CONNECTIVITY OF RANDOM CUBIC SUM GRAPHS

Consider the set SG(Q(k)) of all graphs whose vertices are labeled with nonidentity elements of the group Q(k) = Z(2)(k) so that there is an edge between vertices with labels a and b if and only if the vertex labeled a + b is also in the graph. Note that edges always appear in triangles since a + b = c, b + c = a, and a + c = b are equivalent statements for Q(k). We define the random cubic sum graph SG(Q(k), p) to be the probability space over SG(Q(k)) whose vertex sets are determined by Pr[x is an element of V] = p with these events mutually independent. As p increases from 0 to 1, the expected structure of SG(Q(k), p) undergoes radical changes. We obtain thresholds for some graph properties of SG(Q(k), p) as k -> infinity. As with the classical random graph, the threshold for connectivity coincides with the disappearance of the last isolated vertex.