Revision as of 08:29, 6 July 2025

Graph Theory

Graph theory is a significant branch of mathematics and computer science that studies the properties and applications of graphs, which are structures used to model pairwise relations between objects. Graphs are composed of vertices (also called nodes) and edges that connect pairs of vertices. This article provides a comprehensive overview of graph theory, including its history, theoretical foundations, various types, applications, and its impact on various fields.

Introduction

Graph theory has emerged as a vital area of study in both pure and applied mathematics. It offers a framework to represent and analyze relationships and has applications ranging from computer networking and social sciences to biology and transportation. The foundational concepts of graph theory are simple but powerful; by understanding how vertices and edges interact, one can derive insights applicable in many domains.

The study of graphs allows for the examination of various problems, including the shortest path, connectivity, network flow, and coloring problems, all of which are useful in real-world applications. This article explores these areas in depth while elucidating the historical context, theoretical developments, and practical implications of graph theory.

History

The roots of graph theory can be traced back to the 18th century, with its formal introduction credited to the Swiss mathematician Leonhard Euler. Euler's seminal work in 1736 explored the "Seven Bridges of Königsberg" problem, which asked whether it was possible to traverse the city's seven bridges without crossing any of them more than once. Euler demonstrated that such a path did not exist and, in doing so, laid the groundwork for the formal study of graphs.

In the years that followed, mathematicians began to explore various aspects of graph theory. In the late 19th century, mathematician Gustav Kirchhoff utilized graph theory in his studies of electrical circuits, providing a vital bridge between mathematics and engineering.

The 20th century saw a tremendous expansion in both the theoretical framework and applications of graph theory. With the advent of computers, algorithms developed from graph theory found practical usage in optimizing networks, searching databases, and managing data structures. Key developments included the introduction of various algorithms, such as Dijkstra's algorithm for finding the shortest path and the Ford-Fulkerson method for network flow.

In contemporary research, graph theory continues to evolve, with significant advancements in areas such as random graphs, graph coloring, and spectral graph theory. Major collaborations between mathematicians and computer scientists have facilitated the discovery of new problems and solutions, making graph theory one of the most dynamic areas of research in modern mathematics.

Fundamental Concepts

Graph theory encompasses a wide array of definitions and concepts. Below are several fundamental components that provide a structure to the subject.

Definition of Graphs

A graph is defined as a pair G = (V, E), where V is a set of vertices (or nodes) and E is a set of edges that connect pairs of vertices. There are various types of graphs, including:

Undirected Graphs: In this type of graph, the edges have no orientation. If there is an edge connecting vertices u and v, it implies a connection that is bidirectional.
Directed Graphs (Digraphs): Here, each edge has a direction, indicating a one-way relationship between connected vertices. If an edge from u to v is present, it does not imply an edge from v to u unless explicitly stated.
Weighted Graphs: In weighted graphs, edges are assigned weights (or costs), allowing for the representation of complex relations, such as distances or capacities.
Simple Graphs: These graphs contain no loops (edges connected at both ends to the same vertex) and no multiple edges between any pair of vertices.
Cyclic Graphs and Acyclic Graphs: A cyclic graph contains at least one cycle, whereas an acyclic graph contains no cycles. Directed acyclic graphs (DAGs) play a crucial role in various computational problems, such as task scheduling.

Important Terms

Understanding graph theory also involves familiarization with important terms related to graphs, including:

Degree: The degree of a vertex is the number of edges incident to it. In directed graphs, one can differentiate between in-degree (incoming edges) and out-degree (outgoing edges).
Path and Cycle: A path is a sequence of vertices connected by edges. If the path starts and ends at the same vertex, it forms a cycle.
Connected Graphs: An undirected graph is connected if there is a path between any pair of vertices; otherwise, it is disconnected.
Subgraph: A subgraph is formed by a subset of vertices and edges from a larger graph.
Isomorphism: Two graphs are isomorphic if there exists a one-to-one correspondence between their vertices and edges that preserves adjacency.

Graph Representations

Graphs can be represented in various forms, facilitating their use in algorithms and computations. Common representations include:

Adjacency Matrix: This is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not. For an undirected graph with n vertices, the matrix is n×n.
Adjacency List: An adjacency list is a collection of lists or arrays where each vertex has a list of adjacent vertices. This representation is more space-efficient than an adjacency matrix, especially for sparse graphs.
Edge List: An edge list is a simple representation where the graph is described by listing all its edges, each representing a pair of vertices.

Applications

Graph theory's versatility results in its applications across many fields. This section reviews several of these applications, illustrating the breadth of graph theory's influence.

Computer Science

In computer science, graph theory is a foundation for data structures, algorithms, and computational problems. The following are notable applications:

Network Routing: Algorithms derived from graph theory are essential for optimizing data routing on the Internet. Protocols such as OSPF (Open Shortest Path First) utilize graph-based algorithms to determine the optimal path for data packets.
Social Network Analysis: Graphs model social networks, where individuals are represented by vertices and relationships by edges. Analyzing these graphs allows researchers to study social structures, recommend friends, and measure influences.
Graph Databases: Specialized databases such as Neo4j utilize graph structures to efficiently store and query interconnected data, allowing for rapid association queries and complex relationship exploration.

Transportation and Logistics

Graph theory plays a crucial role in transportation and logistics management. Notable applications include:

Route Planning: Graph models are employed for route optimization in logistics, reducing travel time and costs in delivery systems.
Traffic Flow Analysis: Cities model their traffic systems as graphs, allowing engineers and planners to analyze congestion patterns and optimize traffic signal timing.
Infrastructure Design: Graph theory is applied in designing transportation networks, ensuring efficient connectivity among various nodes (e.g., airports, train stations).

= Biology

In biology, graph theory offers tools for understanding complex biological systems:

Protein-Protein Interaction Networks: Biological graphs can represent interactions among proteins. Analyzing these graphs helps biologists understand cellular processes and the emergence of diseases.
Phylogenetics: Graphs can represent evolutionary relationships via cladograms, aiding researchers in studying the genetic connections among species.

Operations Research

Graph theory provides methodologies for solving optimization problems in operations research. Applications include:

Scheduling Problems: Graph-based models are utilized to schedule tasks in industrial and manufacturing processes efficiently, minimizing delay and resource use.
Project Management: The Critical Path Method (CPM) leverages directed acyclic graphs to help manage project scheduling by identifying key tasks that determine project duration.

Real-world Examples

Graph theory not only serves academic pursuits but heavily informs real-world problems in dynamic contexts. Below are several examples highlighting the practical applications of graph theory.

Telecommunications

Telecommunications networks can be effectively modeled as graphs, where nodes represent switches or routers, and edges represent communication links. Efficient routing algorithms rely on graph theory to ensure the reliability and speed of data transfer across networks, particularly in managing bandwidth and reducing latency.

Epidemiology

During epidemics, the spread of a disease can be represented through graphs where nodes symbolize individuals and edges denote interactions or contacts. Understanding the structure of the underlying network allows public health officials to devise effective strategies for containment and intervention, thereby curbing outbreaks.

Urban Planning

Urban planners utilize graph theory to design efficient public transportation systems. By modeling bus routes and stations as graphs, planners analyze connectivity and accessibility to ensure the urban transport network adequately meets the needs of the population.

Influence and Impact

Graph theory's impact spans numerous disciplines, shaping the course of research and practice in mathematics, computer science, engineering, social sciences, and biology. Its influence extends beyond theoretical exploration due to the rise of data-driven decision-making that relies on insights derived from graph-based models.

Educational Relevance

Graph theory is included in university curricula within mathematics and computer science courses, forming a crucial part of the coursework. As students engage with graph theory, they develop problem-solving skills that are applicable to diverse areas, preparing them for careers in technology, research, and analysis.

Technological Advancements

The rapid advancement of computer technology and data analytics tools further enhances the relevance of graph theory. Machine learning, artificial intelligence, and big data methodologies increasingly apply graph-theoretic principles, leading to breakthroughs in algorithms and applications.

Criticism and Controversies

Though graph theory has garnered substantial attention and application, it is not without its criticisms. Key areas of concern include:

Complexity of Problems: Some graph problems are NP-hard, indicating a lack of efficient algorithms for solving them in a general case. This complexity underlines limitations in applying graph theory to certain practical problems.
Model Limitations: While graphs can model relationships, they may oversimplify complex systems, failing to capture critical nuances inherent in interconnected entities. Critics argue for a more nuanced approach that integrates graph theory with other disciplines.
Resource Consumption: Certain graph algorithms may be resource-intensive, leading to performance bottlenecks in large-scale systems. This criticism emphasizes the importance of developing efficient algorithms to suit varied applications.

References

@@ Line 1: / Line 1: @@
 = Graph Theory =
-Graph Theory is a field of mathematics and computer science that involves the study of graphs, which are mathematical structures used to represent pairwise relationships between objects. A graph is composed of vertices (also called nodes) and edges (connections between the vertices). Graph Theory has applications in various domains including computer science, biology, social sciences, and transportation systems.
+Graph theory is a significant branch of mathematics and computer science that studies the properties and applications of graphs, which are structures used to model pairwise relations between objects. Graphs are composed of vertices (also called nodes) and edges that connect pairs of vertices. This article provides a comprehensive overview of graph theory, including its history, theoretical foundations, various types, applications, and its impact on various fields.
 == Introduction ==
-Graphs are ubiquitous in the representation of data and relationships. They serve as a means to model problems in numerous fields. For instance, social networks can be modeled as graphs where individuals are vertices and relationships are the edges connecting them. Similarly, transportation systems can be represented where intersections are nodes and roads are edges. With its wide-ranging applications, Graph Theory has become a fundamental part of modern mathematics and computational theory.
+Graph theory has emerged as a vital area of study in both pure and applied mathematics. It offers a framework to represent and analyze relationships and has applications ranging from computer networking and social sciences to biology and transportation. The foundational concepts of graph theory are simple but powerful; by understanding how vertices and edges interact, one can derive insights applicable in many domains.
-The core elements of a graph can be defined as follows:
+The study of graphs allows for the examination of various problems, including the shortest path, connectivity, network flow, and coloring problems, all of which are useful in real-world applications. This article explores these areas in depth while elucidating the historical context, theoretical developments, and practical implications of graph theory.
-* A **vertex** (or node) is a fundamental unit of a graph.
-* An **edge** is a connection between two vertices.
-* A **weighted graph** assigns a weight to each edge, which can represent costs, distances, or other quantitative measures.
-* A **directed graph** (or digraph) has edges with a direction, indicating a one-way relationship.
-Graph Theory encompasses various sub-disciplines, including combinatorial graph theory, geometric graph theory, and algebraic graph theory, each focusing on different aspects and properties of graphs.
+== History ==
-== History ==
+The roots of graph theory can be traced back to the 18th century, with its formal introduction credited to the Swiss mathematician Leonhard Euler. Euler's seminal work in 1736 explored the "Seven Bridges of Königsberg" problem, which asked whether it was possible to traverse the city's seven bridges without crossing any of them more than once. Euler demonstrated that such a path did not exist and, in doing so, laid the groundwork for the formal study of graphs.
-The development of Graph Theory dates back to the 18th century, with one of the earliest examples being Leonhard Euler's solution to the **Seven Bridges of Königsberg** problem in 1736. Euler proved that it was impossible to walk through the city of Königsberg by crossing each of its seven bridges exactly once, laying the groundwork for the field of Graph Theory.
+In the years that followed, mathematicians began to explore various aspects of graph theory. In the late 19th century, mathematician Gustav Kirchhoff utilized graph theory in his studies of electrical circuits, providing a vital bridge between mathematics and engineering.
-Following Euler's work, the study of graphs gained momentum in the 19th century, primarily with contributions from mathematicians such as August Ferdinand Möbius and Arthur Cayley. The concept of graph coloring was introduced in the late 19th century as a way to solve problems related to map coloring.
+The 20th century saw a tremendous expansion in both the theoretical framework and applications of graph theory. With the advent of computers, algorithms developed from graph theory found practical usage in optimizing networks, searching databases, and managing data structures. Key developments included the introduction of various algorithms, such as Dijkstra's algorithm for finding the shortest path and the Ford-Fulkerson method for network flow.
-In the 20th century, further advancements were made by Claude Shannon in communication theory and by Paul Erdős and Alfréd Rényi, who developed the field of random graphs. The advent of computer science and digital technology in the mid-to-late 20th century significantly broadened the applications of Graph Theory, leading to extensive research and the development of algorithms for graph processing.
+In contemporary research, graph theory continues to evolve, with significant advancements in areas such as random graphs, graph coloring, and spectral graph theory. Major collaborations between mathematicians and computer scientists have facilitated the discovery of new problems and solutions, making graph theory one of the most dynamic areas of research in modern mathematics.
 == Fundamental Concepts ==
-Graph Theory contains several fundamental concepts and definitions which serve as the foundation for more advanced topics. Here are some key concepts:
+Graph theory encompasses a wide array of definitions and concepts. Below are several fundamental components that provide a structure to the subject.
+=== Definition of Graphs ===
-=== Types of Graphs ===
+A '''graph''' is defined as a pair G = (V, E), where V is a set of vertices (or nodes) and E is a set of edges that connect pairs of vertices. There are various types of graphs, including:
-* A **simple graph** contains no loops (edges connected at both ends to the same vertex) and no multiple edges (two edges connecting the same pair of vertices).
+* '''Undirected Graphs''': In this type of graph, the edges have no orientation. If there is an edge connecting vertices u and v, it implies a connection that is bidirectional.
-* A **complete graph** is a simple graph in which every pair of distinct vertices is connected by a unique edge.
+* '''Directed Graphs (Digraphs)''': Here, each edge has a direction, indicating a one-way relationship between connected vertices. If an edge from u to v is present, it does not imply an edge from v to u unless explicitly stated.
-* A **bipartite graph** can be divided into two disjoint sets such that every edge connects a vertex in one set to a vertex in the other set.
+* '''Weighted Graphs''': In weighted graphs, edges are assigned weights (or costs), allowing for the representation of complex relations, such as distances or capacities.
+* '''Simple Graphs''': These graphs contain no loops (edges connected at both ends to the same vertex) and no multiple edges between any pair of vertices.
+* '''Cyclic Graphs and Acyclic Graphs''': A cyclic graph contains at least one cycle, whereas an acyclic graph contains no cycles. Directed acyclic graphs (DAGs) play a crucial role in various computational problems, such as task scheduling.
-=== Graph Representations ===
+=== Important Terms ===
-Graphs can be represented in several ways:
+Understanding graph theory also involves familiarization with important terms related to graphs, including:
-* An **adjacency matrix** is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not in the graph.
+* '''Degree''': The degree of a vertex is the number of edges incident to it. In directed graphs, one can differentiate between in-degree (incoming edges) and out-degree (outgoing edges).
-* An **adjacency list** is a collection of lists or arrays, where each list corresponds to a vertex and contains the vertices that are adjacent to it.
+* '''Path and Cycle''': A path is a sequence of vertices connected by edges. If the path starts and ends at the same vertex, it forms a cycle.
+* '''Connected Graphs''': An undirected graph is connected if there is a path between any pair of vertices; otherwise, it is disconnected.
+* '''Subgraph''': A subgraph is formed by a subset of vertices and edges from a larger graph.
+* '''Isomorphism''': Two graphs are isomorphic if there exists a one-to-one correspondence between their vertices and edges that preserves adjacency.
-=== Important Properties ===
+=== Graph Representations ===
-Various properties are integral to the study of graphs:
+Graphs can be represented in various forms, facilitating their use in algorithms and computations. Common representations include:
-* The **degree** of a vertex is the number of edges incident to it. In a directed graph, the in-degree and out-degree are distinguished.
+* '''Adjacency Matrix''': This is a square matrix used to represent a finite graph. The elements of the matrix indicate whether pairs of vertices are adjacent or not. For an undirected graph with n vertices, the matrix is n×n.
-* A graph is **connected** if there is a path between every pair of vertices. A disconnected graph consists of multiple connected components.
+* '''Adjacency List''': An adjacency list is a collection of lists or arrays where each vertex has a list of adjacent vertices. This representation is more space-efficient than an adjacency matrix, especially for sparse graphs.
-* A graph is **cyclic** if it contains at least one cycle, while it is **acyclic** if it does not.
+* '''Edge List''': An edge list is a simple representation where the graph is described by listing all its edges, each representing a pair of vertices.
-== Applications of Graph Theory ==
+== Applications ==
-Graph Theory finds applications across multiple domains, influencing research and development in various fields.
+Graph theory's versatility results in its applications across many fields. This section reviews several of these applications, illustrating the breadth of graph theory's influence.
 === Computer Science ===
-In computer science, Graph Theory plays a crucial role in data structures, algorithms, and complexity theory. Common applications include:
+In computer science, graph theory is a foundation for data structures, algorithms, and computational problems. The following are notable applications:
-* **Network design:** Graphs are used to model computer networks, facilitating the understanding of data flows and communication paths.
+* '''Network Routing''': Algorithms derived from graph theory are essential for optimizing data routing on the Internet. Protocols such as OSPF (Open Shortest Path First) utilize graph-based algorithms to determine the optimal path for data packets.
-* **Web page ranking:** Algorithms such as Google’s PageRank utilize graph theoretical concepts to rank web pages based on the structure of hyperlinks.
+* '''Social Network Analysis''': Graphs model social networks, where individuals are represented by vertices and relationships by edges. Analyzing these graphs allows researchers to study social structures, recommend friends, and measure influences.
+* '''Graph Databases''': Specialized databases such as Neo4j utilize graph structures to efficiently store and query interconnected data, allowing for rapid association queries and complex relationship exploration.
-=== Social Sciences ===
-Graph Theory is extensively utilized in social sciences to analyze social networks. It aids researchers in understanding complex relationships, social dynamics, and influence patterns among individuals or groups. Through the use of graphs, sociologists can model behaviours, predict outcomes, and develop frameworks for social interactions.
-=== Biology ===
-In biology, Graph Theory is employed to represent and analyze biological networks, such as metabolic pathways and protein-protein interaction networks. Its application helps in visualizing complex relationships and understanding biological processes.
 === Transportation and Logistics ===
-Graphs are instrumental in optimizing transportation and logistics. They are used to model road networks, airline routes, and shipping pathways, helping in route optimization, traffic management, and logistics planning. Algorithms such as Dijkstra's and the A* algorithm leverage graph structures to find the shortest paths or most efficient routes.
+Graph theory plays a crucial role in transportation and logistics management. Notable applications include:
+* '''Route Planning''': Graph models are employed for route optimization in logistics, reducing travel time and costs in delivery systems.
+* '''Traffic Flow Analysis''': Cities model their traffic systems as graphs, allowing engineers and planners to analyze congestion patterns and optimize traffic signal timing.
+* '''Infrastructure Design''': Graph theory is applied in designing transportation networks, ensuring efficient connectivity among various nodes (e.g., airports, train stations).
-== Algorithms in Graph Theory ==
+=== Biology ==
-The study of Graph Theory has led to the development of numerous algorithms that solve various graph-related problems. Several notable algorithms include:
+In biology, graph theory offers tools for understanding complex biological systems:
+* '''Protein-Protein Interaction Networks''': Biological graphs can represent interactions among proteins. Analyzing these graphs helps biologists understand cellular processes and the emergence of diseases.
+* '''Phylogenetics''': Graphs can represent evolutionary relationships via cladograms, aiding researchers in studying the genetic connections among species.
-=== Depth-First Search (DFS) ===
+=== Operations Research ===
-DFS is an algorithm for traversing or searching through graphs. It explores as far as possible along each branch before backtracking, making it useful for tasks such as pathfinding and topological sorting.
+Graph theory provides methodologies for solving optimization problems in operations research. Applications include:
+* '''Scheduling Problems''': Graph-based models are utilized to schedule tasks in industrial and manufacturing processes efficiently, minimizing delay and resource use.
+* '''Project Management''': The Critical Path Method (CPM) leverages directed acyclic graphs to help manage project scheduling by identifying key tasks that determine project duration.
-=== Breadth-First Search (BFS) ===
+== Real-world Examples ==
-In contrast to DFS, BFS explores all neighboring nodes at the present depth prior to moving on to nodes at the next depth level. It is especially useful in finding the shortest path in unweighted graphs.
+Graph theory not only serves academic pursuits but heavily informs real-world problems in dynamic contexts. Below are several examples highlighting the practical applications of graph theory.
-=== Dijkstra’s Algorithm ===
+=== Telecommunications ===
-Dijkstra's Algorithm is employed to find the shortest path from a starting node to all other nodes in a weighted graph. It is widely used in routing and navigation applications.
+Telecommunications networks can be effectively modeled as graphs, where nodes represent switches or routers, and edges represent communication links. Efficient routing algorithms rely on graph theory to ensure the reliability and speed of data transfer across networks, particularly in managing bandwidth and reducing latency.
-=== Kruskal’s and Prim’s Algorithms ===
+=== Epidemiology ===
-These algorithms are used to find the minimum spanning tree of a connected, weighted graph. They minimize the total edge weight needed to connect all vertices, relevant in the design of efficient networks.
+During epidemics, the spread of a disease can be represented through graphs where nodes symbolize individuals and edges denote interactions or contacts. Understanding the structure of the underlying network allows public health officials to devise effective strategies for containment and intervention, thereby curbing outbreaks.
-== Challenges and Open Problems ==
+=== Urban Planning ===
-Despite extensive research, Graph Theory still poses numerous challenges and open problems. Some notable examples include:
+Urban planners utilize graph theory to design efficient public transportation systems. By modeling bus routes and stations as graphs, planners analyze connectivity and accessibility to ensure the urban transport network adequately meets the needs of the population.
-=== The Graph Isomorphism Problem ===
+== Influence and Impact ==
-The Graph Isomorphism Problem involves determining whether two graphs are isomorphic, meaning they can be transformed into each other by renaming vertices. The difficulty of solving this problem has intrigued mathematicians and computer scientists for decades, as it resides in a grey area between P and NP-complete problems.
+Graph theory's impact spans numerous disciplines, shaping the course of research and practice in mathematics, computer science, engineering, social sciences, and biology. Its influence extends beyond theoretical exploration due to the rise of data-driven decision-making that relies on insights derived from graph-based models.
-=== The Traveling Salesman Problem (TSP) ===
+=== Educational Relevance ===
-TSP seeks to find the shortest possible route that visits a set of cities exactly once and returns to the origin city. Despite its simplicity, TSP is NP-hard, making it one of the most studied problems in combinatorial optimization.
+Graph theory is included in university curricula within mathematics and computer science courses, forming a crucial part of the coursework. As students engage with graph theory, they develop problem-solving skills that are applicable to diverse areas, preparing them for careers in technology, research, and analysis.
-=== The Four Color Theorem ===
+=== Technological Advancements ===
-The Four Color Theorem states that any planar graph can be colored using no more than four colors such that no adjacent vertices share the same color. This theorem is a significant result in Graph Theory, with proofs that rely on extensive computer checks.
+The rapid advancement of computer technology and data analytics tools further enhances the relevance of graph theory. Machine learning, artificial intelligence, and big data methodologies increasingly apply graph-theoretic principles, leading to breakthroughs in algorithms and applications.
-== Influence and Impact ==
-The influence of Graph Theory extends far beyond mathematics. It has transformed multiple fields through its applications and continues to be a vital area of research. Innovative algorithms derived from Graph Theory principles have significant implications for technology and industry.
+== Criticism and Controversies ==
-The advancement of technology, particularly in data science, artificial intelligence, and machine learning, heavily relies on graph-based structures and algorithms. With the advent of big data and networked systems, the relevant applications of Graph Theory are expanding, fostering research on new heuristics and techniques for analysis.
+Though graph theory has garnered substantial attention and application, it is not without its criticisms. Key areas of concern include:
+* '''Complexity of Problems''': Some graph problems are NP-hard, indicating a lack of efficient algorithms for solving them in a general case. This complexity underlines limitations in applying graph theory to certain practical problems.
+* '''Model Limitations''': While graphs can model relationships, they may oversimplify complex systems, failing to capture critical nuances inherent in interconnected entities. Critics argue for a more nuanced approach that integrates graph theory with other disciplines.
+* '''Resource Consumption''': Certain graph algorithms may be resource-intensive, leading to performance bottlenecks in large-scale systems. This criticism emphasizes the importance of developing efficient algorithms to suit varied applications.
-== See Also ==
+== See also ==
-* [[Combinatorics]]
 * [[Network Theory]]
-* [[Computer Science]]
+* [[Combinatorial Optimization]]
-* [[Algorithm Complexity]]
+* [[Algorithms]]
-* [[Discrete Mathematics]]
+* [[Data Structures]]
+* [[Linear Programming]]
 == References ==
-* [https://www.graph-theory.com Graph Theory - Basic Concepts]
+* [https://www.graph-theory.com Graph Theory Online Resources]
-* [https://www.khanacademy.org/math/discrete-math/graph-theory/intro-to-graph-theory/v/introduction-to-graph-theory Khan Academy - Introduction to Graph Theory]
+* [https://www.mathworks.com MathWorks - Graph Theory Algorithms]
-* [https://www.mathworks.com/help/matlab/ref/graph.html MATLAB - Graph and Network Algorithms]
+* [https://www.geeksforgeeks.org GeeksforGeeks - Graph Theory Articles]
-* [https://www.geeksforgeeks.org/graph-data-structure-and-algorithms/ GeeksforGeeks - Graph Data Structure and Algorithms]
+* [https://www.khanacademy.org Khan Academy - Introduction to Graph Theory]
-* [https://www.coursera.org/learn/algorithms-part1 Coursera - Algorithms Part I]
+* [https://en.wikipedia.org/wiki/Graph_theory Wikipedia - Graph Theory]
-* [https://www.wolfram.com/language/guide/graph-theory/ Wolfram - Graph Theory]
+* [https://www.wolfram.com Wolfram Alpha - Graph Theory Functions]
 [[Category:Mathematics]]
 [[Category:Discrete Mathematics]]
 [[Category:Graph Theory]]