Comments on "Line Digraph Iterations and Connectivity Analysis of de Bruijn and Kautz Graphs"

The aim of this note is to present some counterexamples to the results in the paper by Du, Lyuu and Hsu published in this Transactions [1]. Index Terms--- Connectivity, diameter-vulnerability, fault-tolerance, line digraph iteration, spread. We present in this paper, for any d 2, a digraph G 2 L (d; 4) " L ffl (d \Gamma 1; 4) such that LG 62 L ffl (d \Gamma 1; 5) and L 2 G 62 L (d; 6). Therefore, this is a counterexample to Lemma 3.2 and Theorem 3.3 of [1]. The digraphs we consider here are members of a family of bipartite digraphs, called BD(d;n), constructed by Fiol and Yebra [2]. See this paper for the definition and some properties of these digraphs. The digraph BD(d;n) is d-regular and bipartite with partite sets V 0 = f0g \Theta Zn and V 1 = f1g \Theta Zn . One important fact about this family is that the line digraph LBD(d;n) is isomorphic to BD(d; dn). For any vertex (ff; i) of BD(d; d 2 + 1), we have that \Gamma + 2 (ff; i) = V ff \Gamma f(ff; i)g. Then, the..