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Let

More formally,

In what follows the numbers

The quantities

can be computed by means of the formula

Nowadays

The first formal proof of Eq. (1) was given in the book [3]. Eventually, Eq. (1) was much studied; for details see the review [4]. The extension of the right-hand side of (1) to all graphs was named the

Motivated by Eq. (1), in 2001 the

Somewhat more recently, also the

with l being an adjustable parameter. For l = +1 and l = -1, the variable Wiener index reduces, respectively, to the ordinary and to the modified Wiener index.

A further structure-descriptor

where

In Eqs. (1)-(4) the summation goes over all edges of the tree

Studies of the above mentioned structure-descriptors lead to the concept of

It is known [12] that the Wiener index measures the van der Waals surface area of an alkane molecule, which explains the correlations found between

General procedures for constructing pairs of equiseparable trees were developed [11, 16], and it gradually became evident [17] that equiseparable trees and chemical trees occur in large families. In order to gain information on the frequency of the occurrence of equiseparable trees, we examined all trees with up to 20 vertices [17]. We found that with increasing

The results shown in Table 1 suggest that with increasing *n* the ratios *A*(*n*)/*T*(*n*) and *CA*(*n*)/*CT*(*n*) tend to zero. This, in turn, would imply that almost all trees and almost all chemical trees have an equiseparable mate. In what follows we demonstrate that this, indeed, is the case.

We first need some preparations.

is independent of the labelling of the edges of

Since an

Thus the separation sequence is an (

According to a standard result of Combinatorics (see, e. g. [18]), the number of ordered

By the well known Cayley's formula, the number of labelled

Define an auxiliary function

In view of (9) and (10), if the condition

holds, then the limit (7) will also be satisfied, implying the validity of Theorem 1.

Bearing in mind that we arrive at

Applying the Stirling formula [18]

The relation

holds because Therefore also (11) holds. Therefore also (7) holds.

Theorem 2 follows from a somewhat stronger result. Let

Details of the proof of Theorem 3 are omitted. Theorem 2 is a special case of Theorem 3 for D = 3 and D = 4.

[ 2] H. Wiener, Structural determination of paraffin boiling points,

[ 3] I. Gutman, O. E. Polansky,

[ 4] A. A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: theory and applications,

[ 5] I. Gutman, A. A. Dobrynin, The Szeged index – a success story,

[ 6] S. Nikolic, N. Trinajstic, M. Randic, Wiener index revisited,

[ 7] I. Gutman, D. Vukicevic, J. Zerovnik, A class of modified Wiener indices,

[ 8] B. Lucic, A. Milicevic, S. Nikolic, N. Trinajstic, On variable Wiener index,

[ 9] I. G. Zenkevich, Dependence of chromatographic retention indices on the dynamic characteristics of molecules,

[10] I. Gutman, I. G. Zenkevich, Wiener index and vibrational energy,

[11] I. Gutman, B. Arsic, B. Furtula, Equiseparable chemical trees,

[12] I. Gutman, T. Kortvelyesi, Wiener indices and molecular surfaces,

[13] I. Gutman, Y. N. Yeh, S. L. Lee, Y. L. Luo, Some recent results in the theory of the Wiener number,

[14] D. H. Rouvray, The rich legacy of half century of the Wiener index,

[15] I. Gutman, D. Vidovic, B. Furtula, I. G. Zenkevich, Wiener–type indices and internal molecular energy,

[16] I. Gutman, B. Furtula, D. Vukicevic, B. Arsic, Equiseparable molecules and molecular graphs,

[17] O. Miljkovic, B. Furtula, I. Gutman, Statistics of equiseparable trees and chemical trees,

[18] I. Tomescu,

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