Does Graph definition carry also labels?

Question

The slides of my prof. say that two isomorphic graphs, with different labels are the same graph.

This must imply that the definition of a graph doesn't carry the nodes and vertices labels. Is this true ?

Answer 1

That is very true and convenient.

Since two isomorphic graphs are basically the same graph we do not need to consider labeling into the definition of graph.

However, when we assign an adjacency matrix to a graph, labeling comes to play an important role because isomorphic graphs may have different adjacency matrices associated to them.