On the diameter of the generalized undirected de Bruijn graphs UGB(n,m), n2 < m ≤ n3

Abstract

AbstractThe generalized de Bruijn digraph GB(n,m) is the digraph (V,A) where V = {0, 1,…,m − 1} and (i,j) ∈ A if and only if j ≡ in+α (mod m) for some α ∈ {0, 1, 2,…,n− 1}. By replacing each arc of GB(n,m) with an undirected edge and eliminating loops and multi‐edges, we obtain the generalized undirected de Bruijn graph UGB(n,m). In this article, we prove that when 2n2 ≤ m ≤ n3 the diameter of UGB(n,m) is equal to 3. We also show that for pairs (n,m) where n2 <m< 2n2 the diameter of UGB(n,m) can be 2 or 3. © 2008 Wiley Periodicals, Inc. NETWORKS, 2008