Graph Theory: Difference between revisions

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Graph Theory

Introduction

Graph theory is a branch of mathematics and computer science that studies the properties and applications of graphs, which are mathematical structures comprising vertices (or nodes) and edges (or links) connecting pairs of vertices. Graph theory has applications in various fields, such as computer science, biology, sociology, and transportation, making it a crucial discipline for understanding relations and connections within complex systems.

Background or History

Graph theory can trace its origins back to the 18th century. The beginnings of this field can be credited to the Swiss mathematician Leonhard Euler. His investigation into the so-called "Seven Bridges of Königsberg" in 1736 revealed the fundamental principles of graph connectivity and paved the way for further studies in this domain. Euler's work illustrated how traversable paths through a network could be analyzed through an abstract approach, marking a significant milestone in the development of graph theory.

Following Euler's pioneering studies, significant advancements in the field emerged throughout the 19th and 20th centuries. Mathematicians such as Gustav Kirchhoff, who explored electrical circuits and network analysis in the mid-19th century, contributed to the application of graph theory in physics and engineering. Moreover, the work of mathematicians like Arthur Cayley and Pierre de Fermat helped establish the foundational concepts of combinatorial connections in graphs.

The modern formulation and rigorous mathematical framework for graph theory began to evolve in the late 20th century, largely through the contributions of theorists such as Claude Shannon, who utilized graphs in the realm of information theory. Today, graph theory encompasses a wide range of topics and has developed into a critical area of study in both pure and applied mathematics.

Fundamental Concepts

Definitions

At its core, a graph \( G \) is represented as \( G = (V, E) \), where \( V \) is a finite set of vertices and \( E \) is a set of edges connecting those vertices. The edges can be classified as directed or undirected, depending on whether the connections between the vertices have a specific direction. Additionally, graphs can be weighted, where each edge has an associated numerical value, or unweighted, in which edges are treated equally.

In graph theory, various types of graphs are studied, including but not limited to:

**Simple graphs:** Graphs that do not contain multiple edges between the same set of vertices or self-loops.
**Multigraphs:** Graphs that may have multiple edges connecting the same pair of vertices.
**Complete graphs:** Graphs in which every pair of distinct vertices is connected by a unique edge.
**Bipartite graphs:** Graphs whose vertices can be divided into two distinct sets, with edges only connecting vertices from different sets.
**Planar graphs:** Graphs that can be drawn on a plane without any edges crossing.

Understanding these foundational concepts allows mathematicians and computer scientists to analyze various problems that can be modeled as graphs.

Graph Representations

Graphs can be represented in several forms, each suitable for different applications and contexts. The two primary representations of graphs are:

**Adjacency matrix:** A square matrix used to represent a finite graph, where the element \( a_{ij} \) indicates whether there is an edge between vertex \( i \) and vertex \( j \). If the graph is weighted, the matrix elements can represent the weights of the edges.
**Adjacency list:** A collection of lists or arrays that represent the neighbors of each vertex. For each vertex in the graph, there is a corresponding list containing the vertices to which it is directly connected.

Each representation has its own advantages and drawbacks. For example, adjacency matrices are efficient for dense graphs but require more space, while adjacency lists are more space-efficient for sparse graphs.

Types of Problems in Graph Theory

Graph theory addresses various types of problems, including but not limited to connectivity, search, optimization, and traversal. Key problems in graph theory include:

**Shortest path problem:** This involves finding the shortest path or minimal distance between two vertices in a graph. Famous algorithms solving this problem include Dijkstra's algorithm, the Bellman-Ford algorithm, and the Floyd-Warshall algorithm.
**Minimum spanning tree problem:** This problem deals with finding a subset of edges that connects all the vertices in the graph with the minimal total edge weight. Prim's and Kruskal's algorithms are commonly used to solve this problem.
**Graph coloring problem:** A challenging combinatorial problem where the goal is to assign colors to the vertices of a graph such that no two adjacent vertices share the same color. The chromatic number of a graph represents the smallest number of colors needed to achieve this.
**Network flow problem:** This problem focuses on optimizing the flow in a network, represented as a directed graph, where capacities are associated with edges. The Max-Flow Min-Cut Theorem is a key result in this area, facilitating the determination of maximum flow in a network.

These problems are central to various applications and have led to significant advancements in algorithm design and computational theory.

Applications in Computing

Algorithms and Data Structures

Graph theory plays an essential role in the design and optimization of algorithms and data structures. Many algorithms rely on graph-based representations for efficient data manipulation and processing. Some common algorithms in this domain include depth-first search (DFS) and breadth-first search (BFS), which are foundational techniques for exploring graph structures. These algorithms are widely employed in applications such as web crawling, social network analysis, and network routing.

Data structures designed to represent graphs, such as trees and heaps, leverage graph theory to optimize computational efficiency in various programming paradigms. Furthermore, graph databases, which are used to store and query interconnected data, utilize graph structures to enhance performance for specific relationships and hierarchy-based queries.

Computer Networks

In computer networking, graph theory is instrumental in modeling various aspects of communication systems. The connections between devices in a network can be represented as graphs, enabling the analysis of data transmission and network reliability. Problems such as routing, which involves finding optimal paths for data packets, can be efficiently addressed using graph-based algorithms.

Protocols, such as the Open Shortest Path First (OSPF) and Routing Information Protocol (RIP), utilize concepts from graph theory to ensure that data is transferred effectively across the network. By representing networks as graphs, it becomes feasible to optimize performance based on parameters such as latency, bandwidth, and fault tolerance.

Social Network Analysis

Graph theory provides a robust framework for analyzing social networks, where individuals are represented as vertices and relationships or interactions as edges. The analysis of social networks enables researchers to uncover patterns, such as the identification of influential individuals (or hubs), the study of community structures, and the detection of social trends.

Metrics derived from graph theory, such as betweenness centrality, closeness centrality, and eigenvector centrality, are widely used to assess the importance of nodes within networks. Social network analysis has applications in fields like marketing, sociology, and even political science, providing critical insights into group dynamics and information dissemination.

Biology and Ecology

In biology and ecology, graph theory is employed to model various biological networks, such as food webs, metabolic pathways, and protein-protein interaction networks. By representing complex interdependencies as graphs, researchers can study the dynamics of ecosystems and the relationships between different species.

For example, ecological networks can illustrate the flow of energy and nutrients across different trophic levels, enabling the identification of keystone species and the overall health of ecosystems. Similarly, in genomics, graph-based approaches help in analyzing genetic interactions and the relationships between genes, facilitating the understanding of complex biological processes.

Real-world Examples

Transportation Networks

Transportation systems can be effectively represented as graphs, with intersections or stations as vertices and routes or connections as edges. This representation allows for the optimization of travel routes, analysis of traffic patterns, and design of efficient public transport systems. For example, the planning of urban transit systems often employs graph theory to minimize travel time for commuters and improve overall network efficiency.

Graph-theoretical concepts are also applied in logistics and supply chain management, where routes for delivery trucks can be optimized to reduce costs and enhance service delivery. Algorithms designed to identify the shortest or most efficient routes can significantly impact transportation costs and operational efficacy.

Internet and World Wide Web

The structure of the Internet can be modeled as a graph, where websites are represented as vertices and hyperlinks or connections between these sites as edges. This graph representation enables various analyses, including searching and ranking algorithms, such as Google's PageRank, which uses graph properties to assess the importance of web pages based on their links.

Additionally, social media platforms utilize graph theory to analyze relationships between users, facilitating the discovery of content and connections. The understanding of user engagement and the spread of information through social media relies heavily on the principles of graph theory to optimize interaction and content delivery.

Network Security

In the realm of cybersecurity, graph theory provides tools for modeling and analyzing network security protocols. Network vulnerabilities can be represented in graph form, allowing security professionals to visualize potential attack paths and assess the risk of data breaches. Graph-based models enable the formulation of strategies to reinforce security measures and protect sensitive information.

Furthermore, anomaly detection techniques often employ graph theoretical methods to identify unusual patterns in network traffic, enabling proactive measures against potential threats. The application of graph theory in network security underscores its relevance in contemporary digital environments.

Criticism or Limitations

Despite its strengths and vast applications, graph theory has limitations and challenges that researchers continue to address. One of the primary criticisms relates to the computational complexity of certain problems in graph theory. Many fundamental problems, including graph coloring and the traveling salesman problem, are NP-hard, meaning that no known polynomial-time algorithms can solve them efficiently for all instances. This computational difficulty has implications for real-world applications, particularly as datasets grow in size and complexity.

Additionally, the scalability of graph algorithms can pose challenges in large networks, where performance may degrade significantly. Researchers are exploring advanced methodologies, such as heuristics and approximation algorithms, to mitigate these limitations and enhance the applicability of graph theory to large-scale problems.

Furthermore, the interpretability of graph models can be a concern, especially in fields like social sciences, where the underlying assumptions about relationships and connections may not always hold true. This lack of interpretability could lead to misrepresentations or erroneous conclusions when applying graph-theoretical analyses to complex social phenomena.

References

@@ Line 1: / Line 1: @@
-= Graph Theory =
+# 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.
 == Introduction ==
+Graph theory is a branch of mathematics and computer science that studies the properties and applications of graphs, which are mathematical structures comprising vertices (or nodes) and edges (or links) connecting pairs of vertices. Graph theory has applications in various fields, such as computer science, biology, sociology, and transportation, making it a crucial discipline for understanding relations and connections within complex systems.
-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.
+== Background or History ==
+Graph theory can trace its origins back to the 18th century. The beginnings of this field can be credited to the Swiss mathematician Leonhard Euler. His investigation into the so-called "Seven Bridges of Königsberg" in 1736 revealed the fundamental principles of graph connectivity and paved the way for further studies in this domain. Euler's work illustrated how traversable paths through a network could be analyzed through an abstract approach, marking a significant milestone in the development of graph theory.
-The core elements of a graph can be defined as follows:
+Following Euler's pioneering studies, significant advancements in the field emerged throughout the 19th and 20th centuries. Mathematicians such as Gustav Kirchhoff, who explored electrical circuits and network analysis in the mid-19th century, contributed to the application of graph theory in physics and engineering. Moreover, the work of mathematicians like Arthur Cayley and Pierre de Fermat helped establish the foundational concepts of combinatorial connections in graphs.
-* 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.
+The modern formulation and rigorous mathematical framework for graph theory began to evolve in the late 20th century, largely through the contributions of theorists such as Claude Shannon, who utilized graphs in the realm of information theory. Today, graph theory encompasses a wide range of topics and has developed into a critical area of study in both pure and applied mathematics.
-== History ==
+== Fundamental Concepts ==
-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.
+=== Definitions ===
+At its core, a graph \( G \) is represented as \( G = (V, E) \), where \( V \) is a finite set of vertices and \( E \) is a set of edges connecting those vertices. The edges can be classified as directed or undirected, depending on whether the connections between the vertices have a specific direction. Additionally, graphs can be weighted, where each edge has an associated numerical value, or unweighted, in which edges are treated equally.
-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.
+In graph theory, various types of graphs are studied, including but not limited to:
+* **Simple graphs:** Graphs that do not contain multiple edges between the same set of vertices or self-loops.
+* **Multigraphs:** Graphs that may have multiple edges connecting the same pair of vertices.
+* **Complete graphs:** Graphs in which every pair of distinct vertices is connected by a unique edge.
+* **Bipartite graphs:** Graphs whose vertices can be divided into two distinct sets, with edges only connecting vertices from different sets.
+* **Planar graphs:** Graphs that can be drawn on a plane without any edges crossing.
-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.
+Understanding these foundational concepts allows mathematicians and computer scientists to analyze various problems that can be modeled as graphs.
-== Fundamental Concepts ==
-Graph Theory contains several fundamental concepts and definitions which serve as the foundation for more advanced topics. Here are some key concepts:
-=== Types of Graphs ===
-* 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).
-* A **complete graph** is a simple graph in which every pair of distinct vertices is connected by a unique edge.
-* 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.
 === Graph Representations ===
+Graphs can be represented in several forms, each suitable for different applications and contexts. The two primary representations of graphs are:
+* **Adjacency matrix:** A square matrix used to represent a finite graph, where the element \( a_{ij} \) indicates whether there is an edge between vertex \( i \) and vertex \( j \). If the graph is weighted, the matrix elements can represent the weights of the edges.
+* **Adjacency list:** A collection of lists or arrays that represent the neighbors of each vertex. For each vertex in the graph, there is a corresponding list containing the vertices to which it is directly connected.
-Graphs can be represented in several ways:
+Each representation has its own advantages and drawbacks. For example, adjacency matrices are efficient for dense graphs but require more space, while adjacency lists are more space-efficient for sparse graphs.
-* 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.
-* 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.
-=== Important Properties ===
-Various properties are integral to the study of graphs:
-* 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.
-* A graph is **connected** if there is a path between every pair of vertices. A disconnected graph consists of multiple connected components.
-* A graph is **cyclic** if it contains at least one cycle, while it is **acyclic** if it does not.
-== Applications of Graph Theory ==
-Graph Theory finds applications across multiple domains, influencing research and development in various fields.
-=== Computer Science ===
-In computer science, Graph Theory plays a crucial role in data structures, algorithms, and complexity theory. Common applications include:
-* **Network design:** Graphs are used to model computer networks, facilitating the understanding of data flows and communication paths.
-* **Web page ranking:** Algorithms such as Google’s PageRank utilize graph theoretical concepts to rank web pages based on the structure of hyperlinks.
-=== 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.
-== Algorithms in Graph Theory ==
+=== Types of Problems in Graph Theory ===
+Graph theory addresses various types of problems, including but not limited to connectivity, search, optimization, and traversal. Key problems in graph theory include:
+* **Shortest path problem:** This involves finding the shortest path or minimal distance between two vertices in a graph. Famous algorithms solving this problem include Dijkstra's algorithm, the Bellman-Ford algorithm, and the Floyd-Warshall algorithm.
+* **Minimum spanning tree problem:** This problem deals with finding a subset of edges that connects all the vertices in the graph with the minimal total edge weight. Prim's and Kruskal's algorithms are commonly used to solve this problem.
+* **Graph coloring problem:** A challenging combinatorial problem where the goal is to assign colors to the vertices of a graph such that no two adjacent vertices share the same color. The chromatic number of a graph represents the smallest number of colors needed to achieve this.
+* **Network flow problem:** This problem focuses on optimizing the flow in a network, represented as a directed graph, where capacities are associated with edges. The Max-Flow Min-Cut Theorem is a key result in this area, facilitating the determination of maximum flow in a network.
-The study of Graph Theory has led to the development of numerous algorithms that solve various graph-related problems. Several notable algorithms include:
+These problems are central to various applications and have led to significant advancements in algorithm design and computational theory.
-=== Depth-First Search (DFS) ===
+== Applications in Computing ==
-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.
+=== Algorithms and Data Structures ===
+Graph theory plays an essential role in the design and optimization of algorithms and data structures. Many algorithms rely on graph-based representations for efficient data manipulation and processing. Some common algorithms in this domain include depth-first search (DFS) and breadth-first search (BFS), which are foundational techniques for exploring graph structures. These algorithms are widely employed in applications such as web crawling, social network analysis, and network routing.
-=== Breadth-First Search (BFS) ===
+Data structures designed to represent graphs, such as trees and heaps, leverage graph theory to optimize computational efficiency in various programming paradigms. Furthermore, graph databases, which are used to store and query interconnected data, utilize graph structures to enhance performance for specific relationships and hierarchy-based queries.
-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.
+=== Computer Networks ===
+In computer networking, graph theory is instrumental in modeling various aspects of communication systems. The connections between devices in a network can be represented as graphs, enabling the analysis of data transmission and network reliability. Problems such as routing, which involves finding optimal paths for data packets, can be efficiently addressed using graph-based algorithms.
-=== Dijkstra’s Algorithm ===
+Protocols, such as the Open Shortest Path First (OSPF) and Routing Information Protocol (RIP), utilize concepts from graph theory to ensure that data is transferred effectively across the network. By representing networks as graphs, it becomes feasible to optimize performance based on parameters such as latency, bandwidth, and fault tolerance.
-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.
+=== Social Network Analysis ===
+Graph theory provides a robust framework for analyzing social networks, where individuals are represented as vertices and relationships or interactions as edges. The analysis of social networks enables researchers to uncover patterns, such as the identification of influential individuals (or hubs), the study of community structures, and the detection of social trends.
-=== Kruskal’s and Prim’s Algorithms ===
+Metrics derived from graph theory, such as betweenness centrality, closeness centrality, and eigenvector centrality, are widely used to assess the importance of nodes within networks. Social network analysis has applications in fields like marketing, sociology, and even political science, providing critical insights into group dynamics and information dissemination.
-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.
+=== Biology and Ecology ===
+In biology and ecology, graph theory is employed to model various biological networks, such as food webs, metabolic pathways, and protein-protein interaction networks. By representing complex interdependencies as graphs, researchers can study the dynamics of ecosystems and the relationships between different species.
-== Challenges and Open Problems ==
+For example, ecological networks can illustrate the flow of energy and nutrients across different trophic levels, enabling the identification of keystone species and the overall health of ecosystems. Similarly, in genomics, graph-based approaches help in analyzing genetic interactions and the relationships between genes, facilitating the understanding of complex biological processes.
-Despite extensive research, Graph Theory still poses numerous challenges and open problems. Some notable examples include:
+== Real-world Examples ==
-=== The Graph Isomorphism Problem ===
+=== Transportation Networks ===
+Transportation systems can be effectively represented as graphs, with intersections or stations as vertices and routes or connections as edges. This representation allows for the optimization of travel routes, analysis of traffic patterns, and design of efficient public transport systems. For example, the planning of urban transit systems often employs graph theory to minimize travel time for commuters and improve overall network efficiency.
-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-theoretical concepts are also applied in logistics and supply chain management, where routes for delivery trucks can be optimized to reduce costs and enhance service delivery. Algorithms designed to identify the shortest or most efficient routes can significantly impact transportation costs and operational efficacy.
-=== The Traveling Salesman Problem (TSP) ===
+=== Internet and World Wide Web ===
+The structure of the Internet can be modeled as a graph, where websites are represented as vertices and hyperlinks or connections between these sites as edges. This graph representation enables various analyses, including searching and ranking algorithms, such as Google's PageRank, which uses graph properties to assess the importance of web pages based on their links.
-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.
+Additionally, social media platforms utilize graph theory to analyze relationships between users, facilitating the discovery of content and connections. The understanding of user engagement and the spread of information through social media relies heavily on the principles of graph theory to optimize interaction and content delivery.
-=== The Four Color Theorem ===
+=== Network Security ===
+In the realm of cybersecurity, graph theory provides tools for modeling and analyzing network security protocols. Network vulnerabilities can be represented in graph form, allowing security professionals to visualize potential attack paths and assess the risk of data breaches. Graph-based models enable the formulation of strategies to reinforce security measures and protect sensitive information.
-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.
+Furthermore, anomaly detection techniques often employ graph theoretical methods to identify unusual patterns in network traffic, enabling proactive measures against potential threats. The application of graph theory in network security underscores its relevance in contemporary digital environments.
-== Influence and Impact ==
+== Criticism or Limitations ==
+Despite its strengths and vast applications, graph theory has limitations and challenges that researchers continue to address. One of the primary criticisms relates to the computational complexity of certain problems in graph theory. Many fundamental problems, including graph coloring and the traveling salesman problem, are NP-hard, meaning that no known polynomial-time algorithms can solve them efficiently for all instances. This computational difficulty has implications for real-world applications, particularly as datasets grow in size and complexity.
-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.
+Additionally, the scalability of graph algorithms can pose challenges in large networks, where performance may degrade significantly. Researchers are exploring advanced methodologies, such as heuristics and approximation algorithms, to mitigate these limitations and enhance the applicability of graph theory to large-scale problems.
-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.
+Furthermore, the interpretability of graph models can be a concern, especially in fields like social sciences, where the underlying assumptions about relationships and connections may not always hold true. This lack of interpretability could lead to misrepresentations or erroneous conclusions when applying graph-theoretical analyses to complex social phenomena.
-== See Also ==
+== See also ==
+* [[Network theory]]
+* [[Algorithm]]
 * [[Combinatorics]]
-* [[Network Theory]]
+* [[Complexity theory]]
-* [[Computer Science]]
+* [[Data structures]]
-* [[Algorithm Complexity]]
+* [[List of graph algorithms]]
-* [[Discrete Mathematics]]
 == References ==
-* [https://www.graph-theory.com Graph Theory - Basic Concepts]
+* [https://www.mathworks.com/help/matlab/math/graph-theory.html Graph Theory - MATLAB Documentation]
-* [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.geeksforgeeks.org/graph-theory/ Graph Theory - GeeksforGeeks]
-* [https://www.mathworks.com/help/matlab/ref/graph.html MATLAB - Graph and Network Algorithms]
+* [https://www.khanacademy.org/math/linear-algebra/alternate-bases/graph-theory-intro/v/graph-theory-introduction Graph Theory Introduction - Khan Academy]
-* [https://www.geeksforgeeks.org/graph-data-structure-and-algorithms/ GeeksforGeeks - Graph Data Structure and Algorithms]
+* [https://en.wikipedia.org/wiki/Graph_theory Graph Theory - Wikipedia]
-* [https://www.coursera.org/learn/algorithms-part1 Coursera - Algorithms Part I]
+* [https://www.codecademy.com/resources/blog/graph-theory-in-computer-science/ Graph Theory in Computer Science - Codecademy]
-* [https://www.wolfram.com/language/guide/graph-theory/ Wolfram - Graph Theory]
+* [https://www.geeksforgeeks.org/graph-implementation-in-python/ Graph Implementation in Python - GeeksforGeeks]
 [[Category:Mathematics]]
 [[Category:Discrete Mathematics]]
 [[Category:Graph Theory]]