The average distance of spanning trees in terms of independence number

Abstract: 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 $μ(G)$, is defined to be the average of all distances between all pairs of vertices in $G$, i.e., $μ(G) = \binom{n}{2}^{-1} \sum_{{u,v} \subset V(G)}d_G(u,v)$. 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 $α$, where $n > 2 α- 1$, then $G$ has a spanning tree $T$ such that $μ(T) \le α+ 2$. In this paper, we first improve the upper bound to $μ(T) < α+ 1$ for $α\ge 1$, and then we find the bound could be further improved when $α$ becomes larger, so a better upper bound [ μ(T) < \left{ \begin{array}{ll} α+1 & \hbox{if } 1\leα\le 6,\ α+\frac12+\frac{4(α-1)}{α^2} & \hbox{if } α\ge 7, \end{array} \right. ] 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 $α$ are large enough).