Consider a positively weighted connected simple graph with bounded degree $X$ . Denote by $d(x,y)$ the weight on the edge with endpoints $x$ and $y$. Suppose we have the following compatibility axioms:

If $x_0,x_1,\ldots,x_{n-1},x_n$ and $y_0,y_1,\ldots,y_{n-1},y_n$ are two shortest paths in the unlabeled graph connecting the same points (i.e. $x_0=y_0$ and $x_n=y_n$), then $$ \sum d(x_{i-1},x_i)=\sum d(y_{i-1},y_i) $$

If $x_0,x_1,\ldots,x_{n-1},x_n$ is a shortest path in the unlabeled graph connecting $x$ to $y$ and $y_0,y_1,\ldots,y_{m-1},y_m$, with $m>n$, is another path connecting $x$ to $y$, then

$$ \sum d(x_{i-1},x_i)<\sum d(y_{i-1},y_i) $$

Define a metric on the set of vertices just adding the various weights that you encounter moving along a shortest path. Suppose this metric is locally finite. The two axioms above say that this metric is well-defined and that it is in some sense compatible with the graph structure: the shortest-paths are geometrically the same and the distance is additive only along shortest paths.

General question:Has somebody studied these objects?

In particular, I am interested in the following questions:

Question:Suppose that our graph (without labels) is a tree with positive isoperimetric constant. Is it still true that it does not have bi-lipschitz embeddings (with the new metric) into a Hilbert space?

Also, let $\delta_1(X)$ be the best nonnegative constant, if exists, such that every side of a geodesic triangle is contained in the $\delta_1$-neigborhood of the other two sides. (If you like quasi-isometry, let $\delta$ be the infimum of all $\delta_1(Y)$ when $Y$ runs over the quasi-isometric class containing $X$). I am very tempted to say that $X$ has curvature bounded above by $-\frac{1}{\delta(X)}$.

Question:Has been this notion studied before? More specifically, what happens if I take, as graph, the 1-skeleton of a very good triangulation of a compact negatively curved metrizable manifold with labels given by the induced metric?

Thank you in advance,

Valerio