Abstract.
By L(G) we denote a line graph of G.
We set L0(G)=G.
Then for i>0 the i-iterated line graph of G is
Li(G)=L(Li-1(G)).
A graph G is prolific if it is different from a path, a cycle,
and the claw K1,3.
We prove that for every prolific graph G there exists a constant
dG such that for the diameter it holds
diam(Li(G))=diam(Li-1(G))+1
whenever i>dG.
Moreover, we prove that for every prolific graph G there are
constants aG and bG,
such that for the radius it holds
aG< diam(Li(G))-rad(Li(G))
-log2(i)< bG
whenever i>dG.
To prove these results, we give some trivial bounds for the number
of verticees and the minimal and maximal degrees of iterated line graphs.
Moreover, we introduce histories (called butts in this paper).
These histories allow us to represent vertices of Li(G) in
G, and also allow us to count distances between vertices of
Li(G) in G.