"A graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense 'related'."
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.
Graph Fundamentals: This includes understanding the basic terminologies used in Graph Theory such as vertices, edges, degree of a vertex, loops, etc.
Types of Graphs: There are different types of graphs such as undirected, directed, complete, bipartite graphs, etc. It is essential to understand their properties and how they are represented.
Graph Operations: Graph operations include subgraphs, complement graphs, merging, and splitting of graphs, and graph isomorphism. These operations are used to manipulate graphs while preserving some of their properties.
Path and Connectivity: Understanding a path and its properties, connectivity in graphs and its types such as strong, weak, connected components.
Spanning Trees: This includes understanding minimum spanning trees, Kruskal's Algorithm, and Prim's Algorithm. These algorithms are crucial for finding the shortest path in a graph.
Graph Coloring: It is about understanding vertex coloring, edge coloring, chromatic number and how it is applied in real-world applications such as scheduling problems, maps, etc.
Network Flow: This is about understanding network flow, maximum flow, and minimum cut algorithms. These concepts are used in solving problems related to transportation and communication networks.
Graph Embedding: This involves understanding planar graphs, Euler’s theorem, and Kuratowski's theorem. These concepts help in understanding graph drawing and visualization.
Random Graphs: It is crucial to understand random graphs, their properties, and how they are applied in real-world applications such as social network analysis.
Graph Algorithms: Graph algorithms include depth-first search, breadth-first search, shortest path algorithms, topological sort, and Dijkstra's Algorithm.
Graph Databases: It is essential to understand graph databases, how they work, and their different types, including RDF and property graphs.
Graph Applications: Lastly, understanding the application of graph theory in various fields such as computer science, social network analysis, transportation, and logistics is essential.
Simple Vertex: A vertex with degree one or higher.
Isolated Vertex: A vertex with degree zero.
Pendant Vertex: A vertex with degree one.
Cut Vertex: A vertex whose deletion disconnects the graph.
Articulation Point: Same as Cut Vertex.
Complete Vertex: A vertex adjacent to all other vertices in the graph.
Universal Vertex: Same as Complete Vertex.
Bipartition Vertex: A vertex whose deletion results in a bipartite graph.
Separator Vertex: A vertex whose deletion separates the graph into disjoint parts.
Simple Edge: An edge that connects two vertices without intersecting any other edge.
Loop: An edge that connects a vertex to itself.
Parallel Edge: Two or more edges that connect the same two vertices.
Directed Edge: An edge with a direction from one vertex to another vertex.
Weighted Edge: An edge with a value or weight assigned to it.
Bridge Edge: An edge whose deletion disconnects the graph.
Cut Edge: Same as Bridge Edge.
Non-Bridge Edge: An edge that is not a bridge.
Tree Edge: An edge in a spanning tree.
Back Edge: An edge that connects a vertex to an ancestor vertex in a tree.
Forward Edge: An edge that connects a vertex to a descendent vertex in a tree.
Cross Edge: An edge that connects two vertices that are not ancestor or descendant in a tree.
"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)."