Table of Contents
Introduction to Graph Theory
Graph theory is the study of graphs and their properties. A graph consists of a set of vertices (or nodes) and a set of edges connecting them. The edges can be directed or undirected.
Some basic properties of graphs include:
• The degree of a vertex is the number of edges connected to it.
• The shortest path between two vertices is called a shortest path or a path length or a distance.
• The distance between two vertices is the length of the shortest path between them.
• The connectivity of a graph is the number of pairs of vertices that can be connected by a path.
• The degree sequence of a graph is a vector consisting of the degree of each vertex in order from the smallest to the largest.
• The degree of a graph is the sum of the degrees of its vertices.
• The degree of a graph is even if and only if its degree sequence is even.
• The degree of a graph is odd if and only if its degree sequence is odd.
History of Graph Theory
Graph theory is the study of graphs and their properties. Graphs are mathematical structures consisting of vertices (or nodes) and edges connecting them. Graph theory is one of the oldest branches of mathematics, dating back to the 17th century.
The first recorded use of the word “graph” was by Sir William Hamilton in 1852. Hamilton introduced graph theory to study the properties of graphs associated with electrical circuits.
In the early 20th century, graph theory began to be studied more intensively, and many of the basic concepts and results were established.
One of the most important early results in graph theory was the four color theorem, which states that any graph can be colored using four colors so that no two adjacent vertices are the same color.
Since the early days of graph theory, the subject has expanded to include a wide variety of topics, including network theory, transportation theory, and matchings.
Terminologies of Graph Theory
In graph theory, a vertex (plural: vertices) is a point where two or more edges meet. An edge (plural: edges) is a line between two vertices.
A graph is a collection of vertices and edges. The edges in a graph can be directed or undirected. A directed edge has a direction, from one vertex to another. An undirected edge has no direction.
A graph is connected if there is a path between any two vertices in the graph. A graph is disconnected if there is no path between any two vertices in the graph.
A graph is simple if it has no loops or multiple edges. A graph is nonsimple if it has loops or multiple edges.
A graph is connected if there is a path between any two vertices in the graph. A graph is disconnected if there is no path between any two vertices in the graph.
A graph is simple if it has no loops or multiple edges. A graph is nonsimple if it has loops or multiple edges.
A graph is connected if there is a path between any two vertices in the graph. A graph is disconnected if there is no path between any two vertices in the graph.
A graph is simple if it has no loops or multiple edges. A graph is nonsimple if it has loops or multiple edges.
Difference Between Circuit and Cycle in Graph Theory
A circuit is a path in a graph that starts and ends at the same vertex. A cycle is a path in a graph that starts and ends at different vertices.
Graph Theory and Application Question Bank
1. What is a graph?
A graph is a collection of points, called vertices, and the lines connecting them, called edges.
2. What is a connected graph?
A connected graph is a graph in which there exists a path between any two vertices.
3. What is a disconnected graph?
A disconnected graph is a graph in which there is no path between any two vertices.
4. What is a simple graph?
A simple graph is a graph in which each edge connects two different vertices.
5. What is a multigraph?
A multigraph is a graph in which two or more edges may connect the same two vertices.
6. What is a graph isomorphism?
A graph isomorphism is a relationship between two graphs in which one can be transformed into the other by a sequence of graph transformations that preserves the edge relationships.
7. What is a degree sequence?
A degree sequence is a list of the degrees of the vertices in a graph, listed in ascending order.
8. What is a degree?
The degree of a vertex is the number of edges incident to that vertex.
9. What is a path?
A path is a sequence of connected edges between two vertices.
10. What is a cycle?
A cycle is a path that includes
