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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 $e_1,e_2\in C_i$.

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    $\begingroup$ 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. $\endgroup$
    – YCor
    Dec 20, 2021 at 17:05

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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 $2{\cdot}\pi$.

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  • $\begingroup$ I can't manage to track down this generalized Cartan-Hadamard theorem; where could I find it/does it have some other name? $\endgroup$ Dec 21, 2021 at 9:58
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    $\begingroup$ @Carl_Petterson See p. 119 here arxiv.org/abs/1903.08539v1. $\endgroup$ Dec 21, 2021 at 10:12

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