Distance-regular isometric subgraphs of the halved cubes (Q1378299)

A distance regular graph \(\Delta\) is an isometric subgraph of a halved cube iff \(\Delta\) is one of the following graphs: a complete graph, a halved cube, a Johnson graph, a Hamming graph, a Doob graph, a polygon, the icosahedron, a doubled odd graph, the Petersen graph or the dodecahedron. This theorem is a corollary of the next proposition and some known results. Proposition. Suppose that \(\Gamma\) is a distance regular isometric subgraph of a halved cube, that is not a complete graph. Then at least one of the following holds: (1) \(\mu\geq 2\); (2) \(\Gamma\) has valency \(\leq 3\); (3) \(\Gamma\) is an isometric subgraph of a Hamming cube.