Index of parameters of iterated line graphs

Let G be a prolific graph, by which we mean finite connected simple graph is not isomorphic to cycle nor path the star K1,3. The line-graph of G, denoted L(G), defined having its vertex-set equal edge-set and two vertices L(G) are adjacent if corresponding edges in G. For positive integer k, iterated Lk(G) recursively Lk(G)=L(Lk−1(G)). In this paper shall consider fifteen well-known parameters study their behaviour when operation taking iterated. We first show that all these unbounded, is, P=P(G) such parameter on any then P(Lk(G))→∞ k→∞. This idea unboundedness motivated old result van Rooij Wilf says number unbounded only prolific. Following preliminary result, main thrust will value k(P,F), index family graphs with regards given P(G). P(G), ind(P,G)=min⁡{r:P(G)<P(Lr(G)}. Now for F graphs, k(P,F)=max⁡{ind(P,G):G∈F}, k(P,F) smallest k every G∈F, ind(P,G)≤k(P,F). problem determining over classical Chartrand who showed it could require k=|V(G)|−3 iterations guarantee has hamiltonian cycle. twelve considered, exactly determine where some also characterize class realizing extremal k(P,F). example, matching μ, at most 4 sharp, namely k(μ,F)=4 further those ind(μ,G)=4. Interesting open problems remain, particular completing determination three parameters: independence number, independent domination obtain partial results.