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 theory is a fascinating and dynamic field within the branch of mathematics known as combinatorics. It is the study of graphs, which are structures used to model pairwise relationships between objects. A graph is made up of vertices (also called nodes or points) and edges (also called links or lines) that connect these vertices.

What is a graph?

At its most basic, a graph is made up of two sets:

  • Vertices (or nodes): These are the individual points or locations in the graph. They represent entities in real-world scenarios. The set of all vertices is usually denoted as V
  • Edges (or links): These are the connections between vertices. In practice, edges represent relationships, interactions, or paths between the entities represented by the vertices. The set of all edges is denoted as E

Therefore, a graph is defined as G = (V, E), where V is the set of vertices and E is the set of edges.

Types of graphs

There are several types of graphs, each with its own specific properties:

Undirected graph

In an undirected graph, the edges have no orientation. Edge (u, v) is the same as edge (v, u). Such a graph is simply a collection of points connected by lines.

Directed graphs (digraphs)

In a directed graph, each edge has a direction, meaning that the edge (u, v) is not the same as (v, u). These graphs are used extensively in computer networks, where the direction represents flow.

Weighted graphs

In a weighted graph, each edge has a numerical value or weight. These weights can represent different metrics such as distance, cost, or capacity. Weighted graphs are important in problems such as finding the shortest path or minimum spanning tree.

7

Simple graphs

A simple graph is one that has no loops or multiple edges. This means that each edge connects two different vertices, and there is only one edge between any two vertices.

Complete graph

A complete graph is one in which every pair of vertices is connected by an edge. If a complete graph has n vertices, it can be represented as K n.

Graph terminology

To study graphs effectively, it is necessary to understand the specific terminology:

  • Path: A sequence of vertices where each adjacent pair is connected by an edge.
  • Cycle: A path that starts and ends at the same vertex without repeating any edge or vertex.
  • Connected graph: A graph in which there is a path between every pair of vertices.
  • Degree of a vertex: In an undirected graph, it is the number of edges coming to the vertex. In a directed graph, it consists of in-degree (number of edges coming to the vertex) and out-degree (number of edges going out from the vertex).

Graph representation

There are several ways to represent a graph, each with its own advantages and disadvantages:

Adjacency matrix

It is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices in the graph are adjacent or not.

Let G be a graph with n vertices v 1 , v 2 , ..., v n : A = [a ij ] where a ij = [ begin{cases} 1 & text{if there is an edge from } v_{i} text{ to } v_{j}, \ 0 & text{otherwise}. end{cases} ]

Adjacency list

This representation lists all the vertices connected to a vertex. It is often more space-efficient than the adjacency matrix for graphs with a small number of edges (sparse graphs).

Applications of graph theory

Graph theory is widely applicable in many areas:

Social network

Graphs are used to model social networks, where vertices represent users and edges represent relationships such as friendships or followers.

Network analysis

In computer science, graphs are used to study network topology, optimize network traffic, and design network algorithms.

Biology

Diagrams can depict molecular structures in chemistry and biological networks such as food chains or neural networks.

Transportation

Graph theory helps with routing and navigation problems, where intersections are vertices and roads are edges.

Famous problems in graph theory

Graph theory has generated a number of famous problems, many of which have been solved, while others remain unsolved:

Eulerian path

An Eulerian path is one that visits all the edges of a graph exactly once. If such a path exists, the graph is called Eulerian. The famous Seven Bridges of Königsberg problem is an example of finding an Eulerian path.

Hamiltonian path

A Hamiltonian path visits each vertex exactly once. If the path is a cycle, it becomes a Hamiltonian cycle. Determining whether such paths and cycles exist in a given graph is a complicated problem.

Important theorems and concepts

Königsberg bridge problem

The Seven Bridges of Königsberg is a historically important problem that led to the development of graph theory. The city of Königsberg had seven bridges, and the problem was to determine whether one could walk through the city and cross each bridge exactly once.

Königsberg theorem

Euler showed that this was impossible and concluded: you can visit each edge only once if the graph has two vertices of zero or odd degree.

Conclusion

Graph theory remains a vibrant field of research, with wide applications in Internet structure, computational biology, social networks, electrical circuits, and much more. As networks become more complex and widespread, the importance and applicability of graph theory will grow.


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Graph Theory

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