Xiaofeng Guo and M. Randic

Trees with the Same Topological Index JJ

Dept. of Mathematics and Computer Science, Drake University, Des Moines, Iowa

A novel matrix, called Wiener matrix, had been introduced by Randic et. al. in 1994. The Wiener matrix is constructed by generaling the procedure of Wiener for evaluation of Wiener numbers in alkanes. From such matrices some novel structural invariants of potential interest in structure-property studies were obtained. These include higher Wiener numbers, Wiener sequences, and hyper-Wiener number, etc. A novel topological index JJ, analogous to the connectivity index and Balaban's J index, was constructed by considering the the row sums of the Wiener matrix. Application of this index for structure-property relationship was also studied. In the present paper, in particular we investigate the trees with the same JJ index (called JJ-equivalent trees). A construction method for a class of JJ-equivalent trees is given. By using this method, the smallest pairs of non-isomorphic JJ-equivalent trees having 18 vertices were constructed. In addition, there are groups of three, four and six non-isomorphic trees having identical JJ-index. The smallest such trees are of size 28 for triplets and quadruplets and 34 for the last case. It is conjectured that the smallest JJ-equivalent trees have 18 vertices.

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