Graphs which are locally a cube

Distance-Regular Graphs

“…Then L has 240 vectors v with q(v) = 2, and this set «P of 240 vectors is a root system of type Eg. The graph F 4,I is an antipodal 2-cover of the complete tripartite graph K 4,4,4 and one of the two graphs that are locally a cube (see BUSET [152]; the other is 3 X 5, the complement of the 3 X 5 grid on 15 vertices). Its double coset diagram was given in §3.…”

On the local recognition of finite metasymplectic spaces

“…If A is the thin metasymplectic space (i.e., all lines of A have precisely two points and each line is in precisely three triangles), then A is the 24-cell associated with the Weyl group of type F4 and A G 1 is the graph theoretic join of a single vertex graph and the cube, where the latter is viewed as a graph on 8 vertices and 12 edges. A. E. Brouwer and, independently, D. Buset [3] have shown that in this case each connected graph locally isomorphic to A is either isomorphic to d or to the complement E of the 3 x 5 grid.…”

The clique operator on matching and chessboard graphs

“…Another similar proof, but much easier, shows that also G 3,3 is the smallest locally G 2,2 = 2K 2 graph. Buset proves in [4] that there are only two locally G 2,4 graphs, namely, G 3,5 and H 24 , the 1-skeleton of the 24-cell. A stronger unpublished result of Brouwer is mentioned without proof in [4,14], namely, that if m + n ≥ 6 and G is a locally G m,n graph, then G ∼ = G m+1,n+1 save for Buset's exception at m = 2, n = 4, where we can have G ∼ = H 24 , or for m = 3, n = 3 and G ∼ = J (6, 3), a Johnson graph.…”

“…The conjecture was finally settled by Shareshian and Wachs in [24], and this means that H ν n (∆(G n ), Z) = 0 and H ν m,n (∆(G m,n ), Z) = 0. Using the software system GAP [9], together with the Simplicial Homology package [13], we have obtained H ν m,n (∆(K (G m,n )), Z) = 0 for (m, n) = (3, 6), (3,7), (3,8), (4,5), (5,5), and so G m,n K (G m,n ) in those cases. But the general case remains to be done.…”

“…Again, if X = T ∪ { (1, i) : 4 ≤ i ≤ n },we get that X ⊆T and that no edge of K m,n which is outside of X can be disjoint to all edges in X . Then only (2, 2) and (3, 3) could be universal inT , but (2, 2) is not disjoint to (2, 4) ∈ N G m,n [(1, 1), (3, 3)] and (3, 3) is not disjoint to(3,4) ∈ N G m,n [(1, 1),(2,2)]. Therefore the extended triangle of T is not a cone, and G m,n is not clique-Helly in this case.…”