4-cycle properties for characterizing rectagraphs and hypercubes

A (0, 2)-graph is a connected graph, where each pair of vertices has either 0 or 2 common neighbours. These graphs constitute a subclass of (0, λ)-graphs introduced by Mulder in 1979. A rectagraph, well known in diagram geometry, is a triangle-free (0, 2)-graph. (0, 2)-graphs include hypercubes, folded cube graphs and some particular graphs such as icosahedral graph, Shrikhande graph, Klein graph, Gewirtz graph, etc. In this paper, we give some local properties of 4-cycles in (0, λ)-graphs and more specifically in (0, 2)-graphs, leading to new characterizations of rectagraphs and hypercubes.