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# Doro Graph

The Doro graph is a distance-transitive and distance-regular graph on 68 vertices and with valency 12. It is the unique automorphic graph having intersection array (Gordon and Levingston 1981). Its automorphism group is , where denotes the semidirect product of by (Gordon and Levingston 1981).

It has spectrum (van Dam 1996) and is therefore an integral graph.

The Doro graph is implemented in the Wolfram Language as GraphData["DoroGraph"].

Note that Koolen et al. (2023) use the term "Doro graph" to refer to the Hall graph (which was first considered by Doro), which is a different distance-regular graph that has intersection array .

Automorphic Graph, Doro Graph

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## References

Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. "The Even Orthogonal Case; The Doro Graph." §12.1 in Distance Regular Graphs. New York: Springer-Verlag, pp. 211, 225, and 374-379, 1989.Buekenhout, F. and Rowlinson, P. "The Uniqueness of Certain Automorphic Groups." Geom. Dedicata 11, 443-446, 1981.DistanceRegular.org. "Doro Graph from ." http://www.distanceregular.org/graphs/doro68.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 Unpublished.Gordon, L. M. and Levingston, R. "The Construction of Some Automorphic Graphs."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.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 ." 15 Nov 2023. https://arxiv.org/abs/2311.09001.van Dam, E. R. "Graphs with Few Eigenvalues: An Interplay Between Combinatorics and Algebra." Ph.D. dissertation. Tilburg, Netherlands: Tilburg University, pp. 51-52, October 4, 1996.

## Cite this as:

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