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How does Topology relate to Graph Theory? The Wiki article on Topology motivation states that Euler's paper on the "Seven Bridges of Königsberg problem" was "one of the first papers in topology" and "led to the branch of mathematics known as graph theory."

I understand from Wiki that Graph Theory is:

  • $V$ a set of '''vertices'''
  • $E \subseteq \{ \{x, y\} \mid x, y \in V \;\textrm{ and }\; x \neq y \}$, a set of '''edges'''

Given the claim that topology led to graph theory, I try to represent an undirected graph as a topology, but I do not think this is possible.

The Wiki definition for Topology is that open sets require inclusion of $X$, empty set, closure via arbitrary unions and finite intersections. I then consider the following, which is an idea to represent a typical undirected graph:

$$ X = \{a, b, c\} $$ $$\tau \subset \mathcal{P}(X)$$ $$ \tau = \{\{a,b\},\{b,c\}, \{c, a\}\} $$

The conditions for a topology are not met. So how is topology related to graph theory?

Nick
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  • in graph theory, what matters is what is connected to what; the shape of the connection can be deformed; that is like algebraic topology – J. W. Tanner Jul 01 '21 at 22:25
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    Would it help to tell you that the topological model of a graph is a 1-dimensional simplicial complex? – Lee Mosher Jul 01 '21 at 22:51
  • @BobKrueger no, I'm still investigating "bof"'s comment – Nick Jul 07 '21 at 14:20
  • @bof I'm confused on how your suggestion would work using the concept of neighborhoods for each vertex. Did you mean to suggest forming a topological space for 1. vertex, and another for 2. lines from vertex to other vertices. Now I'm thinking of how to manage all this and make inferences to traverse the graph. – Nick Sep 27 '21 at 03:05
  • @bof I’m getting confused on connectedness. As in the OP, using neighborhoods and satisfying the axioms allows a point to be connected or in a neighborhood of all arbitrary unions and intersections that contain it. For instance, I want a graph G={(a,b),(b,c),(c,d)} but as a topology this results in t = {{a, b, c, d},..}. The neighborhood has the wrong connectedness because one must visit a, then b to d in sequence for the connectedness I desire. Am I perhaps thinking of another mathematical theory for this? – Nick Oct 24 '21 at 00:40

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