"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 connectivity, paths, cycles, and algorithms.
Graphs: An introduction to graphs, which are mathematical structures used to model pairwise relations between objects.
Vertices and edges: The basic components of a graph. Vertices (also called nodes) are the objects being modeled, and edges (also called links or lines) represent relations between them.
Types of graphs: There are many types of graphs, including directed and undirected graphs, weighted graphs, bipartite graphs, and more.
Paths and cycles: A path is a sequence of edges connecting two vertices. A cycle is a path that starts and ends at the same vertex.
Connectivity: A graph is connected if there is a path between any two vertices. Otherwise, it is disconnected.
Planar graphs: A graph is planar if it can be drawn on a plane without any edges crossing. Some graphs, such as the complete graph, are not planar.
Coloring: Coloring involves assigning colors to vertices in a graph such that no two adjacent vertices have the same color. The minimum number of colors needed for a given graph is called its chromatic number.
Graph algorithms: There are many algorithms for working with graphs, including Dijkstra's algorithm for finding shortest paths in weighted graphs, breadth-first search and depth-first search for traversing graphs, and more.
Applications of graph theory: Graph theory has many real-world applications, such as modeling social networks, computer networks, transportation networks, and more.
Bipartite Graphs: Bipartite graphs have two sets of vertices, and all edges are between vertices in different sets. They are often used to model relationships between two different types of objects.
Complete Graphs: Complete graphs have every possible edge between vertices. They are often used to model social networks or communication networks.
Directed Graphs: Directed graphs (also called digraphs) have arrows on edges indicating the direction of the relationship between vertices. They are often used to model processes and flowcharts.
Weighted Graphs: Weighted graphs have a weight assigned to each edge, indicating the cost or distance between two vertices. They are often used to model transportation networks or optimization problems.
Eulerian Graphs: Eulerian graphs have a circuit that visits every edge exactly once. They are often used in puzzles and games.
Hamiltonian Graphs: Hamiltonian graphs have a circuit that visits every vertex exactly once. They are often used in the study of optimization problems.
Planar Graphs: Planar graphs can be drawn on a flat surface without any edges crossing. They are often used in the study of geography or maps.
Regular Graphs: Regular graphs have all vertices with the same degree, or number of edges incident to each vertex. They are often used in the study of symmetry and patterns.
Tree Graphs: Tree graphs have a single path between any two vertices. They are often used in the study of hierarchical structures and algorithms.
Random Graphs: Random graphs are generated using a probabilistic model, and are often used in the study of social networks and the internet.
"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."