Armanios-Wells graph

There is a unique distance-regular graph Γ with intersection array {5,4,1,1;1,1,4,5}. It has 32 vertices and spectrum 5¹ 2.236⁸ 1¹⁰ (-2.236)⁸ (-3)⁵. It is known as the Wells graph, or the Armanios-Wells graph.

It is antipodal of diameter 4, the unique double cover without 4-cycles of the folded 5-cube.

For each vertex, the subgraph on the vertices at distance 2 is the dodecahedron. The dodecahedron is itself antipodal, of diameter 5, and antipodal pairs at distance 5 in the dodecahedron are antipodal at distance 4 in the Wells graph.

Group

The full group of automorphisms is 2_-¹⁺⁴:Alt(5) acting distance-transitively with point stabilizer Alt(5).

Independence number

The independence number is 10, and the only 10-cocliques are the lifts of the 5-cocliques in the folded 5-cube.

Chromatic number

The chromatic number is 4.

Characterization by spectrum

Van Dam and Haemers constructed by switching a graph cospectral with the Wells graph, and E. Spence showed by computer search that there are precisely three graphs with the spectrum of the Wells graph.

History

In [BCN] this graph was called the Wells graph. It was constructed by A. L. Wells (1983), but also, presumably earlier, by C. Armanios (1981, 1985).

References

BCN, p. 266.

C. Armanios, Symmetric graphs and their automorphism groups, Ph.D. Thesis, University of Western Australia, 1981.

C. Armanios, A new 5-valent distance transitive graph, Ars Combin. 19A (1985) 77-85.

E. R. van Dam & W. H. Haemers, Spectral Characterizations of Some Distance-Regular Graphs, J. Algebraic Combin. 15 (2002) 189-202.

A. L. Wells, Regular generalized switching classes and related topics, Ph.D. Thesis, University of Oxford, 1983.