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I would like to know if there is any notion of a discrete Riemannian metric on graphs ?. C. Mercat has worked on discrete Riemann Surfaces, but that's not exactly what I am working on. To be more precise, when one defines a weighted Laplacian $\Delta : \mathbb{R}^V \mapsto \mathbb{R}^V $ on a graph $G=(V,E)$ by

$$ \forall v \in V, \quad \delta (f)(v)=\sum_{w \sim v} a_{\{v,w\}} \big( f(v)-f(w) \big),$$ the family $ \big( a_{e} \big)_{e \in E}$ of reals indexed by the set of edges can be viewed as a metric on the graph. But the analogy with the case of riemannian metric on manifolds is not so obvious. For instance, what would mean that two such metrics $ \big( a_{e} \big)_{e \in E}$ and $ \big( b_{e} \big)_{e \in E}$ are conformal ? Thanks in advance, LC.

I would like to know if there is any notion of a discrete Riemannian metric on graphs ? C. Mercat has worked on discrete Riemann Surfaces, but that's not exactly what I am working on. To be more precise, when one defines a weighted Laplacian $\Delta : \mathbb{R}^V \mapsto \mathbb{R}^V $ on a graph $G=(V,E)$ by

$$ \forall v \in V, \quad \delta (f)(v)=\sum_{w \sim v} a_{\{v,w\}} \big( f(v)-f(w) \big),$$ the family $ \big( a_{e} \big)_{e \in E}$ of reals indexed by the set of edges can be viewed as a metric on the graph. But the analogy with the case of riemannian metric on manifolds is not so obvious. For instance, what would mean that two such metrics $ \big( a_{e} \big)_{e \in E}$ and $ \big( b_{e} \big)_{e \in E}$ are conformal ? Thanks in advance, LC.

I would like to know if there is any notion of a discrete Riemannian metric on graphs. C. Mercat has worked on discrete Riemann Surfaces, but that's not exactly what I am working on. To be more precise, when one defines a weighted Laplacian $\Delta : \mathbb{R}^V \mapsto \mathbb{R}^V $ on a graph $G=(V,E)$ by

$$ \forall v \in V, \quad \delta (f)(v)=\sum_{w \sim v} a_{\{v,w\}} \big( f(v)-f(w) \big),$$ the family $ \big( a_{e} \big)_{e \in E}$ of reals indexed by the set of edges can be viewed as a metric on the graph. But the analogy with the case of riemannian metric on manifolds is not so obvious. For instance, what would mean that two such metrics $ \big( a_{e} \big)_{e \in E}$ and $ \big( b_{e} \big)_{e \in E}$ are conformal ? Thanks in advance, LC.

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Does there exist a notion of discrete riemannian metric on graph?

I would like to know if there is any notion of a discrete Riemannian metric on graphs ? C. Mercat has worked on discrete Riemann Surfaces, but that's not exactly what I am working on. To be more precise, when one defines a weighted Laplacian $\Delta : \mathbb{R}^V \mapsto \mathbb{R}^V $ on a graph $G=(V,E)$ by

$$ \forall v \in V, \quad \delta (f)(v)=\sum_{w \sim v} a_{\{v,w\}} \big( f(v)-f(w) \big),$$ the family $ \big( a_{e} \big)_{e \in E}$ of reals indexed by the set of edges can be viewed as a metric on the graph. But the analogy with the case of riemannian metric on manifolds is not so obvious. For instance, what would mean that two such metrics $ \big( a_{e} \big)_{e \in E}$ and $ \big( b_{e} \big)_{e \in E}$ are conformal ? Thanks in advance, LC.