Graph Theory | Definition and Terminology | Simple Graph | Multi-Graph | Pseudo-Graph |Graph Example

Summary

In this lecture, Dr. Anuj Kumar, Assistant Professor of Mathematics, introduces the concepts of Graph Theory and Trees as part of Discrete Mathematics. The discussion starts with the basics of graphs, covering essential definitions such as vertices, edges, and types of graphs like simple, multigraphs, and pseudographs. The video also explores important concepts like loops, parallel edges, and adjacency in graphs. Dr. Kumar explains how graphs are represented and categorized, offering examples to illustrate these concepts. The lecture sets the stage for further discussions on degrees of vertices and key theorems in future videos.

Takeaways

• 📚 Introduction: The speaker, Dr. Anuj Kumar, an assistant professor of mathematics, introduces the topic of discrete mathematics, focusing first on graph theory.
• 🧩 Graph Basics: A graph is defined using vertices (nodes) and edges (connections), and any point or figure can be considered a graph when expressed in terms of vertices and edges.
• 🔢 Graph Notation: Graphs are typically denoted using 'G' and are written in the form G(V, E), where 'V' represents the set of vertices and 'E' the set of edges.
• 🔍 Vertex and Edge Sets: The cardinality (total count) of vertices and edges can be found and defined using specific notations.
• 🌳 Graph Types: The speaker explains different types of graphs, such as simple graphs, which contain no loops or multiple edges, and multigraphs, which contain parallel edges but no loops.
• 🔄 Self-Loops and Parallel Edges: A self-loop occurs when an edge connects a vertex to itself. Parallel edges occur when two or more edges connect the same pair of vertices.
• 📊 Defining Graph Properties: The speaker uses examples to demonstrate the properties of graphs, such as vertex sets and edge connections, to define and illustrate graph characteristics.
• 🧮 Cardinality Explanation: Cardinality refers to the number of vertices (for the set V) and edges (for the set E) in a graph, and it helps determine graph properties.
• 💡 Loop and Parallel Edge Graphs: The speaker categorizes graphs based on the presence of loops and parallel edges, defining pseudographs as those that contain both.
• 🔍 Next Topic - Degree of Vertices: In the next video, the speaker plans to discuss the degree of vertices and how to find it using graph definitions, including incident edges and adjacency.

Q & A

What is the basic definition of a graph in graph theory?

A graph is defined as a collection of vertices (or nodes) and edges. The vertices represent points, while the edges represent connections between these points. Graphs are often denoted as G = (V, E), where V is the set of vertices and E is the set of edges.

What are vertices and edges in a graph?

Vertices, also known as nodes, are the fundamental units of a graph, typically represented by points. Edges are the lines or connections that join two vertices, representing relationships or links between them.

How can one represent the set of vertices and edges in a graph?

The set of vertices is denoted as V = {V1, V2, ..., Vn}, where each V represents a vertex. The set of edges is denoted as E = {E1, E2, ..., Em}, where each E connects two vertices.

What is the cardinality of a graph?

The cardinality of a graph refers to the number of vertices and edges it contains. The cardinality of the vertex set V is the total number of vertices, and the cardinality of the edge set E is the total number of edges.

What are self-loops and parallel edges in a graph?

A self-loop is an edge that connects a vertex to itself. Parallel edges, or multiple edges, occur when two or more edges connect the same pair of vertices.

What is a simple graph?

A simple graph is a graph that has no self-loops or parallel edges. It consists of distinct vertices connected by edges, with each edge joining two different vertices.

What is a multigraph?

A multigraph is a graph that allows for parallel edges but does not have self-loops. It may have multiple edges connecting the same pair of vertices.

What is a pseudograph?

A pseudograph is a type of graph that can contain both self-loops and parallel edges. It is a more generalized form of a graph compared to simple graphs and multigraphs.

What is an incident edge in graph theory?

An edge is said to be incident on a vertex if it connects that vertex to another. For example, in a graph where an edge connects V1 and V2, the edge is incident on both V1 and V2.

What does it mean for two vertices to be adjacent?

Two vertices are considered adjacent if they are connected by an edge. This means there is a direct relationship or connection between them through that edge.