Are there counter examples to the conjecture, that in a complete, finite and symmetric weighted graph $G\left(V,E,\omega\right),\ E=\lbrace \lbrace i,j\rbrace\subset V\times V\rbrace,\ \omega:E\ni e\mapsto\mathbb{R_0^+}$ the edges constituting to the solution of $$\min\frac{\sum_{i,j} \alpha_{ij}\omega(e_{ij})}{\sum_{i,j} \alpha_{ij}}$$ $$\sum_{i\ne u}\alpha_{iu}=0,\quad \sum_{i\ne u}\alpha_{ui}=1 $$ $$\sum_{j\ne v}\alpha_{jv}=1,\quad \sum_{j\ne v}\alpha_{vj}=0 $$ $$\sum_{i\notin\lbrace j,k,u,v\rbrace}\alpha_{ik}\ -\sum_{j\notin\lbrace i,k,u,v\rbrace}\alpha_{kj}=\ 0$$ $$\alpha_{ij}\in\lbrace 0,1\rbrace$$
i.e. to the path connecting $u$ to $v$ with minimal average edge-weight are elements of the set of edges constituting to the minimum spanning tree (MST)?
Edit to further explain, as requested by Brendan McKay:
the graphs, that this question relates to, are undirected and shall have no cycles of negative length; it is further required, that the graph is connected and if not, the question relates to the connected components.
a simple path connecting two distinct vertices $u$ and $v$ in such a graph is characterized
- by the set $P_{uv}$ of edges constituting to that connecting simple path,
- by the cardinality $C_{uv}$ of that set of edges.
by the sum of weights $L_{uv}$ of $P_{uv}$, i.e. the path's length
the average edge-length of $P_{uv}$ is then defined as $\frac{L_{uv}}{C_{uv}}$.
This must not be confused with the average path-length, which is also a the subject of research!
for explaining, what is conjectured, further notation is introduced:
$P_{uv}^E$, $L_{uv}^E$ and $C_{uv}^E$ shall be the set of edges, its sum of weights and its cardinality of a path between $u$ and $v$ consisting of edges from the entire edge-set $E$ of $G$, whereas for $P_{uv}^{\mathrm{MST}}$, $L_{uv}^{\mathrm{MST}}$ and $C_{uv}^{\mathrm{MST}}$ the set of edges shall be restricted to $G$'s minimum spanning tree MST.
Conjecture: $$\frac{L_{uv}^{\mathrm{MST}}}{C_{uv}^{\mathrm{MST}}}\ \le\ \frac{L_{uv}^E}{C_{uv}^E}\ \forall u,v\in G$$
