greater than one are illustrated above.
It has graph diameter 5, graph radius 5, and chromatic number 3. Its graph spectrum is 
.
It is the line graph of the dodecahedral graph.
