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Hall Graph

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The Hall graph is a distance-transitive and distance-regular graph on 65 vertices having intersection array {10,6,4,1;1,2,5} (Hall 1980). It is denoted Gamma^((3)) by Hall (1980) and was originally considered by Doro.

It is an integral graph with graph spectrum (-3)^(25)0^(26)5^(13)10^1.

It is one of the three locally Petersen graphs (Hall 1980), and is denoted L_2(25).25 by Brouwer et al. (1989, p. 224).

The Hall graph is implemented in the Wolfram Language as GraphData["HallGraph"].

Koolen et al. use the term "Doro graph" to refer to the Hall graph, though that term also refers to a different distance-regular graph graph with intersection array {12,10,3;1,3,8}.

See also

Conway-Smith Graph, Distance-Transitive Graph, Doro Graph, Locally Petersen Graph

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References

Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. Distance Regular Graphs. New York: Springer-Verlag, p. 211 and 224, 1989.DistanceRegular.org. "Hall Graph from PSigmaL(2,25)." http://www.distanceregular.org/graphs/hall65.html.Doro, S. "Two New Distance-Transitive Graphs." Unpublished.Gordon, L. M. and Levingston, R. "The Construction of Some Automorphic Graphs." Geom. Dedicata 10, 261-267, 1981.Hall, J. I. "Locally Petersen Graphs." J. Graph Th. 4, 173-187, 1980.Koolen, J. H.; Yu, K.; Liang, X.; Choi, H.; and Markowsky, G. "Non-Geometric Distance-Regular Graphs of Diameter at Least 3 With Smallest Eigenvalue at Least -3." 15 Nov 2023. https://arxiv.org/abs/2311.09001.

Referenced on Wolfram|Alpha

Hall Graph

Cite this as:

Weisstein, Eric W. "Hall Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HallGraph.html

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