Graph edges total distance

Say one has the following graph

One has $$8$$ (black) vertices.

Say one has a $$8 \times 1$$ vector of distances between each black vertex and the red vertex. By distances, I mean that all my vertices are members of a metric space, e.g. a geographical space. Let's denote this vector by $$\boldsymbol{d}$$.

Say one has a $$8 \times 8$$ matrix in which each element represents the percentage of edge-distance in common between two vertices (a fortiori with entries equal to $$1$$ on the main diagonal and a priori asymmetric since two vertices are very likely to share different percentages). Let's denote this matrix by $$\boldsymbol{D}$$.

Do you have any reference which explains how to compute the total distance over edges when having only $$\boldsymbol{d}$$ and $$\boldsymbol{D}$$ in hand?

Starting with a very simple case, and complexifying it progressively, I ended with

$$|(\boldsymbol{D}^{-1})^{'} \boldsymbol{d}|$$

Where the operator $$|.|$$ stands for the L$$1$$-norm, i.e. the sum of all the absolute elements' value of $$(\boldsymbol{D}^{-1})^{'} \boldsymbol{d}$$ (although the absolute-value of entries is not needed since everything is already positive).

Am I right by computing this (total graph) distance like this ?

• Do you by "total distance" mean the number of edges in the graph? – wonce Sep 23 '17 at 16:26
• @wonce. No I really mean that all my vertices are members of a metric space. As if each of the vertices were a location in a geographical world. – keepAlive Sep 23 '17 at 18:51
• Are you assuming your graph is a tree (so there's exactly one path from any vertex to any other vertex)? – Gerry Myerson Jan 29 at 8:30
• @GerryMyerson Yes it is a tree. – keepAlive Jan 29 at 9:09