The following Penrose graphs have been constructed using an inflation/deflation procedure.
One possibility to start with, but there are others as well.
vertices: 16
faces: 10+1
edges: 25
genus: 0
characteristic polynomial:
x4 (20 - 13x2 + x4) (4 - 6x2 + x4)2
dual's characteristic polynomial:
(-4 - 2x + x2) (1 - 4x - 4x2 + x3 + x4)2
chromatic number: 3
lattice group:
(C5 × (((C5 × ((C5 × C5) : C2)) : C2) : C3)) :C2
Already more interesting.
vertices: 41
faces: 30+1
edges: 70
genus: 0
characteristic polynomial:
x11 (-1 + x)4 (1 + x)4 (-24 + 83x2 - 20x4 + x6) (576 - 552x2 + 179x4 - 23x6 + x8)2
dual's characteristic polynomial:
(-1 + x + x2) (8 + 8x - 5x2 - 3x3 + x4) (-41 - 100x + 284x2 + 694x3 + 40x4 - 661x5 - 258x6 + 218x7 + 119x8 - 27x9 - 19x10 + x11 + x12)
chromatic number: 4
lattice group (lossy):
Z(2) × Z(5) 15 × A(15)
After another round of inflation and deflation.
vertices: 116
faces: 90+1
edges: 205
genus: 0
characteristic polynomial:
dual's characteristic polynomial:
chromatic number: 4
lattice group (lossy):
Z(2) 2 × Z(5) 53 × A(53)
We didn't wait long enough until the layouting was settled.
vertices: 321
faces: 260+1
edges: 580
genus: 0
characteristic polynomial:
dual's characteristic polynomial:
chromatic number: 4
lattice group (lossy):
