VertexTransitive Graphs
Last update=21 Feb, 2009 A graph is vertextransitive if every vertex can be mapped to any other vertex by some automorphism, that is, it is symmetric. The graphs listed here were produced by B.D. McKay's graph generation software. They can also be downloaded from G. Royle's Combinatorial Data website, where they are stored in a compressed "g6format". They are available here in G&G binary and text formats. The G&G binary files contain nice drawings of the graphs, illustrating their structure. These drawings are also shown in the G&G Overview. The vertextransitive graphs up to 16 vertices can mostly be described with a simple notation, which is here described. Only the connected vertex transitive graphs are contained here. (But the disconnected ones are here too, disguised as complements of connected graphs.) Notation:
Graph Names, Automorphism GroupsThe vertextransitive graphs are named as VT12_38[96]=~OctahedronxK2. This means VertexTransitive Graph number 38 on 12 points. The automorphism group has order 96 (the number in square brackets). This graph is isomorphic to the complement of OctahedronxK_{2}. The numbering is the same as that of B.D. MacKay and G. Royle.
Download:Note: Stuffit Expander may be necessary to unpack these files. It is a free download.
The 6point Transitive GraphsThere are 5 connected vertextransitive graphs on 6 vertices:C_{6}, K_{3,3}, Prism3=~C_{6}, Octahedron=C_{6}(2)=~3K_{2}, and K_{6}. The connected vertextransitive graphs on 6 vertices.
The 7point Transitive GraphsThere are 3 connected vertextransitive graphs on 7 vertices:C_{7}, C_{7}(2)=~C_{7}, and K_{7}. The connected vertextransitive graphs on 7 vertices.
The 8point Transitive GraphsThere are 10 connected vertextransitive graphs on 8 vertices:C_{8}, C_{8}^{+}, Cube, K_{4,4}, C_{8}(2)=~C_{8}^{+}, ~Cube, ~C_{8}, ~2C_{4}, ~4K_{2}, K_{8}. The connected vertextransitive graphs on 8 vertices.
The 9point Transitive GraphsThere are 7 connected vertextransitive graphs on 9 vertices:C_{9}, C_{9}(3), C_{3}xC_{3}=L(K_{3,3}), ~C_{9}(3), ~3K_{3}, ~C_{9}, K_{9}. The connected vertextransitive graphs on 9 vertices.
The 10point Transitive GraphsThere are 18 connected vertextransitive graphs on 10 vertices:C_{10}, Petersen, C_{10}^{+}, Prism5=C_{5}xK_{2}, ~K_{5}xK_{2}, C_{10}(4), C_{10}(2), K_{5,5}, C_{10}(2,5), C_{10}(4,5), K_{5}xK_{2}, ~Petersen, ~Prism5, ~C_{10}^{+}, ~2C_{5}, ~C_{10}, ~5K_{2}, K_{10}. The connected vertextransitive graphs on 10 vertices.
The 11point Transitive GraphsThere are 7 connected vertextransitive graphs on 11 vertices:C_{11}, C_{11}(3), C_{11}(2), C_{11}(2,5)=~C_{11}(2), C_{11}(4,5)=~C_{11}(3), ~C_{11}, K_{11}. The connected vertextransitive graphs on 11 vertices.
The 12point Transitive GraphsThere are 64 connected vertextransitive graphs on 12 vertices:The connected vertextransitive graphs on 12 vertices.
The 13point Transitive GraphsThere are 13 connected vertextransitive graphs on 13 vertices:C_{13}, C_{13}(5), C_{13}(3), C_{13}(2), C_{13}(3,4)=Paley(13), C_{13}(2,5), C_{13}(2,6), ~C_{13}(2,5), ~C_{13}(5), ~C_{13}(2), ~C_{13}(3), ~C_{13}, K_{11}. The connected vertextransitive graphs on 13 vertices.
The 14point Transitive GraphsThere are 51 connected vertextransitive graphs on 14 vertices:The connected vertextransitive graphs on 14 vertices.
The 15point Transitive GraphsThere are 44 connected vertextransitive graphs on 15 vertices:The connected vertextransitive graphs on 15 vertices.
The 16point Transitive GraphsThere are 272 connected vertextransitive graphs on 16 vertices.The connected vertextransitive graphs on 16 vertices.
The 17point Transitive GraphsThere are 35 connected vertextransitive graphs on 17 vertices.The connected vertextransitive graphs on 17 vertices. Three are selfcomplementary, including the Paley graph.
The 19point Transitive GraphsThere are 59 connected vertextransitive graphs on 19 vertices.The connected vertextransitive graphs on 19 vertices.
Locate to the ftp server to download the text form of these graphs. The text format does not contain drawings of the graphs. It contains adjacencies only.
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