Discrete Math Notes:
Chapter 6: Graphs and Trees
6.3 Graph isomorphism
Isomorphic, a correspondence between the vertex sets of each graph such that there is an edge between two vertices of one graph if and only if there is an edge between the corresponding vertices of the second graph.
preserved under isomorphism, whenever two graphs are isomorphic, one graph has the property if and only if the other graph also has the property.
degree sequence, a list of the degrees of all of the vertices in non-increasing order.
Theorem 6.3.1: Vertex degree preserved under isomorphism, Let's assume that two graphs, G and G'. Let f be an isomorphism from G to G'. For each vertex v in G, the degree of vertex v in G is equal to the degree of vertex f(v) in G'.
Overview:
The two graphs' drawings indicate a connection between the two sets of vertices, making it simple to check that each edge in the left graph corresponds to an edge in the right graph, and vice versa. Isomorphism is defined as a correspondence between the vertex sets of two graphs such that an edge between two vertices of one graph exists only if and only if an edge exists between the corresponding vertices of the other graph. Although the graphs are not identical, the vertices can be relabeled to make them so. If f graph G is isomorphic to graph G', G has a degree d vertex if and only if G' contains a degree d vertex. When two graphs are isomorphic, a property is said to be conserved if one graph has the property if and only if the other graph also has the property.