Minimum distance–unbalancedness of the merged graph of $C_3$ and a tree

For a graph $G$, let $n_G(u,v)$ be the number of vertices of $G$ that are strictly closer to $u$ than to $v$. The distance–unbalancedness index $\mathrm{u}\mathrm{B}(G)$ is defined as the sum of $|n_G(u,v)-n_G(v,u)|$ over all unordered pairs of vertices $u$ and $v$ of $G$. In this paper, we show that the minimum distance–unbalancedness of the merged graph $C_3\cdot T$ is $(n+2)(n-3)$, where $C_3\cdot T$ is obtained by attaching a tree $T$ to the cycle $C_3$.