Isomorphism vs equality of graphs

A graph is a set of vertices and edges. Isomorphic graphs look the same but aren't. For example, the persons in a household can be turned into a graph by decalring that there is an edge $ab$ whenever $a$ is parent or child of $b$. So the graph of the inhabitants of a certin house in Evergreen terrace, Springfield, consists of five vertices where each of "Marge" and "Homer" is directly connected with each of "Bart", "Lisa", "Maggie". We would obtain an isomorphic graph with any other typical couple-with-three-kids household, but not the same (i.e. not all these graphs will contain a vertex named Homer).