Transitive Graphs

This page lists all the transitive graphs on up to 31 vertices, divided into every natural category that you could possibly want. The data was prepared by Brendan McKay and Gordon Royle and thereby subsumes the various previous works of these authors in which the catalogue of transitive graphs was extended to 19 (McKay's thesis), 24 (Royle's thesis) and then to 26 (McKay & Royle). The current extension has been made possible by the work of Alexander Hulpke who has constructed all the transitive groups of degree up to 31. Using these groups a complete recomputation of the graphs was performed. Fortunately for us, the recomputation confirmed that our original numbers were correct.

Warning:

At this stage the numbers are guaranteed correct only up to 26 vertices. The transitive groups on 24, 27, 28 and 30 vertices have not yet been checked by Hulpke. As 24 vertices is covered by the earlier work, only 27, 28 and 30 are not checked. However an error in the groups can only affect the graphs if somehow a minimal transitive group has been omitted, so this is unlikely.



The Graphs

All graphs are in the graph6 format.

You may prefer to download the entire lot as a single file

  1. Transitive Graphs up to 31 vertices [32Mb]
  2. Non-Cayley Transitive Graphs up to 31 vertices [81K]

Transitive graphs on up to 31 vertices
n Transitive Cayley Non-Cayley Connected Transitive Connected Cayley Connected Non-Cayley
1 1 1 0 1 1 0
2 2 2 0 1 1 0
3 2 2 0 1 1 0
4 4 4 0 2 2 0
5 3 3 0 2 2 0
6 8 8 0 5 5 0
7 4 4 0 3 3 0
8 14 14 0 10 10 0
9 9 9 0 7 7 0
10 22 20 2 18 16 2
11 8 8 0 7 7 0
12 74 74 0 64 64 0
13 14 14 0 13 13 0
14 56 56 0 51 51 0
15 48 44 4 44 40 4
16 286 278 8 272 264 8
17 36 36 0 35 35 0
18 380 376 4 365 361 4
19 60 60 0 59 59 0
20 1214 1132 82 1190 1110 80
21 240 240 0 235 235 0
22 816 816 0 807 807 0
23 188 188 0 187 187 0
24 15506 15394 112 15422 15310 112
25 464 464 0 461 461 0
26 4236 4104 132 4221 4089 132
27 1434 1434 0 1425 1425 0
28 25850 25784 66 25792 25726 66
29 1182 1182 0 1181 1181 0
30 46308 45184 1124 46236 45118 1118
31 2192 2192 0 2191 2191 0

Transitive Graphs by Degree

The following tables list them all by degree.

Transitive graphs on 1 vertices

Degree 0
Cayley 1
Conn. Cayley 1

Transitive graphs on 2 vertices

Degree 0 1
Cayley 1 1
Conn. Cayley 0 1

Transitive graphs on 3 vertices

Degree 0 2
Cayley 1 1
Conn. Cayley 0 1

Transitive graphs on 4 vertices

Degree 0 1 2 3
Cayley 1 1 1 1
Conn. Cayley 0 0 1 1

Transitive graphs on 5 vertices

Degree 0 2 4
Cayley 1 1 1
Conn. Cayley 0 1 1

Transitive graphs on 6 vertices

Degree 0 1 2 3 4 5
Cayley 1 1 2 2 1 1
Conn. Cayley 0 0 1 2 1 1

Transitive graphs on 7 vertices

Degree 0 2 4 6
Cayley 1 1 1 1
Conn. Cayley 0 1 1 1

Transitive graphs on 8 vertices

Degree 0 1 2 3 4 5 6 7
Cayley 1 1 2 3 3 2 1 1
Conn. Cayley 0 0 1 2 3 2 1 1

Transitive graphs on 9 vertices

Degree 0 2 4 6 8
Cayley 1 2 3 2 1
Conn. Cayley 0 1 3 2 1

Transitive graphs on 10 vertices

Degree 0 1 2 3 4 5 6 7 8 9
Transitive 1 1 2 3 4 4 3 2 1 1
Cayley 1 1 2 2 4 4 2 2 1 1
Non-Cayley 0 0 0 1 0 0 1 0 0 0
Conn. Transitive 0 0 1 3 3 4 3 2 1 1
Conn. Cayley 0 0 1 2 3 4 2 2 1 1
Conn. Non-Cayley 0 0 0 1 0 0 1 0 0 0

Transitive graphs on 11 vertices

Degree 0 2 4 6 8 10
Cayley 1 1 2 2 1 1
Conn. Cayley 0 1 2 2 1 1

Transitive graphs on 12 vertices

Degree 0 1 2 3 4 5 6 7 8 9 10 11
Cayley 1 1 4 7 11 13 13 11 7 4 1 1
Conn. Cayley 0 0 1 4 10 12 13 11 7 4 1 1

Transitive graphs on 13 vertices

Degree 0 2 4 6 8 10 12
Cayley 1 1 3 4 3 1 1
Conn. Cayley 0 1 3 4 3 1 1

Transitive graphs on 14 vertices

Degree 0 1 2 3 4 5 6 7 8 9 10 11 12 13
Cayley 1 1 2 3 6 6 9 9 6 6 3 2 1 1
Conn. Cayley 0 0 1 3 5 6 8 9 6 6 3 2 1 1

Transitive graphs on 15 vertices

Degree 0 2 4 6 8 10 12 14
Transitive 1 3 8 12 12 8 3 1
Cayley 1 3 7 11 11 7 3 1
Non-Cayley 0 0 1 1 1 1 0 0
Conn. Transitive 0 1 7 12 12 8 3 1
Conn. Cayley 0 1 6 11 11 7 3 1
Conn. Non-Cayley 0 0 1 1 1 1 0 0

Transitive graphs on 16 vertices

Degree 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Transitive 1 1 3 7 16 27 40 48 48 40 27 16 7 3 1 1
Cayley 1 1 3 7 15 26 39 47 47 39 26 15 7 3 1 1
Non-Cayley 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0
Conn. Transitive 0 0 1 4 13 25 39 47 48 40 27 16 7 3 1 1
Conn. Cayley 0 0 1 4 12 24 38 46 47 39 26 15 7 3 1 1
Conn. Non-Cayley 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0

Transitive graphs on 17 vertices

Degree 0 2 4 6 8 10 12 14 16
Cayley 1 1 4 7 10 7 4 1 1
Conn. Cayley 0 1 4 7 10 7 4 1 1

Transitive graphs on 18 vertices

Degree 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Transitive 1 1 4 7 16 24 38 45 54 54 45 38 24 16 7 4 1 1
Cayley 1 1 4 7 16 23 38 45 53 53 45 38 23 16 7 4 1 1
Non-Cayley 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0
Conn. Transitive 0 0 1 5 12 23 36 45 53 54 45 38 24 16 7 4 1 1
Conn. Cayley 0 0 1 5 12 22 36 45 52 53 45 38 23 16 7 4 1 1
Conn. Non-Cayley 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 0 0 0

Transitive graphs on 19 vertices

Degree 0 2 4 6 8 10 12 14 16 18
Cayley 1 1 4 10 14 14 10 4 1 1
Conn. Cayley 0 1 4 10 14 14 10 4 1 1

Transitive graphs on 20 vertices

Degree 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Transitive 1 1 4 11 28 47 83 115 149 168 168 149 115 83 47 28 11 4 1 1
Cayley 1 1 4 8 24 43 76 107 142 160 160 142 107 76 43 24 8 4 1 1
Non-Cayley 0 0 0 3 4 4 7 8 7 8 8 7 8 7 4 4 3 0 0 0
Conn. Transitive 0 0 1 7 24 43 80 113 148 167 168 149 115 83 47 28 11 4 1 1
Conn. Cayley 0 0 1 5 20 39 74 105 141 159 160 142 107 76 43 24 8 4 1 1
Conn. Non-Cayley 0 0 0 2 4 4 6 8 7 8 8 7 8 7 4 4 3 0 0 0

Transitive graphs on 21 vertices

Degree 0 2 4 6 8 10 12 14 16 18 20
Cayley 1 3 11 29 48 56 48 29 11 3 1
Conn. Cayley 0 1 10 28 48 56 48 29 11 3 1

Transitive graphs on 22 vertices

Degree 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Cayley 1 1 2 3 11 18 38 52 79 94 109 109 94 79 52 38 18 11 3 2 1 1
Conn. Cayley 0 0 1 3 9 18 36 52 78 94 108 109 94 79 52 38 18 11 3 2 1 1

Transitive graphs on 23 vertices

Degree 0 2 4 6 8 10 12 14 16 18 20 22
Cayley 1 1 5 15 30 42 42 30 15 5 1 1
Conn. Cayley 0 1 5 15 30 42 42 30 15 5 1 1

Transitive graphs on 24 vertices

Degree 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
Transitive 1 1 6 20 74 167 373 652 1064 1473 1858 2064 2064 1858 1473 1064 652 373 167 74 20 6 1 1
Cayley 1 1 6 20 74 166 368 645 1055 1462 1847 2052 2052 1847 1462 1055 645 368 166 74 20 6 1 1
Non-Cayley 0 0 0 0 0 1 5 7 9 11 11 12 12 11 11 9 7 5 1 0 0 0 0 0
Conn. Transitive 0 0 1 11 60 152 359 640 1057 1469 1857 2063 2064 1858 1473 1064 652 373 167 74 20 6 1 1
Conn. Cayley 0 0 1 11 60 151 354 633 1048 1458 1846 2051 2052 1847 1462 1055 645 368 166 74 20 6 1 1
Conn. Non-Cayley 0 0 0 0 0 1 5 7 9 11 11 12 12 11 11 9 7 5 1 0 0 0 0 0

Transitive graphs on 25 vertices

Degree 0 2 4 6 8 10 12 14 16 18 20 22 24
Cayley 1 2 9 25 57 86 104 86 57 25 9 2 1
Conn. Cayley 0 1 8 25 57 86 104 86 57 25 9 2 1

Transitive graphs on 26 vertices

Degree 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Transitive 1 1 2 5 16 29 71 117 204 286 397 466 523 523 466 397 286 204 117 71 29 16 5 2 1 1
Cayley 1 1 2 4 16 27 68 110 198 278 384 452 511 511 452 384 278 198 110 68 27 16 4 2 1 1
Non-Cayley 0 0 0 1 0 2 3 7 6 8 13 14 12 12 14 13 8 6 7 3 2 0 1 0 0 0
Conn. Transitive 0 0 1 5 13 29 67 117 201 286 396 466 522 523 466 397 286 204 117 71 29 16 5 2 1 1
Conn. Cayley 0 0 1 4 13 27 64 110 195 278 383 452 510 511 452 384 278 198 110 68 27 16 4 2 1 1
Conn. Non-Cayley 0 0 0 1 0 2 3 7 6 8 13 14 12 12 14 13 8 6 7 3 2 0 1 0 0 0

Transitive graphs on 27 vertices

Degree 0 2 4 6 8 10 12 14 16 18 20 22 24 26
Cayley 1 3 16 54 128 223 292 292 223 128 54 16 3 1
Conn. Cayley 0 1 13 52 127 223 292 292 223 128 54 16 3 1

Transitive graphs on 28 vertices

Degree 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Transitive 1 1 4 10 34 79 204 392 748 1202 1803 2373 2898 3176 3176 2898 2373 1803 1202 748 392 204 79 34 10 4 1 1
Cayley 1 1 4 9 33 77 199 391 747 1196 1800 2372 2893 3169 3169 2893 2372 1800 1196 747 391 199 77 33 9 4 1 1
Non-Cayley 0 0 0 1 1 2 5 1 1 6 3 1 5 7 7 5 1 3 6 1 1 5 2 1 1 0 0 0
Conn. Transitive 0 0 1 6 28 73 195 383 742 1196 1800 2371 2897 3175 3176 2898 2373 1803 1202 748 392 204 79 34 10 4 1 1
Conn. Cayley 0 0 1 5 27 71 190 382 741 1190 1797 2370 2892 3168 3169 2893 2372 1800 1196 747 391 199 77 33 9 4 1 1
Conn. Non-Cayley 0 0 0 1 1 2 5 1 1 6 3 1 5 7 7 5 1 3 6 1 1 5 2 1 1 0 0 0

Transitive graphs on 29 vertices

Degree 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28
Cayley 1 1 7 26 73 143 217 246 217 143 73 26 7 1 1
Conn. Cayley 0 1 7 26 73 143 217 246 217 143 73 26 7 1 1

Transitive graphs on 30 vertices

Degree 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
Transitive 1 1 6 15 52 105 259 473 935 1483 2337 3183 4165 4853 5286 5286 4853 4165 3183 2337 1483 935 473 259 105 52 15 6 1 1
Cayley 1 1 6 11 43 90 238 445 890 1429 2277 3115 4078 4764 5204 5204 4764 4078 3115 2277 1429 890 445 238 90 43 11 6 1 1
Non-Cayley 0 0 0 4 9 15 21 28 45 54 60 68 87 89 82 82 89 87 68 60 54 45 28 21 15 9 4 0 0 0
Conn. Transitive 0 0 1 10 40 100 244 471 922 1482 2329 3183 4162 4853 5285 5286 4853 4165 3183 2337 1483 935 473 259 105 52 15 6 1 1
Conn. Cayley 0 0 1 7 32 85 225 443 878 1428 2270 3115 4075 4764 5203 5204 4764 4078 3115 2277 1429 890 445 238 90 43 11 6 1 1
Conn. Non-Cayley 0 0 0 3 8 15 19 28 44 54 59 68 87 89 82 82 89 87 68 60 54 45 28 21 15 9 4 0 0 0

Transitive graphs on 31 vertices

Degree 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Cayley 1 1 7 31 91 201 335 429 429 335 201 91 31 7 1 1
Conn. Cayley 0 1 7 31 91 201 335 429 429 335 201 91 31 7 1 1


Home Gordon Royle, gordon@cs.uwa.edu.au, January 1997