Distances in Iterated Line Graphs

For a connected graph \(G\) that is not a cycle, a path or a claw, let its \(k\)-iterated line graph have the diameter \(diam_k\), and the radius \(r_k\). Then \(diam_{k+1} = diam_k + 1\) for sufficiently large \(k\). Moreover, \(\{r_k\}\) also tends to infinity and the sequence \(\{diam_k – r_k – \sqrt{2\log_2 k}\}\) is bounded.