My concern is about finding a mathematical model in order to describe graphs as combinatorial structures (with operations like edge addition, deletion and so on), and as elements in the Euclidean space.
Dealing with combinatorics for graphs is easy, Matroids are very good tool for this. They can describe many operations from node contraction to edge deletion and graph union and so on. Matroids actually perform all these operations in the most suitable way
Matroids and... metric?
How about Matroids and metric when vertices in a graph are points in the Euclidean space? How is it possible to use a metric in Matroids?
Some problems in literature are like finding minimal paths or considering nodes whose distance is the lowest from a given point and so on... Is there any known procedure to do this?
Another example is describing graphs construction using a k-NN approach (nearest neighbors). This is quite an issue for example. Can Matroids deal with a metric?