Distance-transitive graph

The Biggs-Smith graph, the largest 3-regular distance-transitive graph.
Graph families defined by their automorphisms
distance-transitivedistance-regularstrongly regular
symmetric (arc-transitive)t-transitive, t  2skew-symmetric
(if connected)
vertex- and edge-transitive
edge-transitive and regularedge-transitive
vertex-transitiveregular(if bipartite)
Cayley graphzero-symmetricasymmetric

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.


Early works

External links

This article is issued from Wikipedia - version of the 1/31/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.