Distance-regular graph
In the mathematical field of graph theory, a distance-regular graph is a regular graph such that for any two vertices v and w, the number of vertices at distance j from v and at distance k from w depends only upon j, k, and the distance between v and w.
| Graph families defined by their automorphisms | ||||
|---|---|---|---|---|
| distance-transitive | → | distance-regular | ← | strongly regular |
| ↓ | ||||
| symmetric (arc-transitive) | ← | t-transitive, t ≥ 2 | skew-symmetric | |
| ↓ | ||||
| (if connected) vertex- and edge-transitive |
→ | edge-transitive and regular | → | edge-transitive |
| ↓ | ↓ | ↓ | ||
| vertex-transitive | → | regular | → | (if bipartite) biregular |
| ↑ | ||||
| Cayley graph | ← | zero-symmetric | asymmetric | |
Some authors exclude the complete graphs and disconnected graphs from this definition.
Every distance-transitive graph is distance-regular. Indeed, distance-regular graphs were introduced as a combinatorial generalization of distance-transitive graphs, having the numerical regularity properties of the latter without necessarily having a large automorphism group.
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