On the Diameter of the Generalized Graphs UG(B)(n, m), n(2) < m <= n(3)

Abstract

The generalized de Bruijn digraph G(B)(n, m) is the digraph (V, A) where V = {0, 1, ..., m-1) and (i, j) is an element of A if and only if j equivalent to in + alpha (mod m) for some alpha is an element of {0, 1, 2, ..., n-1}. By replacing each arc of G(B)(n, m) with an undirected edge and eliminating loops and multi-edges, we obtain the generalized undirected de Bruijn graph UG(B)(n, m). In this article, we prove that when 2n(2) <= m <= n(3) the diameter of UG(B)(n, m) is equal to 3. We also show that for pairs (n, m) where n(2) < m < 2n(2) the diameter of UG(B) (n, m) can be 2 or 3. (C) 2008 Wiley Periodicals, Inc. NETWORKS, Vol. 52(4), 180-182 2008