The average distance of spanning trees in terms of independence number

Let $G$ be a connected graph with vertex set $V(G)$, and denote by $d_G(u,v)$ the distance from $u$ to $v$ in $G$, for any $u,v\in V(G)$. The average distance of an $n$-vertex connected graph $G$, denoted by $\mu (G)$, is defined to be the average of all distances between all pairs of vertices in $G$, i.e., $\mu (G)={\left(\genfrac{}{}{0px}{}{n}{2}\right)}^{-1}{\sum }_{\{u,v\}\subset V(G)}{d}_G(u,v)$. The problem of finding a spanning tree of minimum average distance is known to be NP-hard, so establishing an upper bound for the minimum average distance among all spanning trees is of particular interest. Mukwembi (J. Graph Theory, 2014) showed that if $G$ is a connected graph of order $n$ with independence number $\alpha$, where $n>2\alpha -1$, then $G$ has a spanning tree $T$ such that $\mu (T)\le \alpha +2$. In this paper, we first improve the upper bound to $\mu (T)<\alpha +1$ for $\alpha \ge 1$, and then we find the bound could be further improved when $\alpha$ becomes larger, so a better upper bound $$\mu (T)<\{\begin{array}{ll}\alpha +1& \text{if }1\le \alpha \le 6,\\ \alpha +\frac{1}{2}+\frac{4(\alpha -1)}{{\alpha }^2}& \text{if }\alpha \ge 7,\end{array}$$ is established later. In the end, we give a remark to indicate our new upper bound is best possible in the sense of asymptotics (when $n$ and $\alpha$ are large enough).