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# Wells Graph

The Wells graph, sometimes also called the Armanios-Wells graph, is a quintic graph on 32 nodes and 80 edges that is the unique distance-regular graph with intersection array . It is also distance-transitive. It is a double cover of the complement of the Clebsch graph (Brouwer et al. 1989, p. 266).

It is implemented in the Wolfram Language as GraphData["WellsGraph"].

It has graph diameter 4, girth 5, graph radius 4, and is Hamiltonian and nonplanar. It has chromatic number 4, edge connectivity 5, and vertex connectivity 5.

It has graph spectrum (van Dam and Haemers 2003).

There are three distinct graphs having the spectrum of the Wells graph (van Dam and Haemers 2003).

Distance-Regular Graph, Quintic Graph

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## References

Armanios, C. "Symmetric Graphs and Their Automorphism Groups." Ph.D. thesis. Perth, Australia: University of Western Australia, 1981.Armanios, C. "A New 5-Valent Distance Transitive Graph." Ars Combin. 19A, 77-85, 1985.Brouwer, A. E. "Armanios-Wells Graph." http://www.win.tue.nl/~aeb/drg/graphs/Wells.html.Brouwer, A. E.; Cohen, A. M.; and Neumaier, A. "Covers of Cubes and Folded Cubes--The Wells Graph." §9.2E in Distance Regular Graphs. New York: Springer-Verlag, pp. 266-267, 1989.DistanceRegular.org. "Armanios-Wells Graph." http://www.distanceregular.org/graphs/armanios-wells.html.van Dam, E. R. and Haemers, W. H. "Spectral Characterizations of Some Distance-Regular Graphs." J. Algebraic Combin. 15, 189-202, 2003.Wells, A. L. "Regular Generalized Switching Classes and Related Topics." Ph.D. thesis. Oxford, England: University of Oxford, 1983.

## Cite this as:

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