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 one-dimensional $CW$ -complex.