Distance-transitive graph

The Biggs-Smith graph, the largest 3-regular distance-transitive graph.
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

In the mathematical field of graph theory, a distance-transitive graph is a graph such that, given any two vertices v and w at any distance i, and any other two vertices x and y at the same distance, there is an automorphism of the graph that carries v to x and w to y. Distance-transitive graphs were first defined in 1971 by Norman L. Biggs and D. H. Smith.

A distance-transitive graph is interesting partly because it has a large automorphism group. Some interesting finite groups are the automorphism groups of distance-transitive graphs, especially of those whose diameter is 2.

Examples

Some first examples of families of distance-transitive graphs include:

Classification of cubic distance-transitive graphs

After introducing them in 1971, Biggs and Smith showed that there are only 12 finite connected trivalent distance-transitive graphs. These are:

Graph name Vertex count Diameter Girth Intersection array
Tetrahedral graph or complete graph K4413{3;1}
complete bipartite graph K3,3624{3,2;1,3}
Petersen graph1025{3,2;1,1}
Cubical graph834{3,2,1;1,2,3}
Heawood graph1436{3,2,2;1,1,3}
Pappus graph1846{3,2,2,1;1,1,2,3}
Coxeter graph2847{3,2,2,1;1,1,1,2}
Tutte–Coxeter graph3048{3,2,2,2;1,1,1,3}
Dodecahedral graph2055{3,2,1,1,1;1,1,1,2,3}
Desargues graph2056{3,2,2,1,1;1,1,2,2,3}
Biggs-Smith graph10279{3,2,2,2,1,1,1;1,1,1,1,1,1,3}
Foster graph90810{3,2,2,2,2,1,1,1;1,1,1,1,2,2,2,3}

Relation to distance-regular graphs

Every distance-transitive graph is distance-regular, but the converse is not necessarily true.

In 1969, before publication of the Biggs–Smith definition, a Russian group led by Georgy Adelson-Velsky showed that there exist graphs that are distance-regular but not distance-transitive. The smallest distance-regular graph that is not distance-transitive is the Shrikhande graph, with 16 vertices and degree 6. The only graph of this type with degree three is the 126-vertex Tutte 12-cage. Complete lists of distance-transitive graphs are known for some degrees larger than three, but the classification of distance-transitive graphs with arbitrarily large vertex degree remains open.

References

Early works
Surveys