Probability of diameter two for Steinhaus graphs

Abstract

A Steinhaus graph is a graph with n vertices whose adjacency matrix ( a i,j ) satisfies the condition that a i,j ≡ a i − 1, j − 1 + a i − 1, j (mod 2) for each 1 < i < j ≤ n . It is clear that a Steinhaus graph is determined by its first row. In “Almost all Steinhaus graphs have diameter two”, J. Graph Theory 16 (1992) 213–219 it is shown that almost all Steinhaus graphs have diameter two. Here we generalize to the case where the j th entry of the first row has probability p j of being 1. Under reasonable conditions it is shown that the probability measure of the set of Steinhaus graphs with diameter two approaches 1 as the number of vertices in the graph approaches infinity.