Characterization of Laborde-Mulder graphs (extended odd graphs)

Abstract

The identification of diametrical vertices in the d-dimensional hypercube ( d ⩾ 3) leads to a (0, 2)-graph of degree d on 2 d−1 vertices and of diameter ⌊ d 2 ⌋ namely the extended odd graph (or Laborde-Mulder graph) for odd values of d, and the half-cube for even values of d. In this paper we prove that the diameter of a (0, 2)-graph of degree d on 2 d−1 vertices is at least ⌊ d 2 ⌋ , and when d is odd the equality holds if and only if the graph is a Laborde-Mulder graph.