# When is a graph a $\operatorname{CAT}(\kappa)$ space?

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 conditions are given which guarantee that $$d_G$$ has non-positive curvature (in the sense of Alexandrov); 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 $$e_1,e_2\in C_i$$.

• My rough guess is that a metric graph with smallest cycle of length $r$ is CAT($\kappa$) iff the round 2-sphere with equator of size $r$ (i.e., radius $r/2\pi$) has curvature $\le\kappa$. But I'm really unsure.
– YCor
Dec 20, 2021 at 17:05

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 $$2{\cdot}\pi$$.