A graph is identified by its vertices and its edges . A graph also admits a natural topology, called the graph topology, by identifying every edge with the unit interval 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 . And the desired topological realization of the graph is just the geometric realization of .

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.