Given an $n \times n$ shortest path distance matrix $D$. And a complete graph $G(D)$ on $n$ nodes, where edge $(i, j)$ has weight $D_{ij}$. Furthermore, the distance matrix $D$ is computed for a connected network $T$ of $n$ nodes and $n-1$ edges - i.e. the shortest path distance between two nodes in $T$.
About the network $T$:
- There are exactly $n$ nodes and $n-1$ edges.
- The network is one connected component, thus it is a tree.
- All edges have positive length.
My problem is that if I compute a minimum spanning tree $T'$ of $G(D)$ using, for instance, Kruskal's algorithm will the resulting tree $T'= T$?
- I tried to argue that they are indeed equal, because if we consider an edge $e$ has been added in a step of Kruskal's algorithm then this edge must have been a light edge thus it is also the shortest path distance.
I am not sure if this argument holds. How can I argue that $T' = T$?