On the Diameter of the Generalized Undirected De Bruijn Graphs

Abstract

The generalized de Bruijn digraph denoted by 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 ni + alpha (mod m) for some alpha is an element of {0, 1, . . . , n - 1}. By replacing each arc of G(B)(n, m) with an undirected edge and eliminating loops and multi-edges, we obtain a generalized undirected de Bruijn graph UG(B)(n, m). In this paper, we prove that the diameter of UG(B)(n, m) is equal to 3 whenever n >= 2 and n(2) + (root 5+1/2)n <= m <= 2n(2).