Difference between revisions of "Template:Graph families defined by their automorphisms"
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imported>David Eppstein (add a couple more) |
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<tr><td style="vertical-align:middle;">[[distance-transitive graph|distance-transitive]]</td><td style="vertical-align:middle;"><math>\boldsymbol{\rightarrow}</math></td><td style="vertical-align:middle;">[[distance-regular graph|distance-regular]]</td><td style="vertical-align:middle;"><math>\boldsymbol{\leftarrow}</math></td><td style="vertical-align:middle;">[[strongly regular graph|strongly regular]]</td></tr> | <tr><td style="vertical-align:middle;">[[distance-transitive graph|distance-transitive]]</td><td style="vertical-align:middle;"><math>\boldsymbol{\rightarrow}</math></td><td style="vertical-align:middle;">[[distance-regular graph|distance-regular]]</td><td style="vertical-align:middle;"><math>\boldsymbol{\leftarrow}</math></td><td style="vertical-align:middle;">[[strongly regular graph|strongly regular]]</td></tr> | ||
<tr><td><math>\boldsymbol{\downarrow}</math></td><td></td><td></td></tr> | <tr><td><math>\boldsymbol{\downarrow}</math></td><td></td><td></td></tr> | ||
| − | <tr><td style="vertical-align:middle;">[[symmetric graph|symmetric (arc-transitive)]]</td><td style="vertical-align:middle;"><math>\boldsymbol{\leftarrow}</math></td><td style="vertical-align:middle;">[[symmetric graph|''t''-transitive, ''t'' ≥ 2]]</td></tr> | + | <tr><td style="vertical-align:middle;">[[symmetric graph|symmetric (arc-transitive)]]</td><td style="vertical-align:middle;"><math>\boldsymbol{\leftarrow}</math></td><td style="vertical-align:middle;">[[symmetric graph|''t''-transitive, ''t'' ≥ 2]]</td><td></td><td>[[Semi-symmetric graph|semi-symmetric]]</td></tr> |
| − | <tr><td><math>\boldsymbol{\downarrow}</math></td><td></td><td></td></tr> | + | <tr><td><math>\boldsymbol{\downarrow}</math></td><td></td><td></td><td></td><td><math>\boldsymbol{\downarrow}</math></td></tr> |
<tr><td style="vertical-align:middle;"><sub>(if connected)</sub><br/>[[half-transitive graph|vertex- and edge-transitive]]</td><td style="vertical-align:middle;"><math>\boldsymbol{\rightarrow}</math></td><td style="vertical-align:middle;">[[semi-symmetric graph|edge-transitive and regular]]</td><td style="vertical-align:middle;"><math>\boldsymbol{\rightarrow}</math></td><td style="vertical-align:middle;">[[edge-transitive graph|edge-transitive]]</td></tr> | <tr><td style="vertical-align:middle;"><sub>(if connected)</sub><br/>[[half-transitive graph|vertex- and edge-transitive]]</td><td style="vertical-align:middle;"><math>\boldsymbol{\rightarrow}</math></td><td style="vertical-align:middle;">[[semi-symmetric graph|edge-transitive and regular]]</td><td style="vertical-align:middle;"><math>\boldsymbol{\rightarrow}</math></td><td style="vertical-align:middle;">[[edge-transitive graph|edge-transitive]]</td></tr> | ||
<tr><td><math>\boldsymbol{\downarrow}</math></td><td></td><td><math>\boldsymbol{\downarrow}</math></td><td></td><td><math>\boldsymbol{\downarrow}</math></td></tr> | <tr><td><math>\boldsymbol{\downarrow}</math></td><td></td><td><math>\boldsymbol{\downarrow}</math></td><td></td><td><math>\boldsymbol{\downarrow}</math></td></tr> | ||
<tr><td style="vertical-align:middle;">[[vertex-transitive graph|vertex-transitive]]</td><td style="vertical-align:middle;"><math>\boldsymbol{\rightarrow}</math></td><td style="vertical-align:middle;">[[regular graph|regular]]</td><td style="vertical-align:middle;"><math>\boldsymbol{\rightarrow}</math></td><td style="vertical-align:middle;"><sub>(if bipartite)</sub><br/>[[biregular graph|biregular]]</td></tr> | <tr><td style="vertical-align:middle;">[[vertex-transitive graph|vertex-transitive]]</td><td style="vertical-align:middle;"><math>\boldsymbol{\rightarrow}</math></td><td style="vertical-align:middle;">[[regular graph|regular]]</td><td style="vertical-align:middle;"><math>\boldsymbol{\rightarrow}</math></td><td style="vertical-align:middle;"><sub>(if bipartite)</sub><br/>[[biregular graph|biregular]]</td></tr> | ||
<tr><td><math>\boldsymbol{\uparrow}</math></td><td></td><td></td></tr> | <tr><td><math>\boldsymbol{\uparrow}</math></td><td></td><td></td></tr> | ||
| − | <tr><td style="vertical-align:middle;">[[Cayley graph]]</td><td></td><td>[[Skew-symmetric graph|skew-symmetric]]</td><td></td><td>[[Asymmetric graph|asymmetric]]</td></tr> | + | <tr><td style="vertical-align:middle;">[[Cayley graph]]</td><td style="vertical-align:middle;"><math>\boldsymbol{\leftarrow}</math><td style="vertical-align:middle;">[[Zero-symmetric graph|zero-symmetric]]</td><td></td><td></td></tr> |
| + | <tr><td> </td><td></td><td></td><td></td><td></td><td></td></tr> | ||
| + | <tr><td></td><td></td><td>[[Skew-symmetric graph|skew-symmetric]]</td><td></td><td>[[Asymmetric graph|asymmetric]]</td></tr> | ||
</table><!-- | </table><!-- | ||
--> | --> | ||
Revision as of 19:54, 11 January 2015
| Graph families defined by their automorphisms | |||||
| distance-transitive | <math>\boldsymbol{\rightarrow}</math> | distance-regular | <math>\boldsymbol{\leftarrow}</math> | strongly regular | |
| <math>\boldsymbol{\downarrow}</math> | |||||
| symmetric (arc-transitive) | <math>\boldsymbol{\leftarrow}</math> | t-transitive, t ≥ 2 | semi-symmetric | ||
| <math>\boldsymbol{\downarrow}</math> | <math>\boldsymbol{\downarrow}</math> | ||||
| (if connected) vertex- and edge-transitive | <math>\boldsymbol{\rightarrow}</math> | edge-transitive and regular | <math>\boldsymbol{\rightarrow}</math> | edge-transitive | |
| <math>\boldsymbol{\downarrow}</math> | <math>\boldsymbol{\downarrow}</math> | <math>\boldsymbol{\downarrow}</math> | |||
| vertex-transitive | <math>\boldsymbol{\rightarrow}</math> | regular | <math>\boldsymbol{\rightarrow}</math> | (if bipartite) biregular | |
| <math>\boldsymbol{\uparrow}</math> | |||||
| Cayley graph | <math>\boldsymbol{\leftarrow}</math> | zero-symmetric | |||
| skew-symmetric | asymmetric | ||||