As written here and here: given a graph $G=(V,E)$ there is a connection between its Laplacian , $L$ to the continuous Laplace-Beltrami operator $\Delta_g$ defined on a Riemannian manifold $(M,g)$ .
($L = W - D$ where $W$ is a weight matrix that measures distance between the graph nodes, $D$ is a diagonal matrix that sums the weight assigned to each node and $g$ is the metric of the manifold, $M$)
Is the metric $g$ linked to the graph? if so how can I extract it?