"In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects."
The study of graphs and networks, including their representation, properties, and applications.
Graphs: A graph is a collection of points (also called vertices or nodes) connected by lines (also called edges). Graph theory is the study of graphs and their properties.
Types of Graphs: There are many types of graphs, such as simple graphs, directed graphs, weighted graphs, bipartite graphs, complete graphs, and others.
Graph Representations: There are multiple ways to represent a graph, including adjacency matrix, adjacency list, edge list, and others.
Graph Traversal: Graph traversal is the process of visiting all the nodes of a graph. It includes popular algorithms such as Breadth-First Search (BFS) and Depth-First Search (DFS).
Shortest Paths: The shortest path problem aims to find the shortest path between two nodes in a graph. There exist many algorithms for this task, including Dijkstra's algorithm, Bellman-Ford algorithm, and Floyd-Warshall algorithm.
Graph Coloring: Graph coloring is the assignment of colors to vertices of a graph so that no two adjacent vertices have the same color. This area of research has many applications, such as scheduling.
Minimum Spanning Trees: A minimum spanning tree is a subgraph of a graph that includes all the nodes and has the minimum possible weight for the sum of all its edges. This problem is solvable through Kruskal's algorithm and Prim's algorithm.
Planar Graphs: A planar graph is a graph that can be drawn on a plane without any edges crossing. Planar graphs have specific properties that make them important in areas such as circuit design.
Graph Connectivity: Connectivity refers to the ability to reach any node in a graph from any other node. In connected graphs, there is a path between any two vertices. This concept is related to graph cuts and algorithms like Ford-Fulkerson algorithm.
Network Flow: Network flow is a problem that involves finding the maximum amount of flow that can be sent through a network. This concept is heavily used in transportation, communication, and logistics.
Random Graphs: Random graphs are graphs generated by statistical models, such as the Erdős–Rényi model. Analysis of random graphs provides insights into graph properties and characteristics.
Basic Graph Theory: It deals with the properties and characteristics of graphs, including the study of the shortest path, connectivity, coloring and matching, structures and decomposition, and the theory of trees.
Combinatorial Graph Theory: It is the study of graphs from a combinatorial viewpoint, involving the enumeration and classification of graph properties, such as the number of edges, vertices, and subgraphs.
Geometric Graph Theory: It deals with the study of graphs in the context of their geometric properties, such as planarity, convexity, and embeddings.
Algorithmic Graph Theory: It is the study of the design and analysis of algorithms for solving graph problems, including the problems of graph coloring, shortest path, connectivity, and matching.
Probability Graph Theory: It deals with the study of graphs in the context of probability theory, including the problems of random graph generation, graph properties, and random walks on graphs.
Topological Graph Theory: It is the study of the topology and classification of graphs, including the study of graph embeddings and their invariants.
Algebraic Graph Theory: It deals with the study of graphs in the context of algebraic structures, such as groups, rings, and vector spaces, including the study of graph spectra, graph cohomology, and graph automorphisms.
Spectral Graph Theory: It is the study of the eigenvalues and eigenvectors of matrices associated with graphs, including the study of the Laplacian and adjacency matrices.
Graph Theory in Computer Science: It deals with the application of graph theory in computer science, including the study of graph algorithms, network optimization, and the modeling of systems and data structures.
Graph Theory in Social Networks: It deals with the application of graph theory in the analysis of social networks, including the study of network structures, community detection, graph clustering, and the analysis of network dynamics.
Graph Theory in Operations Research: It deals with the application of graph theory in operations research, including the study of network flow, transportation, and optimization problems.
Graph Theory in Biology: It deals with the application of graph theory in the context of biological systems, including the study of graph models for protein-protein interactions, gene regulation, and metabolic pathways.
Graph Theory in Chemistry: It deals with the application of graph theory in chemistry, including the study of molecular graph theory, chemical graph theory, and topological indices.
Graph Theory in Physics: It deals with the application of graph theory in the context of physics, including the study of graph models for networks, quantum systems, and phase transitions.
"A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines)."
"Graphs are one of the principal objects of study in discrete mathematics."
"A graph in this context is made up of vertices (also called nodes or points) which are connected by edges (also called links or lines)."
"A distinction is made between undirected graphs, where edges link two vertices symmetrically..."
"...and directed graphs, where edges link two vertices asymmetrically."
"A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically."
"Graphs are one of the principal objects of study in discrete mathematics."
"...graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects."
"...graphs, which are mathematical structures used to model pairwise relations between objects."
"Graphs are one of the principal objects of study in discrete mathematics."
"...edges link two vertices symmetrically..."
"...edges link two vertices asymmetrically."
"A distinction is made between undirected graphs...and directed graphs..."
"A graph in this context is made up of vertices (also called nodes or points)..."
"A graph in this context is made up of...edges (also called links or lines)."
"Graphs are one of the principal objects of study in discrete mathematics."
"...graphs, which are mathematical structures used to model pairwise relations between objects."
"A distinction is made between undirected graphs...and directed graphs..."
"Graphs are one of the principal objects of study in discrete mathematics."