Maximally distance-unbalanced trees

For a graph G, and two distinct vertices u v of let $$ n_{{G(u,v)}} n G ( , ) be the number that are closer in to than v. Miklavi? Šparl arXiv:2011.01635v1 define distance-unbalancedness $${{\mathrm{uB}}}(G)$$ uB as sum $$|n_G(u,v)-n_G(v,u)|$$ | - over all unordered pairs G. positive integers up 15, they determine trees T fixed order with smallest largest values $${\mathrm{uB}}(T)$$ respectively. While value is achieved by star $$K_{1,n-1}$$ K 1 for these n, which we then proved general (Minimum trees, J Math Chem, https://doi.org/10.1007/s10910-021-01228-4 ), structure maximizing remained unclear. 15 at least, were subdivided stars. Contributing problems posed Šparl, show $$\begin{aligned} \max \Big \{{\mathrm{uB}}(T):T \text{ } tree n\Big \} =\frac{n^3}{2}+o(n^3) \end{aligned}$$ max { : = 3 2 + o \{{\mathrm{uB}}(S(n_1,\ldots ,n_k)):1+n_1+\cdots +n_k=n\Big =\left( \frac{1}{2}-\frac{5}{6k}+\frac{1}{3k^2}\right) n^3+O(kn^2), S … k ? 5 6 O where $$S(n_1,\ldots ,n_k)$$ such removing its center vertex leaves paths orders $$n_1,\ldots ,n_k$$ .