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 \(V = \{0,1,\ldots,m-1\}\) and \((i,j) \in A\) if and only if \(j \equiv ni + \alpha \pmod{m}\) for some \(\alpha \in \{0,1,\ldots,n-1\}\). 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 \(n \geq 2\) and \(n^2 + (\frac{\sqrt{5}+1}{2})\leq m \leq 2n^2.\)