DIAMETER IN PATH GRAPHS
A. Belan, P. Jurica · 1999
If G is a graph, then its path graph, P k (G), has vertex set identical with the set of paths of length k in G, with two vertices adjacent in P k (G) if and only if the corresponding paths are consecutive in G. We construct bounds on the diameter of every componen t of P k (G) in form diam(G) +f(k), where f(k) is a function depending only on k. We have a general lower bound with f(k) = k; upper bound for trees with f(k) = k(k 2); and an upper bound for graphs with large diameter with f(k) = k 2 2, if 2 k 4. All bounds are best possible.