Graph Isomorphism and Connectivity - A Comprehensive Tutorial

Welcome to the tutorial on Graph Isomorphism and Connectivity in Discrete Mathematics. Graph isomorphism deals with comparing graphs for structural equivalence, while graph connectivity focuses on the robustness of a graph's connections.

Graph Isomorphism

Graph Isomorphism refers to the process of determining whether two graphs are structurally identical, even if their vertex and edge labels are different. Let's illustrate this with a Python code example:

# Using NetworkX library to check graph isomorphism
import networkx as nx

# Create two graphs
G1 = nx.Graph([(1, 2), (2, 3)])
G2 = nx.Graph([(A, B), (B, C)])

# Check if G1 and G2 are isomorphic
isomorphic = nx.is_isomorphic(G1, G2)

if isomorphic:
    print("The graphs are isomorphic.")
else:
    print("The graphs are not isomorphic.")
        

This code demonstrates the use of the NetworkX library in Python to check if two graphs are isomorphic, regardless of the vertex labels.

Graph Connectivity

Graph Connectivity measures the degree to which a graph is connected. A graph can be classified as:

• Connected Graph: Every pair of vertices is connected by a path.
• Disconnected Graph: Contains one or more pairs of vertices with no path between them.
• Strongly Connected Graph: In directed graphs, every pair of vertices has both forward and backward paths.

Steps to Determine Isomorphism

1. Label Vertices: Assign labels to the vertices of the graphs.
2. Generate Canonical Form: Convert the graph into a canonical form for easy comparison.
3. Compare Structures: Compare the structures of the canonical forms to determine isomorphism.

Common Mistakes with Graph Isomorphism and Connectivity

• Forgetting to label vertices before comparing structures.
• Confusing graph isomorphism with graph similarity.
• Assuming a disconnected graph is necessarily not isomorphic to a connected graph.

Frequently Asked Questions

Q1: Can isomorphic graphs have different numbers of vertices or edges?

A1: No, isomorphic graphs have the same number of vertices and edges.

Q2: Are two disconnected graphs always non-isomorphic?

A2: Not necessarily, disconnected graphs can be isomorphic if their connected components are isomorphic.

Q3: How does graph connectivity affect network reliability?

A3: Higher graph connectivity indicates more robust and reliable networks.

Q4: Is graph isomorphism computationally difficult?

A4: Yes, graph isomorphism is generally considered a challenging computational problem.

Q5: Can a directed graph be isomorphic to an undirected graph?

A5: No, isomorphism preserves the type of edges (directed or undirected).

Summary

Understanding graph isomorphism and connectivity is crucial for analyzing and comparing graphs in various applications. By following the steps to determine isomorphism and grasping the concept of connectivity, you can effectively work with graphs and make informed decisions in network design and analysis.