Sharper upper bound for the asymmetry term on trees

Determine a sharper upper bound for the asymmetry term \(\mathcal A(T)\) among trees of fixed order.

Background

The paper defines the asymmetry term for a finite connected graph by $\mathcal A(G)=\frac12\sum_{\{u,v\}}|R_d(u)-R_d(v)|$, where $R_d(u)$ is the degree-weighted effective-resistance sum. For a tree $T$ with $n\geq3$ vertices, it proves the upper bound $\mathcal A(T)\leq(n-2)W(T)$, where $W(T)$ is the Wiener index. The conclusion explicitly identifies finding a sharper bound over trees of a fixed order as an unresolved question.