I'm reading through this paper here downloads.hindawi.com/journals/mpe/2013/815035.pdf where they say "Since a graph can be equipped with a topology to turn it into a a one-dimensional space, we can directly apply persistent homology to a graph filtration." This must be obvious but given a graph what topology on the sets of vertices and edges can produce a 1-dimensional space?
2026-05-04 19:17:19.1777922239
What's the 1-dimensional topology of a graph?
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I believe they're talking about the 'natural' topology of a graph as a realisation of a complex whose $0$-cells are the vertices of the graph, and whose $1$-cells are the edges of the graph.