Distance-transitive graph

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

Graph families defined by their automorphisms
distance-transitive	$\boldsymbol{\rightarrow}$	distance-regular	$\boldsymbol{\leftarrow}$	strongly regular
$\boldsymbol{\downarrow}$
symmetric (arc-transitive)	$\boldsymbol{\leftarrow}$	t-transitive, t ≥ 2		skew-symmetric
$\boldsymbol{\downarrow}$
(if connected) vertex- and edge-transitive	$\boldsymbol{\rightarrow}$	edge-transitive and regular	$\boldsymbol{\rightarrow}$	edge-transitive
$\boldsymbol{\downarrow}$		$\boldsymbol{\downarrow}$		$\boldsymbol{\downarrow}$
vertex-transitive	$\boldsymbol{\rightarrow}$	regular	$\boldsymbol{\rightarrow}$	(if bipartite) biregular
$\boldsymbol{\uparrow}$
Cayley graph	$\boldsymbol{\leftarrow}$	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.

A distance transitive graph is vertex transitive and symmetric as well as distance regular.

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.

Distance-transitive graphs were first defined in 1971 by Norman L. Biggs and D. H. Smith, who showed that there are only 12 finite trivalent distance-transitive graphs. These are:

Graph name	Vertex count	Diameter	Girth	Intersection array
complete graph K4	4	1	3	{3;1}
complete bipartite graph K3,3	6	2	4	{3,2;1,3}
Petersen graph	10	2	5	{3,2;1,1}
Graph of the cube	8	3	4	{3,2,1;1,2,3}
Heawood graph	14	3	6	{3,2,2;1,1,3}
Pappus graph	18	4	6	{3,2,2,1;1,1,2,3}
Coxeter graph	28	4	7	{3,2,2,1;1,1,1,2}
Tutte–Coxeter graph	30	4	8	{3,2,2,2;1,1,1,3}
Graph of the dodecahedron	20	5	5	{3,2,1,1,1;1,1,1,2,3}
Desargues graph	20	5	6	{3,2,2,1,1;1,1,2,2,3}
Biggs-Smith graph	102	7	9	{3,2,2,2,1,1,1;1,1,1,1,1,1,3}
Foster graph	90	8	10	{3,2,2,2,2,1,1,1;1,1,1,1,2,2,2,3}

Independently in 1969 a Russian group led by Georgy Adelson-Velsky showed that there exist graphs that are distance-regular but not distance-transitive. The only graph of this type with degree three is the 126-vertex Tutte 12-cage. The smallest distance-regular graph that is not distance-transitive is the Shrikhande graph. 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.

The simplest asymptotic family of examples of distance-transitive graphs is the Hypercube graphs. Other families are the folded cube graphs and the square rook's graphs. All three of these families have arbitrarily high degree.

References

Early works

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Surveys

• , chapter 20.
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• , chapter 7.
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• , section 4.5.
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External links

• Weisstein, Eric W., "Distance-Transitive Graph", MathWorld.