# On inverse Wiener interval problem of trees

## TL;DR

This paper proves the size of the set of Wiener indices for trees on n vertices and confirms two conjectures, showing that these sets have cubic growth in n with specific constants.

## Contribution

It establishes the exact asymptotic size of Wiener index sets for trees, resolving two previously posed conjectures in the literature.

## Key findings

- Cardinality of Wiener index sets is approximately (1/6)n^3 for even n
- Cardinality of Wiener index sets is approximately (1/12)n^3 for odd n
- Both conjectures about the inverse Wiener interval problem are proven

## Abstract

The Wiener index W(G) of a simple connected graph G is defined as the sum of distances over all pairs of vertices in a graph. We denote by W[T_{n}] the set of all values of Wiener index for a graph from class T_{n} of trees on n vertices. The largest interval of contiguous integers (contiguous even integers in case of odd n) is denoted by W^{int}[T_{n}]. In this paper we prove that both sets are of the cardinality (1/6)n^3+O(n^2) in the case of even n, while in the case of odd n we prove that the cardinality of both sets equals (1/(12))n^3+O(n^2) solving thus two conjectures posed in literature.