Let's take an $n\times n$ mesh (with $N=n^2$ unknowns) and think about whether you can enumerate them in such a way that you end up with a bandwidth less than $m=n=\sqrt{N}$? You get this bandwidth with a 5-point stencil if you enumerate the first row left to right, then the next row left to right, etc. In that case, each degree of freedom $i$ couples with $...


I mean I think you clearly showed the graph $G$ can induce a metric, namely define $d_G(u,v)$ as the shortest path from $u \in V(G)$ to $v \in V(G)$ and you can readily show the three properties you define. The graph $G_c$ is not even needed in this discussion, as far as I can tell. Here is a link to some document that describes a similar claim.


Maybe this is not a full answer to your question. However, I am currently developing my own unstructured mesh generator and found this toolbox quite helpful. Perhaps there are some algorithms which will help you. Matlab or Github Short (not complete) overview of the genetic algorithms for the TPS problem: Solves the classic Traveling Salesman Problem (TSP)...

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