Trees with the same topological index JJ

In the present paper we investigate the trees with the same JJ index (calle d JJ-equivalent trees). The topological index JJ is derived from the so cal led Wiener matrix introduced by Randic el al., in 1994. The Wiener matrix i s constructed by generalizing the procedure of Wiener for evaluation of Wie ner numbers in alkanes. From such matrices several novel structural invaria nts of potential interest in structure-property studies were obtained. Thes e include higher Wiener numbers, Wiener sequences, and hyper-Wiener number, etc. The topological index JJ is constructed by considering the row sums o f the Wiener matrix. A construction method for a class of JJ-equivalent tre es is given. By using this method we construct the smallest pairs of non-is omorphic JJ-equivalent trees which have 18 vertices. In addition we report on groups of 3, 4, and 6 non-isomorphic JJ-equivalent trees. The smallest s uch trees are of size 28 for triples and quadruples, and 34 for the group o f 6 elements.