A bound on connectivity of iterated line graphs

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For simple connected graphs that are neither paths nor cycles, we define l(G) = max{m: G has a divalent path of length m that is not both of length 2 and in a K3}, where a divalent path is a path whose internal vertices have degree two in G. Let G be a graph and Ln(G) be its n-th iterated line graph of G. We use (Formula Presented) and κ(G) for the essential edge connectivity and vertex connectivity of G, respectively. Let G be a simple connected graph that is not a path, a cycle or K1,3, with l(G) = l ≥ 1. We prove that (i) for integers (Formula Presented) (ii) for integers (Formula Presented). The bounds are best possible.