# Named Graphs

When I first became interested in regular maps, I did not know what they were called. But as each one involves a symmetric graph, I realised that I could try to find them by choosing some symmetric graph, maybe from Wikipedia's "gallery of named graphs", and trying to find a way to embed it regularly in a compact 2-manifold.

The page lists some symmetric graphs, and for each which I know how to embed regularly in some manifold, gives a link to a page where I show it as a regular map.

Some named graphs can be embedded nicely in a manifold, but not in a fully regular way. For example, the Franklin graph can be embedded in the projective plane, where it is seen to be a truncated hemi-octahedron, which is not face-transitive. A few of these are listed in the main table of this page, but I have not tried to list irregular named graphs systematically.

## Named Graphs identified with Regular Maps

Name of graph
Picture of graph Valency Vertices Edges Full symmetry
group of graph
Rotational
symmetry group of
regular map
Schläfli symbol
Picture of regular map Faces of regular map
Cube 3 8 12 48

S4×C2

24

S4

{4,3} 6 squares
Cycle graph 2 n n 2n

D2n

n

Cn

{5,2} 2 pentagons
Desargues graph 3 20 30 240

S5×C2

60

A5

C6:{10,3}10 6 decagons
Dodecahedron 3 12 30 120

S5

60

A5

{3,5} 12 pentagons
Dyck graph 3 32 48 192 96

{6,3}(4,4) 16 hexagons
96

S3{8,3} 12 octagons
F26A graph 3 26 39 78 78 {6,3}(2,4) 13 hexagons
Franklin graph 3 12 18 48

S4×C2

26

S4

Not regular. Can be embedded in the projective plane, as a truncated hemi-octahedron, having .. .. 3 squares and 4 hexagons
Heawood graph 3 14 21 336

PGL(2,7)

21

C7⋊C3

S3:{14,3} Not quite regular.

3 14-gons

Icosahedron 3 20 30 120

A5×C2

60

A5

{3,5} 20 triangles
K1 0 0 0 1

S1

1

S1

{0,0} 1 point-bounded face
K2 1 2 1 2

S2

2

S2

{2,1} 1 digon
K3 2 3 3 6

S3

6

S3

{3,2} 2 triangles
K4 3 4 6 24

S4

12

A4

{3,3} 4 triangles
K5 4 5 10 120

S5

20

C5⋊C4

{4,4}(2,1) 5 squares
K6 5 6 15 720

S6

60

A5

C1:{3,5}, the hemi-icosahedron 10 triangles
K7 6 7 21 5,040

S7

42

C7⋊C6

{3,6}(1,3) 14 triangles
K8 7 8 28 40,320

S8

56

(C2×C2×C2)⋊C7

S7:{7,7} and its dual S7:{7,7} 8 heptagons
KnSee a list of complete graphs embedded as regular maps.
K3,3

Utility
or
Thomsen graph

3 6 9 72

(S3×S3)⋊C2

18

D6×C3

{6,3}(0,2) 3 hexagons
K4,4 4 4 8 1,152

(S4×S4)⋊C2

16

C22⋊C4

{4,4}(2,2) 8 squares
16 S3{8,4|4} 4 octagons
K5,5 4 10 25 28,800 50 S6:{10,5} 5 hexagons
K6,6 5 12 36 1,036,000 72 S4:{4,6} 18 squares
K7,7 6 14 49 50,803,200 98 S15:{14,7} 7 14-gons
K2,2,2 4 6 12 96

(S2×S2×S2)⋊S3

12

A4

{3,4} 8 triangles
K3,3,3 6 9 27 1,296

(S3×S3×S3)⋊S3

54

{3,6}(3,3) 18 triangles
K4,4,4 8 12 48 82,944

(S4×S4×S4)⋊S3

96

S3:{3,8}, the dual of the Dyck map 12 octagons
K5,5,5 10 15 75 10,368,000 96

S6:{3,10} 50 triangles
Klein graph 3 56 84 336

PGL(2,7)

168

PSL(2,7)

S3:{7,3}, the Klein map 24 heptagons
Möbius-Kantor graph 3 16 24 96 48

S2:{8,3} 6 octagons
Nauru graph 3 24 36 144

S4×D6

72

{6,3}(0,4) 12 hexagons
Octahedron See K2,2,2 above
Paley order-13 graph 6 13 39 78 78 {3,6}(2,4) 26 triangles
Pappus graph 3 18 27 216 54

{6,3}(3,3) 9 hexagons
Petersen graph 3 10 15 120

S5

60

A5

C1:{5,3}, the hemidodecahedron 6 pentagons
Shrikhande graph 6 16 48 192 96

{3,6}(4,4) 32 triangles
Tetrahedron See K4 above

## Named Graphs not identified with Regular Maps

These either don't exist as regular maps, or do but I don't know how.

 Biggs-Smith graph 3 102 153 2448PSL(2,17) ? Clebsch graphalso called the Greenwood-Gleason graph 5 16 40 1920 (20 squares) Coxeter graph 3 28 42 336PGL(2,7) (12 heptagons) 10-crown 4 10 20 80 (8 pentagons, but this graph is bipartite) Double-star snark 3 30 45 ? 3 30-gons Foster graph 3 90 135 4320 ? Hall-Janko graph ? 100 ? ? ? Higman-Sims graph 22 100 1100 88,704,000HS⋊C2) ? Hoffman-Singleton graph 7 50 175 252,000PSU(3,52)⋊C2 ? Horton graph 3 96 144 96S4×C2×C2 Not the same as {6,3}(0,8) Tutte-Coxeter graph 3 30 45 1,440Aut(S6) ?

## Named Graphs known not to be Regular

More on Regular Maps

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