This might be a naive question to those who are good at graph theory. After a long time search without finding a clear answer, I would like to know if anyone could help me out on this.
I am studying a problem related to the Wiener Index which calculates the sum of distances over all un-ordered vertex pairs in a graph, as:
$$WI(G)=\sum_{u\neq v} d_G(u,v)$$
Whereas my exact problem requires me to replace the topological distance $d_G(u,v)$ with a product of the weights of the edges $w(e)$ along the walk $\mathcal{L}_{uv}$ from $u$ to $v$, i.e.: $$d_G'(u,v)=\prod_{e_{ab}\in E(\mathcal{L}_{uv})}w(e_{ab})$$ I would like to know if there is any existing research on such an index (since I have no idea what it should be called, let's say $Index(G)$) that: $$Index(G)=\sum_{u\neq v} d'_G(u,v)$$
P.S.: Since, in my study, the weight of an edge $w(e)$ is always between $0$ to $1$, $d'_G(u,v)$ of close pairs usually contributes more to $Index(G)$ than that of the distant pairs. This seems to be similar to the property of Haray Index, but I would like to know if there is an exact same thing or a conversion of my problem to an existing research.
Many thanks.