What does graph isomorphism refer to?

Graph isomorphism refers to the relationship between two graphs that can be transformed into one another through a relabeling of their vertices. This means that there exists a one-to-one correspondence between the vertices of the two graphs such that if two vertices are connected by an edge in one graph, their corresponding vertices in the other graph are also connected by an edge. Essentially, isomorphic graphs have the same structure or connectivity, even if their vertex names differ.

This concept is crucial in many areas of graph theory and computer science, as it helps in identifying when two graphs are fundamentally the same despite possibly being labeled differently or appearing distinct at first glance. Understanding graph isomorphism is essential for solving various problems related to network analysis, data representation, and pattern recognition.