A Characterization of Halved Cubes

The vertex set of a halved cube \(Q’_d\) consists of a bipartition vertex set of a cube \(Q_d\) and two vertices are adjacent if they have a common neighbour in the cube. Let \(d \geq 5\). Then it is proved that \(Q’_d\) is the only connected, \(\binom{d}{3}\)-regular graph on \(2^d\) vertices in which every edge lies in two \(d\)-cliques and two \(d\)-cliques do not intersect in a vertex.