Complete solution of equation W(L3(T))=W(T) W ( L 3 ( T ) ) = W ( T ) for the Wiener index of iterated line graphs of trees
Let G G be a graph. Denote by Li(G) L i ( G ) its i i -iterated line graph and denote by W(G) W ( G ) its Wiener index. In Knor etal. (in press) we show that there is an infinite class T T of trees T T satisfying W(L3(T))=W(T) W ( L 3 ( T ) ) = W ( T ) , which disproves a conjecture of Dobrynin and Entringer. In this paper we prove that except the trees of T T , there is no non-trivial tree T T satisfying W(L3(T))=W(T) W ( L 3 ( T ) ) = W ( T ) . Consequently, for a tree T T and i?3 i ? 3 , the equation W(Li(T))=W(T) W ( L i ( T ) ) = W ( T ) holds if and only if T?T T ? T and i=3 i = 3 .
wiener index iterated line graph tree