Define the terms 'graph isomorphism' and 'graph automorphism' in Graph Theory.

In Graph Theory, the terms 'graph isomorphism' and 'graph automorphism' refer to important concepts related to the study of graphs.

1. Graph Isomorphism:
Graph isomorphism is a concept that compares two graphs to determine if they are structurally identical. In other words, two graphs G and H are said to be isomorphic if there exists a one-to-one correspondence between their vertices such that the adjacency relationships between vertices are preserved. This means that for every vertex in G, there is a corresponding vertex in H, and vice versa, and the edges between corresponding vertices are the same in both graphs.

Formally, let G = (V, E) and H = (W, F) be two graphs. G and H are isomorphic if there exists a bijective function f: V -> W such that for any two vertices u and v in V, (u, v) is an edge in G if and only if (f(u), f(v)) is an edge in H.

Determining graph isomorphism is a challenging problem in computer science, and no efficient algorithm has been found yet. It falls under the class of problems known as NP (nondeterministic polynomial time) problems.

2. Graph Automorphism:
Graph automorphism refers to a special type of isomorphism where a graph is isomorphic to itself. In other words, an automorphism of a graph is an isomorphism from the graph to itself. It is a symmetry operation that preserves the structure of the graph.

Formally, let G = (V, E) be a graph. An automorphism of G is a bijective function f: V -> V such that for any two vertices u and v in V, (u, v) is an edge in G if and only if (f(u), f(v)) is an edge in G.

Graph automorphisms are important in understanding the symmetries and properties of graphs. They can be used to simplify the study of graphs by identifying equivalent vertices and reducing the search space for certain graph algorithms.

Determining graph automorphisms is also a challenging problem, and finding all automorphisms of a graph is generally computationally expensive. However, there are efficient algorithms available for certain classes of graphs, such as trees and regular graphs.

In summary, graph isomorphism compares two graphs for structural equivalence, while graph automorphism focuses on the symmetry and self-isomorphism of a single graph. Both concepts play significant roles in the analysis and classification of graphs in Graph Theory.