"A graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense 'related'."
An introduction to graphs, which are mathematical structures used to model pairwise relations between objects.
Graph types and properties: Understanding the different types of graphs such as directed, undirected, weighted, unweighted, simple, and multigraphs, and their basic properties.
Degrees and adjacency: Understanding the concepts of degrees of vertices and adjacency matrices to represent graphs.
Connectivity and components: Understanding concepts like connected components, strongly connected components, cut vertices, and bridges.
Trees: Understanding what trees are, their properties, and their various applications in graph theory.
Spanning trees: Understanding the concept of spanning trees as a subgraph that is a tree and includes all the vertices of the original graph.
Planar graphs: Understanding what planar graphs are and their basic properties, including Euler's formula.
Graph coloring: Understanding the concept of vertex coloring and edge coloring along with the various algorithms.
Cliques and independent sets: Understanding the concept of cliques and independent sets, their properties, and their applications.
Graph algorithms: Understanding various graph algorithms like Dijkstra's algorithm, Bellman-Ford algorithm, and Kruskal's algorithm.
Random graphs: Understanding the concept of random graphs and their properties, including the famous Erdős-Rényi model.
Hypergraphs: Understanding the concept of hypergraphs, their properties, and applications in various fields.
Spectral graph theory: Understanding the role of eigenvalues and eigenvectors in determining graph properties like connectivity, clustering, and community structure.
Matroid theory: Understanding the concept of matroids and their applications in various fields, including optimization and combinatorial geometry.
Network flows: Understanding the concept of network flows and their use in transportation, logistics, and telecommunications.
Applications of graph theory: Studying the various applications of graph theory in different fields, including computer networks, social networks, bioinformatics, and game theory.
Simple Graph: A graph with no loops or multiple edges.
Complete Graph: A graph where every vertex is adjacent to all other vertices.
Bipartite Graph: A graph where vertices can be split into two independent sets, with no edges between vertices in the same set.
Directed Graph: A graph where edges have a direction, indicating a one-way relationship between vertices.
Weighted Graph: A graph where edges have assigned weights, usually representing some cost or distance between vertices.
Subgraph: A graph that is a subset of another graph, with fewer vertices and edges.
Connected Graph: A graph where every vertex is connected to at least one other vertex.
Disconnected Graph: A graph where at least two vertices are not connected to any other vertex.
Tree Graph: A connected graph with no cycles.
Spanning Tree: A subgraph of a connected graph that includes all of the original vertices and is also a tree.
Eulerian Graph: A graph where a single path can visit every edge exactly once.
Hamiltonian Graph: A graph where a single path can visit every vertex exactly once.
Planar Graph: A graph that can be drawn on a plane without any edges crossing.
Isomorphic Graph: Two graphs that have the same number of vertices and edges, and the same adjacency relationship between vertices.
Regular Graph: A graph where all vertices have the same degree, or number of incident edges.
Clique Graph: A subgraph of a graph where every vertex is connected to every other vertex.
Line Graph: A graph where every vertex represents an edge, and vertices are adjacent if their corresponding edges share an endpoint.
Complement Graph: A graph where every missing edge in the original graph is added, and every existing edge is removed.
Wheel Graph: A graph where one vertex is connected to every other vertex, forming a cycle, with a single vertex connected to each of the vertices in the cycle.
"The objects correspond to mathematical abstractions called vertices (also called nodes or points)."
"Each of the related pairs of vertices is called an edge (also called a link or line)."
"Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges."
"Graphs are one of the objects of study in discrete mathematics."
"The edges may be directed or undirected."
"If the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected."
"Any person A can shake hands with a person B only if B also shakes hands with A."
"If an edge from a person A to a person B means that A owes money to B, then this graph is directed."
"Owing money is not necessarily reciprocated."
"The word 'graph' was first used in this sense by J. J. Sylvester in 1878."
"Due to a direct relation between mathematics and chemical structure (what he called a chemico-graphical image)."
"The edges may be directed or undirected."
"The objects correspond to mathematical abstractions called vertices (also called nodes or points)."
"Each of the related pairs of vertices is called an edge (also called a link or line)."
"Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges."
"Graphs are one of the objects of study in discrete mathematics."
"The nature of the relationship between the objects determines whether the edges are directed or undirected."
"A directed graph is one where the edges have a specific direction, indicating a relationship or flow from one object to another."
"An undirected graph is one where the edges do not have a specific direction and represent a symmetric relationship between objects (such as shaking hands)."