In computer science we refer to the "shape" of a graph as the graph's topology. Here is an example of this usage.
In computer science a graph topology T is a a set of unlabled graphs which all share a given property or set of properties. A graph A has topology T if an isomorphism exists between A and one of the graphs in T.
I am, however, aware of the fact that mathematicians DO NOT use the word topology in this way. Is there another word that graph theorists use?