Distance-transitive graph: Difference between revisions
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{{No footnotes|date=September 2021}}[[Image:BiggsSmith.svg|thumb|right|The [[Biggs-Smith graph]], the largest 3-regular distance-transitive graph.]] |
{{Short description|Graph where any two nodes of equal distance are isomorphic}} |
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{{No footnotes|date=September 2021}}[[Image:BiggsSmith.svg|thumb|right|The [[Biggs-Smith graph]], the largest [[Cubic graph|3-regular]] distance-transitive graph.]] |
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{{Graph families defined by their automorphisms}} |
{{Graph families defined by their automorphisms}} |
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In the [[mathematics|mathematical]] field of [[graph theory]], a '''distance-transitive graph''' is a [[Graph (discrete mathematics)|graph]] such that, given any two vertices |
In the [[mathematics|mathematical]] field of [[graph theory]], a '''distance-transitive graph''' is a [[Graph (discrete mathematics)|graph]] such that, given any two [[Vertex (graph theory)|vertices]] {{mvar|v}} and {{mvar|w}} at any [[Distance (graph theory)|distance]] {{mvar|i}}, and any other two vertices {{mvar|x}} and {{mvar|y}} at the same distance, there is an [[Graph automorphism|automorphism]] of the graph that carries {{mvar|v}} to {{mvar|x}} and {{mvar|w}} to {{mvar|y}}. Distance-transitive graphs were first defined in 1971 by [[Norman L. Biggs]] and D. H. Smith. |
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A distance-transitive graph is interesting partly because it has a large [[automorphism group]]. Some interesting [[finite group]]s are the automorphism groups of distance-transitive graphs, especially of those whose diameter is 2. |
A distance-transitive graph is interesting partly because it has a large [[automorphism group]]. Some interesting [[finite group]]s are the automorphism groups of distance-transitive graphs, especially of those whose diameter is 2. |
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* The [[Johnson graph]]s. |
* The [[Johnson graph]]s. |
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* The [[Grassmann graph]]s. |
* The [[Grassmann graph]]s. |
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* The [[Hamming graph|Hamming Graphs]]. |
* The [[Hamming graph|Hamming Graphs]] (including [[Hypercube graph]]s). |
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* The [[folded cube graph]]s. |
* The [[folded cube graph]]s. |
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* The square [[rook's graph]]s. |
* The square [[rook's graph]]s. |
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* The [[hypercube graph]]s. |
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* The [[Livingstone graph]]. |
* The [[Livingstone graph]]. |
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== Classification of cubic distance-transitive graphs == |
== Classification of cubic distance-transitive graphs == |
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After introducing them in 1971, [[Norman L. Biggs|Biggs]] and Smith showed that there are only 12 finite [[cubic graph|trivalent]] distance-transitive graphs. These are: |
After introducing them in 1971, [[Norman L. Biggs|Biggs]] and Smith showed that there are only 12 finite connected [[cubic graph|trivalent]] distance-transitive graphs. These are: |
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Latest revision as of 11:39, 8 March 2024
This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. (September 2021) |
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[edit]
Some first examples of families of distance-transitive graphs include:
- The Johnson graphs.
- The Grassmann graphs.
- The Hamming Graphs (including Hypercube graphs).
- The folded cube graphs.
- The square rook's graphs.
- The Livingstone graph.
Classification of cubic distance-transitive graphs[edit]
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 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} |
Cubical graph | 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} |
Dodecahedral graph | 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} |
Relation to distance-regular graphs[edit]
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[edit]
- Early works
- Adel'son-Vel'skii, G. M.; Veĭsfeĭler, B. Ju.; Leman, A. A.; Faradžev, I. A. (1969), "An example of a graph which has no transitive group of automorphisms", Doklady Akademii Nauk SSSR, 185: 975–976, MR 0244107.
- Biggs, Norman (1971), "Intersection matrices for linear graphs", Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969), London: Academic Press, pp. 15–23, MR 0285421.
- Biggs, Norman (1971), Finite Groups of Automorphisms, London Mathematical Society Lecture Note Series, vol. 6, London & New York: Cambridge University Press, MR 0327563.
- Biggs, N. L.; Smith, D. H. (1971), "On trivalent graphs", Bulletin of the London Mathematical Society, 3 (2): 155–158, doi:10.1112/blms/3.2.155, MR 0286693.
- Smith, D. H. (1971), "Primitive and imprimitive graphs", The Quarterly Journal of Mathematics, Second Series, 22 (4): 551–557, doi:10.1093/qmath/22.4.551, MR 0327584.
- Surveys
- Biggs, N. L. (1993), "Distance-Transitive Graphs", Algebraic Graph Theory (2nd ed.), Cambridge University Press, pp. 155–163, chapter 20.
- Van Bon, John (2007), "Finite primitive distance-transitive graphs", European Journal of Combinatorics, 28 (2): 517–532, doi:10.1016/j.ejc.2005.04.014, MR 2287450.
- Brouwer, A. E.; Cohen, A. M.; Neumaier, A. (1989), "Distance-Transitive Graphs", Distance-Regular Graphs, New York: Springer-Verlag, pp. 214–234, chapter 7.
- Cohen, A. M. Cohen (2004), "Distance-transitive graphs", in Beineke, L. W.; Wilson, R. J. (eds.), Topics in Algebraic Graph Theory, Encyclopedia of Mathematics and its Applications, vol. 102, Cambridge University Press, pp. 222–249.
- Godsil, C.; Royle, G. (2001), "Distance-Transitive Graphs", Algebraic Graph Theory, New York: Springer-Verlag, pp. 66–69, section 4.5.
- Ivanov, A. A. (1992), "Distance-transitive graphs and their classification", in Faradžev, I. A.; Ivanov, A. A.; Klin, M.; et al. (eds.), The Algebraic Theory of Combinatorial Objects, Math. Appl. (Soviet Series), vol. 84, Dordrecht: Kluwer, pp. 283–378, MR 1321634.