In topology, a branch of mathematics, a graph is a topological space which arises from a usual graph
by replacing vertices by points and each edge
by a copy of the unit interval
That is, as topological spaces, graphs are exactly the simplicial 1-complexes and also exactly the one-dimensional CW complexes.
[1] Thus, in particular, it bears the quotient topology of the set under the quotient map used for gluing.
is the 0-skeleton (consisting of one point for each vertex
are the closed intervals glued to it, one for each edge
[1] The topology on this space is called the graph topology.
which is also a graph and whose nodes are all contained in the 0-skeleton of
is a subgraph if and only if it consists of vertices and edges from
is called a tree if it is contractible as a topological space.
[1] This can be shown equivalent to the usual definition of a tree in graph theory, namely a connected graph without cycles.