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graph topology
A graph $(V,E)$ is identified by its vertices $V=\{v_{1},v_{2},\ldots\}$ and its edges $E=\{\{v_{i},v_{j}\},\{v_{k},v_{l}\},\ldots\}$. A graph also admits a natural topology, called the graph topology, by identifying every edge $\{v_{i},v_{j}\}$ with the unit interval $I=[0,1]$ and gluing them together at coincident vertices.
This construction can be easily realized in the framework of simplicial complexes. We can form a simplicial complex $G=\left\{\{v\}\mid v\in V\right\}\cup E$. And the desired topological realization of the graph is just the geometric realization $G$ of $G$.
Viewing a graph as a topological space has several advantages:

The notion of graph isomorphism becomes that of simplicial (or cell) complex isomorphism.

The notion of a connected graph coincides with topological connectedness.

A connected graph is a tree if and only if its fundamental group is trivial.
Remark: A graph is/can be regarded as a onedimensional $CW$complex.
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