## Abstract

For a graph *G*, and two distinct vertices *u* and *v* of *G*, let \( n_{{G(u,v)}} \) be the number of vertices of *G* that are closer in *G* to *u* than to *v*. Miklavič and Šparl (arXiv:2011.01635v1) define the distance-unbalancedness \({{\mathrm{uB}}}(G)\) of *G* as the sum of \(|n_G(u,v)-n_G(v,u)|\) over all unordered pairs of distinct vertices *u* and *v* of *G*. For positive integers *n* up to 15, they determine the trees *T* of fixed order *n* with the smallest and the largest values of \({\mathrm{uB}}(T)\), respectively. While the smallest value is achieved by the star \(K_{1,n-1}\) for these *n*, which we then proved for general *n* (Minimum distance-unbalancedness of trees, J Math Chem, https://doi.org/10.1007/s10910-021-01228-4), the structure of the trees maximizing the distance-unbalancedness remained unclear. For *n* up to 15 at least, all these trees were subdivided stars. Contributing to problems posed by Miklavič and Šparl, we show

and

where \(S(n_1,\ldots ,n_k)\) is the subdivided star such that removing its center vertex leaves paths of orders \(n_1,\ldots ,n_k\).

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## References

- 1.
T. Došlic, I. Martinjak, R. Škrekovski, S. Tipuric Spuževic, I. Zubac, Mostar index. J. Math. Chem.

**56**, 2995–3013 (2018) - 2.
K. Handa, Bipartite graphs with balanced \((a, b)\)-partitions. Ars Combin.

**51**, 113–119 (1999) - 3.
J. Jerebic, S. Klavžar, D.F. Rall, Distance-balanced graphs. Ann. Comb.

**12**, 71–79 (2008) - 4.
M. Kramer, D. Rautenbach, Minimum distance-unbalancedness of trees. J. Math. Chem. (2021). https://doi.org/10.1007/s10910-021-01228-4

- 5.
Š Miklavič, P. Šparl, \(\ell \)-distance-balanced graphs. Discret. Appl. Math.

**244**, 143–154 (2018) - 6.
Š Miklavič, P.: Šparl, Distance-unbalancedness of graphs. arXiv:2011.01635v1

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Kramer, M., Rautenbach, D. Maximally distance-unbalanced trees.
*J Math Chem* **59, **2261–2269 (2021). https://doi.org/10.1007/s10910-021-01287-7

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### Keywords

- Distance-unbalancedness
- Distance-balanced graph
- Mostar index