Editing Graph isomorphism (section)

==Motivation==
The formal notion of "isomorphism", e.g., of "graph isomorphism", captures the informal notion that some objects have "the same structure" if one ignores individual distinctions of "atomic" components of objects in question. Whenever individuality of "atomic" components (vertices and edges, for graphs) is important for correct representation of whatever is modeled by graphs, the model is refined by imposing additional restrictions on the structure, and other mathematical objects are used: [[digraph (mathematics)|digraph]]s, [[labeled graph]]s, [[colored graph]]s, [[rooted tree]]s and so on. The isomorphism relation may also be defined for all these generalizations of graphs:  the isomorphism bijection must preserve the elements of structure which define the object type in question: [[arc (graph theory)|arc]]s, labels, vertex/edge colors, the root of the rooted tree, etc.

The notion of "graph isomorphism" allows us to distinguish [[graph properties]] inherent to the structures of graphs themselves from properties associated with graph representations: [[graph drawing]]s, [[graph (data structure)|data structures for graphs]], [[graph labeling]]s, etc. For example, if a graph has exactly one [[cycle (graph theory)|cycle]], then all graphs in its isomorphism class also have exactly one cycle. On the other hand, in the common case when the vertices of a graph are (''represented'' by) the [[integer]]s 1, 2,... ''N'', then the expression
:<math>\sum_{v \in V(G)} v\cdot\text{deg }v</math>
may be different for two isomorphic graphs.