# On the Wiener inverse interval problem

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Matjaž Krnc

Institute of Mathematics, Physics and Mechanics

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Riste Škrekovski

Faculty of Mathematics and Physics

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**Minisymposium:**
CHEMICAL GRAPH THEORY

**Content:**
The well studied Wiener index $W(G)$ of a graph $G$ is equal to
the sum of distances between all pairs of vertices of $G$. Denote
by $\mathrm{Im}\left(n\right)$ the set of all values of the Wiener Index over all connected
graphs on $n$ vertices. Also, let $\mathcal{I}_{n}$ be the largest interval
which is fully contained in $\mathrm{Im}\left(n\right)$.
In the minisymposium I will outline our results on the cardinality of $\mathcal{I}_{n}$ and $\mathrm{Im}\left(n\right)$. In particular, I will show that $\mathcal{I}_{n}$ and $\mathrm{Im}\left(n\right)$
are of cardinality $\frac{1}{6}n^{3}+O\left(n^{2}\right)$, and that they both start at $\binom{n}{2}$.
We will also discuss other properties of $\mathrm{Im}\left(n\right)$ and $\mathcal{I}_{n}$ and state some open problems.