Let $G:=(E,V,W)$ be a weighted graph and let $d\_G$ be its graph metric, defined by on any two edges $e\_1,e\_2\in E$ by
$$
d\_G(e\_1,e\_2)\triangleq 
\inf\_{\gamma}\, \sum\_{v\in \gamma} W(v),\qquad\tag{0}\label{0}
$$
where the infimum is taken over all sequences of vertices $\gamma=(v\_1,\dots,v\_t)$ connecting $e\_1$ to $e\_2$; where $W:V\rightarrow \mathbb{R}$ is the *weight function* defined on unweighted graph $(E,V)$.


In Ballmann's book [Lectures on spaces of nonpositive curvature](https://people.mpim-bonn.mpg.de/hwbllmnn/archiv/NPC0606.pdf) conditions are given which guarantee that $d\_G$ has non-positive curvature (*[in the sense of Alexandrov](https://en.wikipedia.org/wiki/Alexandrov_space)*); namely, there is some $c>0$ satisfying:
$$
\inf\_{v}\, W(v)\geq c.\tag{1}\label{1}
$$
My question is, when is $(E,d\_G)$ a $\operatorname{CAT}(\kappa)$ space, for some $\kappa\in \mathbb{R}$?




---


My hypothesis is that there exist some partition $C\_1,\dotsc,C\_N$ of $E$ into open balls such that \eqref{1} holds; where the infimum in \eqref{0} is taken over all paths $\gamma$ contained in $C\_i$; where $e\_1,e\_2\in C\_i$.



When is a graph a $\operatorname{CAT}(\kappa)$ space?
The generalized Cartan--Hadamard theorem states that a length space is CAT(1) if it is locally CAT(1) + any closed curve of length $<2{\cdot}\pi$ is null-homotopic in class of curves of length $<2{\cdot}\pi$.


If the space is a graph, then the latter is equivalent to saying that any cycle has length at least $2{\cdot}\pi$.


