Convergence of sequences of iterated triangular line graphs

Abstract

The triangular line graph T(G) of a graph G is the graph with vertex set E(G), with two distinct vertices e and f of T(G) adjacent if and only if the edges e and f belong to a common copy of K₃ in G. For n ⩾ 1, the nth iterated triangular line graph Tⁿ(G) of a graph G is defined as T(Tⁿ⁻¹(G)), where T°(G) = G. In [4] it is shown that the sequence of iterated triangular line graphs of a graph G converges to r disjoint copies of K₃, for some r ⩾ 0. Here we determine how many iterations are required for convergence, and how many disjoint copies of K₃ are obtained.