Heredity of the index of convergence of the line digraph

Let D be a digraph, and let L(D) and k(D) denote the nth iterated line digraph of D and the index of convergence of D, respectively. We prove that if D contains neither sources nor sinks, then: (1) k(L(D))= k(D)=0 if every connected component of D is a directed cycle. (2) k(L(D))= k(D)+ n if there exists at least one connected component of D that is not a directed cycle. Finally, we prove that if D has no sources or no sinks then k(D)6 k(L(D))6 k(D) + 1. ? 2003 Elsevier B.V. All rights reserved. MSC: 05C50