"A complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge."
Complete graphs have every possible edge between vertices. They are often used to model social networks or communication networks.
Introduction to Graph Theory: This includes basic definitions of a graph, vertices, edges, degrees, etc., along with an introduction to different types of graphs.
Properties of Complete Graphs: This includes an introduction to complete graphs and their properties, including the number of edges, vertices, and degrees of each vertex.
Isomorphism of Graphs: Isomorphism is the study of the properties of graphs that remain unchanged irrespective of its labels. It is a necessary topic to learn because it helps to understand the structural similarities between graphs and establishes the relationships between different graphs.
Connectedness of Graphs: Connectedness is a measure of how connected a graph is with its vertices and edges. In this topic, you'll learn about the different types of connectedness, such as strongly connected graphs, weakly connected graphs, and disconnected graphs.
Hamiltonian Cycles and Paths: A Hamiltonian cycle is a path in a graph visiting each vertex once. This topic explores the properties of Hamiltonian cycles and paths in complete graphs.
Eulerian Circuits: An Eulerian circuit is a path that visits every edge in a graph exactly once. This topic explores the properties of Eulerian circuits in complete graphs.
Planar Graphs: A planar graph is a graph that can be drawn in the plane without any edges crossing over each other. This topic explores the properties of planar graphs and their relationship to complete graphs.
Chromatic Number: The chromatic number of a graph is the minimum number of colors needed to color the vertices of a graph so that no adjacent vertices have the same color. This topic explores the chromatic number of complete graphs.
Ramsey Theory: Ramsey Theory is the study of how order arises out of chaos. In particular, it deals with the existence of substructures of high order in large, disordered structures. This topic explores the Ramsey numbers of complete graphs, which are the smallest numbers that guarantee a certain type of order in certain graph structures.
Special Types of Complete Graphs: Topics such as Turan's theorem, Kneser's theorem, and Erdos-Ko-Rado's theorem focus on different variations of complete graphs and their properties. This includes analyzing the relationships between sets of vertices within complete graphs, finding bounds on their size, and exploring the existence of certain types of graphs within them.
Simple Complete Graph: A graph in which every pair of distinct vertices is connected by an edge.
Hypergraph: A variant of a complete graph, in which edges can connect more than two vertices.
Weighted Complete Graph: A graph in which each edge has a weight or cost assigned to it.
Infinite Complete Graph: A graph with an infinite number of vertices, wherein every possible connection between vertices exists.
Directed Complete Graph: A graph where edges have directions, meaning one vertex (the “source”) is the start point, and the other vertex (the “target”) is the end point of an edge.
Bipartite Complete Graph: A graph where vertices are split into two disjoint sets (left and right), with each vertex in one set connected to every vertex in the other set.
Turán Graph: A complete graph where edge density is minimized; that is, the complete graph has the minimum number of edges possible for a given number of vertices.
Star Graph: A complete graph in which all vertices except one are connected to a central vertex.
Wheel Graph: A complete graph where one vertex is connected to every other vertex, and all other vertices are connected in a cycle.
Complete k-partite Graph: A complete graph where vertices are grouped into k disjoint sets, with every vertex in one set connected to every vertex in every other set.
"A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction)."
"Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg."
"Leonhard Euler's 1736 work on the Seven Bridges of Königsberg."
"Drawings of complete graphs, with their vertices placed on the points of a regular polygon, had already appeared in the 13th century..."
"...in the work of Ramon Llull."
"Such a drawing is sometimes referred to as a mystic rose."
A complete graph connects every pair of distinct vertices with a unique edge, making it a fundamental concept in graph theory.
Complete graphs are undirected.
No, every pair of distinct vertices in a complete graph is connected by an edge.
No, a complete digraph has a pair of unique edges between every pair of distinct vertices.
Complete graphs were not introduced by a specific individual, but rather developed as a fundamental concept in graph theory.
Euler's work on the Seven Bridges of Königsberg marked the beginning of graph theory as a mathematical field.
Yes, drawings of complete graphs with vertices placed on a regular polygon were already present in the 13th century.
A drawing of a complete graph with vertices placed on a regular polygon is sometimes referred to as a mystic rose.
Yes, in a complete graph, every pair of distinct vertices is connected by a unique edge, ensuring full connectivity.
Yes, a complete digraph can have cycles since there are directed edges in both directions between every pair of distinct vertices.
Leonhard Euler is often referred to as the founder of modern graph theory.
Yes, in a complete graph, every pair of distinct vertices is connected by a unique edge, making the graph symmetrical.
Complete graphs can be utilized to model scenarios with complete interaction or communication between all members, such as social networks or transportation systems.