EXTREMAL PROBLEMS IN THE CONSTRUCTION OF DISTRIBUTED LOOP NETWORKS

Let G(N, A) be the Cayley digraph associated with Z/(N) and A, where N is a positive integer and A is a subset of {1, 2,..., N - 1}. Let N(d , k) be the maximum N such that the diameter of G(N, A) is less than o r equal to d for some A = {a(1), a(2),..., a(k)} with 1 = a(1) < a(2) < ... < a(k). An exact formula for N(d, 2) is given, and N(d, k) is es timated for k greater than or equal to 3. These results provide new bo unds for minimal diameter in the construction of loop networks. A rela tion between this problem and the postage stamp problem in additive nu mber theory is established to enhance the study of these problems.