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 {5,4,1,1;1,1,4,5}. 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 (-3)^5(-sqrt(5))^81^(10)(sqrt(5))^85^1 (van Dam and Haemers 2003).

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

See also

Distance-Regular Graph, Quintic Graph

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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.", 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, "Armanios-Wells Graph." 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.

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