Day 07Part 02- Discrete- Parallel edge and Self loop , Simple graph , Multi graph , Pseudo graph.

Day 07Part 02- Discrete- Parallel edge and Self loop , Simple graph , Multi graph , Pseudo graph.

Parallel Edge, Self Loop, Simple Graph, Multigraph, and Pseudograph in Discrete Mathematics

Graphs are fundamental structures in Discrete Mathematics and Computer Science, used to model relationships between objects. Let’s explore different types of graphs and their properties.

1. Basic Terminology in Graph Theory

A graph $G=(V,E)$ consists of:

• V (Vertices or Nodes): The set of points.

• E (Edges or Links): The set of connections between nodes.

2. Parallel Edges (Multiple Edges)

Definition: Parallel edges (also called multiple edges) occur when two or more edges connect the same pair of vertices.
Example: In a railway network, two different train routes may connect the same two cities.

Graph Representation:

Allowed in: Multigraphs and Pseudographs
Not allowed in: Simple Graphs

3. Self-Loop

Definition: A self-loop is an edge that connects a vertex to itself.
Example: In a social network, a person liking their own post can be represented as a self-loop.

Graph Representation:

(A has a self-loop)

Allowed in: Pseudographs
Not allowed in: Simple Graphs and Multigraphs

4. Simple Graph

Definition: A simple graph is a graph that does not have:
 Parallel edges
 Self-loops

Example: A friendship network, where each edge represents a unique connection between two people.

Graph Representation:

Characteristics:

• Each edge connects two distinct vertices.

• At most one edge between any two vertices.

5. Multigraph

Definition: A multigraph is a graph that allows parallel edges but does not allow self-loops.
Example: A road network where two different roads exist between the same two cities.

Graph Representation:

Allowed: Parallel edges
Not allowed: Self-loops

6. Pseudograph

Definition: A pseudograph is a generalized graph that allows both parallel edges and self-loops.
Example: A computer network where a server may have multiple direct connections to the same device, and it may also loop back to itself for redundancy.

Graph Representation:

Allowed: Parallel edges
Allowed: Self-loops

7. Comparison Table

8. Applications in Computer Science

Simple Graphs → Social networks, communication systems
Multigraphs → Transport networks (railway, road, airline routes)
Pseudographs → Computer networks, self-feedback loops in circuits

Conclusion

• Parallel Edges: Multiple connections between two nodes.

• Self-Loop: A node connected to itself.

• Simple Graph: No self-loops, no parallel edges.

• Multigraph: Allows parallel edges, but no self-loops.

• Pseudograph: Allows both parallel edges and self-loops.

Would you like practice questions or Python code to implement graphs?

Day 07Part 02- Discrete- Parallel edge and Self loop , Simple graph , Multi graph , Pseudo graph.

Introduction to Graph Theory

Graph theory: Introduction

Graph Types and Applications

Graph Theory

Here’s a clear explanation for Day 07 Part 02: Discrete Mathematics – Graph Types and Concepts like parallel edges, self-loops, simple graphs, multigraphs, and pseudographs.

---

Key Concepts in Graph Theory (Discrete Mathematics)

---

1. Parallel Edge

• Also called: Multiple edge

• Definition: Two or more edges that connect the same pair of vertices.

• Example: Edges (A, B), (A, B) → two edges between A and B.

• Occurs in: Multigraphs and Pseudographs

• Not allowed in a simple graph

---

2. Self-Loop

• Definition: An edge that connects a vertex to itself.

• Notation: (A, A)

• Occurs in: Pseudographs

• Not allowed in simple graphs or multigraphs

---

3. Simple Graph

• Definition: A graph with:

  • No parallel edges

  • No self-loops

• Edges: At most one edge between any two distinct vertices

• Most common in basic graph theory

• Edge = unordered pair of distinct vertices

---

4. Multigraph

• Definition: A graph that:

  • Allows parallel edges

  • Does not allow self-loops

• Useful to represent multiple relationships between the same pair of nodes

---

5. Pseudograph

• Definition: A graph that:

  • Allows both parallel edges and self-loops

• More generalized form of graph

---

Summary Table

Graph Type	Parallel Edges	Self-Loops
Simple Graph	Not allowed	Not allowed
Multigraph	Allowed	Not allowed
Pseudograph	Allowed	Allowed

---

Example:

Let’s consider a graph with:

• Vertices: A, B, C

• Edges: (A, B), (A, B), (B, C), (C, C)

This graph has:

• Parallel edges → (A, B), (A, B)

• A self-loop → (C, C)

So it’s a Pseudograph.

---

Let me know if you want a diagram or examples coded in Python/Graphviz.

Day 07Part 02- Discrete- Parallel edge and Self loop , Simple graph , Multi graph , Pseudo graph.

Discrete Mathematics – Graphs