The adjacency matrix of a graph is a square matrix that represents the edges of the graph. If two graphs have the same adjacency matrix, then they have the same number of vertices, the same number of edges, and the same pattern of edges. Therefore, they are isomorphic.

A complete graph is a graph in which every vertex is connected to every other vertex. A star graph is a graph in which one vertex is connected to all other vertices. Both of these graphs have the same adjacency matrix, so they are isomorphic.

A cycle graph is a graph in which the vertices are arranged in a cycle, and each vertex is connected to the two adjacent vertices. A path graph is a graph in which the vertices are arranged in a line, and each vertex is connected to the two adjacent vertices. These two graphs have different adjacency matrices, so they are not isomorphic.

The fastest known algorithm for graph isomorphism is the Weisfeiler-Lehman algorithm, which has a time complexity of O(2^n). This algorithm is based on the idea of repeatedly refining the vertex labels of the graph until the labels of isomorphic graphs are identical.

The chromatic number of a graph is the minimum number of colors needed to color the vertices of the graph so that no two adjacent vertices have the same color. The chromatic number is not preserved under graph isomorphism. For example, the cycle graph C_5 has a chromatic number of 3, while the star graph S_5 has a chromatic number of 2.

The subgraph isomorphism problem is the problem of determining whether a given graph is a subgraph of another given graph. This problem is NP-complete, which means that it is unlikely that there is a polynomial-time algorithm for solving it.

The maximum common subgraph problem is the problem of finding the largest subgraph that is common to two given graphs. This problem is NP-complete, which means that it is unlikely that there is a polynomial-time algorithm for solving it.

The graph automorphism problem is the problem of finding all of the automorphisms of a given graph. An automorphism of a graph is a permutation of the vertices of the graph that preserves the adjacency relationships between the vertices. This problem is NP-complete, which means that it is unlikely that there is a polynomial-time algorithm for solving it.

Graph isomorphism is used in a variety of applications in computer science, including circuit design, chemical structure analysis, and protein folding. In circuit design, graph isomorphism is used to determine whether two circuits are equivalent. In chemical structure analysis, graph isomorphism is used to determine whether two molecules have the same structure. In protein folding, graph isomorphism is used to determine the three-dimensional structure of a protein.

Graph isomorphism is used in a variety of applications in mathematics, including group theory, number theory, and algebraic geometry. In group theory, graph isomorphism is used to study the structure of groups. In number theory, graph isomorphism is used to study the properties of numbers. In algebraic geometry, graph isomorphism is used to study the geometry of algebraic varieties.

Graph isomorphism is used in a variety of applications in biology, including protein folding, DNA sequencing, and RNA structure prediction. In protein folding, graph isomorphism is used to determine the three-dimensional structure of a protein. In DNA sequencing, graph isomorphism is used to assemble the sequence of nucleotides in a DNA molecule. In RNA structure prediction, graph isomorphism is used to predict the three-dimensional structure of an RNA molecule.

Graph isomorphism is used in a variety of applications in chemistry, including chemical structure analysis, drug design, and materials science. In chemical structure analysis, graph isomorphism is used to determine whether two molecules have the same structure. In drug design, graph isomorphism is used to design new drugs that are similar to existing drugs but have improved properties. In materials science, graph isomorphism is used to study the structure of materials and to design new materials with improved properties.

Graph isomorphism is used in a variety of applications in physics, including quantum mechanics, statistical mechanics, and condensed matter physics. In quantum mechanics, graph isomorphism is used to study the structure of atoms and molecules. In statistical mechanics, graph isomorphism is used to study the behavior of large systems of particles. In condensed matter physics, graph isomorphism is used to study the structure and properties of solids.

Graph isomorphism is used in a variety of applications in engineering, including circuit design, mechanical engineering, and civil engineering. In circuit design, graph isomorphism is used to determine whether two circuits are equivalent. In mechanical engineering, graph isomorphism is used to study the structure of machines and to design new machines. In civil engineering, graph isomorphism is used to study the structure of bridges and buildings and to design new bridges and buildings.

Graph isomorphism is used in a variety of applications in computer graphics, including 3D modeling, animation, and rendering. In 3D modeling, graph isomorphism is used to create 3D models of objects. In animation, graph isomorphism is used to create animations of objects. In rendering, graph isomorphism is used to generate images of objects.