PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Combinatorics → Graph Theory ↓


Graph Isomorphisms


In the field of graph theory, a major branch of combinatorics, graphs are fundamental objects used to model pairwise relationships between objects. They consist only of nodes (or vertices) and edges (connections between nodes). Graph isomorphism is an important concept that allows us to determine that two graphs are essentially the same, even if they look different at first glance.

Understanding graph isomorphism

Two graphs are said to be isomorphic if there is a one-to-one correspondence between their vertex sets and edge sets, preserving connectivity. In simple terms, you can think of isomorphic graphs as the same graph drawn in different ways.

A graph isomorphism is a mapping f between the vertex sets of two graphs G and H :
F : V(G) → V(H)
such that G contains an edge (u,v) if and only if H contains an edge (f(u),f(v)) .

Consider the following example:

A B C D 1 2 3 4

Although the graphs above look different because of their layout, they are actually isomorphic. The mapping can be as follows:

  • a → 1
  • b → 2
  • c → 3
  • D → 4

Under this mapping, vertex combinations and their corresponding edges are preserved.

Properties of isomorphic graphs

Isomorphic graphs have several important properties. Here are some important properties:

  • Equal number of vertices: If two graphs are isomorphic, then they must have the same number of vertices.
  • Same number of edges: Isomorphic graphs have even number of edges.
  • Vertex degree: The degree (number of edges connected to a vertex) sequence of a graph must match the degree sequence of another graph.
  • Structure: The overall connectivity and pattern of connections will remain unchanged.

Identifying symmetry

Identifying whether two graphs are similar can sometimes be simple and sometimes very challenging. Here is a more structured approach to determine graph similarity:

1. Compare vertex and edge counts

The first step is to make sure that both graphs have the same number of vertices and edges. If this is not the case, they cannot be similar.

2. Compare vertex degree sequences

An effective strategy is to compare the degree sequences of the vertices in both graphs, which should be the same.

3. Analyze local structures

Look deeper into connectivity patterns beyond just vertex degrees. Consider the presence of subgraphs, cycles, and other structures to find recognizable patterns.

4. Attempt to create a map

Finally, try to create vertex mappings explicitly. Every mapping attempt should ensure that the connectivity between two graphs is preserved.

Working example

Let us work through some scenarios to better understand how to identify graph isomorphism.

Example 1: Simple path graph

Consider a two-path graph with three vertices:

A B C 1 2 3

Both path graphs have 3 vertices and 2 edges arranged linearly. Both vertices-degree sequences are [1, 2, 1]. They are actually isomorphic with a simple mapping:

  • a → 1
  • b → 2
  • c → 3

Example 2: Complete graph

Now consider two complete graphs with four vertices:

A B C D 1 2 3 4

In these graphs every vertex connects to every other vertex resulting in the vertex-degree sequence [3, 3, 3, 3]. These graphs are similar in structure and are isomorphic.

Challenges in graph isomorphism

The graph isomorphism problem, determining whether two graphs are isomorphic, is computationally challenging. This problem has puzzled mathematicians and computer scientists for decades. Although it is considered neither easy nor difficult to prove, recent progress has continually pushed the boundaries of our understanding of graph isomorphism.

Although it is relatively easy to manually analyze small graphs for isomorphism, algorithmic solutions become necessary when the size of the graph grows. Researchers have developed several algorithmic approaches to tackle this problem, each with its own strengths and weaknesses.

Algorithmic approach

  • Group theoretic approach: These methods use symmetries in graphs based on group theoretic concepts.
  • Coloring algorithms: Coloring techniques repeatedly differentiate vertices to help identify graph symmetries.
  • Canonical labeling: This involves labeling each graph in a standardized way and comparing the labels for similarity.

Conclusion

Understanding graph isomorphism is important in the study of graph theory and combinatorics. It helps identify underlying similarities between graphs by looking at their superficial differences. These insights benefit many applications ranging from chemistry, biology, network analysis, and computer science, especially in data structure optimization and software engineering.


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Graph Isomorphisms

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