Graph Theory with AI Applications – VOL-1

Foundations, Algorithms, and Modern Neural Approaches

Author: Anshuman Mishra
Publisher: Published by Anshuman Mishra
Publication Year: 2025
Volume: VOL-1

About the Book

Graph Theory with AI Applications: Foundations, Algorithms, and Modern Neural Approaches – VOL-1 presents a comprehensive introduction to Graph Theory and demonstrates how graph-based mathematical models and algorithms are increasingly important in Artificial Intelligence, Machine Learning, Robotics, Social Network Analysis, Navigation, Recommendation Systems, Network Design, and Intelligent Computing.

Graphs provide a powerful way to represent relationships and connections. Computers use graph structures to model everything from road networks and communication systems to social relationships, knowledge structures, recommendation systems, and complex AI environments.

This volume begins with the mathematical foundations of Graph Theory and progressively introduces classical graph algorithms, pathfinding techniques, network flow methods, social network analysis, community detection, link prediction, and graph mining.

The book is designed to provide both a strong theoretical foundation and an understanding of practical AI-oriented applications.

What You Will Learn

The book covers:

Fundamentals of Graph Theory
Graph terminology and properties
Directed and undirected graphs
Weighted and unweighted graphs
Labeled graphs
Static and dynamic graphs
Trees and forests
Cliques and independent sets
Cut vertices
Bipartite graphs
Matching
Graph isomorphism and automorphism
Graph representations
Adjacency matrices
Incidence matrices
Laplacian matrices
Spectral properties of graphs
Eigenvalues and graph connectivity
BFS
DFS
Bidirectional Search
Topological Sorting
Connected Component Detection
Shortest Path Algorithms
Dijkstra’s Algorithm
Bellman-Ford Algorithm
Floyd-Warshall Algorithm
Johnson’s Algorithm
AI Pathfinding
A* Search
Heuristics and admissibility
IDA*
Jump Point Search
Minimum Spanning Trees
Kruskal’s Algorithm
Prim’s Algorithm
Disjoint Set Union
Network Flows
Max Flow-Min Cut
Ford-Fulkerson Algorithm
Edmonds-Karp Algorithm
Social Network Analysis
Centrality Measures
PageRank
Community Detection
Modularity Optimization
Girvan-Newman Algorithm
Spectral Clustering
Label Propagation
Link Prediction
Graph Mining
Graph Embeddings
Fraud Detection
Recommendation Systems

PART I – FOUNDATIONS OF GRAPH THEORY

Chapter 1: Introduction to Graph Theory

The first chapter introduces the fundamental concept of graphs and explains why graph structures are important in computer science and Artificial Intelligence.

Topics include:

What is a Graph?
Historical Evolution of Graph Theory
Graphs in Modern Computing and AI
Directed and Undirected Graphs
Weighted and Unweighted Graphs
Labeled Graphs
Static and Dynamic Graphs
Real-World Applications of Graphs

The chapter establishes the foundation for understanding how complex relationships can be represented mathematically.

Chapter 2: Graph Terminologies and Basic Properties

This chapter introduces the fundamental terminology used throughout Graph Theory.

Topics include:

Degree
Path
Cycle
Connectivity
Trees
Forests
Cliques
Independent Sets
Cut Vertices
Bipartite Graphs
Matching
Isomorphism
Automorphism
Graph Representations

Different representation methods, including matrices, lists, and adjacency structures, are also introduced.

Chapter 3: Matrix Representations and Computations

Graphs can be represented mathematically using matrices. This chapter explores the important matrix-based representations used in graph algorithms and AI.

Topics include:

Adjacency Matrix
Incidence Matrix
Laplacian Matrix
Spectral Properties
Eigenvalues
Graph Connectivity
Applications of Graph Matrices in AI

Graph matrices form an important bridge between classical Graph Theory, linear algebra, spectral analysis, and modern AI techniques.

Chapter 4: Graph Traversal Algorithms

Graph traversal is one of the fundamental algorithmic operations in Graph Theory.

This chapter covers:

Breadth-First Search (BFS)
Depth-First Search (DFS)
Bidirectional Search
Topological Sorting
Connected Component Detection
AI Applications of Graph Traversal

Traversal techniques are important in search, planning, navigation, network analysis, and intelligent systems.

PART II – CLASSICAL GRAPH ALGORITHMS AND PATHFINDING

Chapter 5: Shortest Path Algorithms

Shortest path algorithms are among the most widely used graph algorithms in computing.

This chapter introduces:

Dijkstra’s Algorithm
Bellman-Ford Algorithm
Floyd-Warshall Algorithm
Johnson’s Algorithm
Robotics Applications
Navigation Applications
AI Planning Applications

These techniques provide the algorithmic foundation for route planning, navigation systems, robotics, network optimization, and intelligent path selection.

Chapter 6: Advanced Pathfinding in AI

AI systems often need to search through large spaces to find efficient paths. This chapter focuses on AI-oriented pathfinding techniques.

Topics include:

Introduction to AI Pathfinding
A* Search Algorithm
Heuristics
Admissibility
Iterative Deepening A* (IDA*)
Jump Point Search
Pathfinding in Games
Autonomous Systems

The chapter explains how graph search can be combined with heuristic information to improve intelligent pathfinding.

Chapter 7: Minimum Spanning Trees

Minimum Spanning Tree algorithms provide efficient solutions for connecting nodes while minimizing total connection cost.

This chapter covers:

Kruskal’s Algorithm
Prim’s Algorithm
Cycle Detection
Disjoint Set Union (DSU)
Network Design Applications
Clustering Applications

These methods have applications in communication networks, infrastructure design, clustering, and optimization.

Chapter 8: Network Flows and Cuts

Network flow models are important for understanding the movement of resources through networks.

Topics include:

Max Flow-Min Cut Theorem
Ford-Fulkerson Algorithm
Edmonds-Karp Algorithm
Scheduling Applications
Matching Applications
AI Resource Allocation

Network flow techniques provide useful foundations for resource distribution, scheduling, matching, transportation, and network optimization.

PART III – SOCIAL NETWORK ANALYSIS WITH GRAPH THEORY

Chapter 9: Social Networks and Graph Structures

Social networks can naturally be represented as graphs where users, organizations, or entities form nodes and their interactions form edges.

This chapter introduces:

Graph-Based Social Interaction Modeling
Homophily
Influence
Small-World Phenomenon
Community Structures

The chapter demonstrates how graph models can help analyze complex social relationships and interaction patterns.

Chapter 10: Centrality Measures in Social Networks

Centrality measures help identify important or influential nodes in a network.

Topics include:

Degree Centrality
Betweenness Centrality
Closeness Centrality
Eigenvector Centrality
PageRank
AI-Driven Influence Prediction

These measures have applications in social network analysis, recommendation systems, influence analysis, information networks, and intelligent decision-making.

Chapter 11: Community Detection Algorithms

Large networks often contain groups of nodes that are more strongly connected to one another. Community detection algorithms help identify these structures.

This chapter explores:

Modularity Optimization
Girvan-Newman Algorithm
Spectral Clustering
Label Propagation
Social Recommendation
AI Applications

Community detection has applications in social media analysis, recommendation systems, customer segmentation, knowledge networks, and fraud analysis.

Chapter 12: Link Prediction & Graph Mining

Link prediction focuses on predicting possible future or missing connections within a graph.

This chapter introduces:

Similarity-Based Methods
Probabilistic Models
Matrix Factorization
Graph Embeddings
Link Prediction
Graph Mining

Applications include:

Fraud Detection
Friend Recommendation
Social Recommendation
Network Analysis
Relationship Prediction

The chapter provides an important foundation for understanding how graph structures can be combined with data-driven AI techniques.

Key Highlights

✓ Complete Graph Theory foundations
✓ Directed and undirected graphs
✓ Weighted and dynamic graphs
✓ Graph matrices and spectral properties
✓ BFS and DFS
✓ Shortest path algorithms
✓ Dijkstra, Bellman-Ford and Floyd-Warshall
✓ AI pathfinding with A*
✓ IDA* and Jump Point Search
✓ Minimum Spanning Trees
✓ Kruskal and Prim algorithms
✓ Network Flow and Min-Cut
✓ Social Network Analysis
✓ Centrality measures
✓ PageRank
✓ Community Detection
✓ Spectral Clustering
✓ Link Prediction
✓ Graph Mining
✓ Graph Embeddings
✓ AI applications of Graph Theory

Real-World Applications

Graph Theory has applications across a wide range of modern technologies.

This volume explores applications in:

Artificial Intelligence
Machine Learning
Robotics
Autonomous Systems
Navigation
Route Planning
Social Networks
Recommendation Systems
Fraud Detection
Network Design
Communication Networks
Resource Allocation
Scheduling
Game AI
Knowledge Networks
Graph Mining

Why Graph Theory Matters in AI

Many Artificial Intelligence problems can be represented as relationships between entities.

For example:

Users → Social Networks
Locations → Road Networks
Web Pages → Web Graphs
Products → Recommendation Networks
Entities → Knowledge Graphs
Robots → Navigation Graphs
Transactions → Fraud Detection Networks

Understanding Graph Theory therefore provides an important mathematical and algorithmic foundation for modern AI systems.

Who Should Read This Book?

This volume is suitable for:

BCA students
MCA students
B.Sc. Computer Science students
Engineering students
Computer Science students
Artificial Intelligence students
Machine Learning learners
Data Science students
Mathematics students
Researchers
University teachers
AI/ML professionals
Algorithm enthusiasts

It can be used as a textbook, reference book, teaching resource, self-learning guide, or foundation for research in Graph Theory and AI.

Conclusion

Graph Theory with AI Applications – VOL-1 provides a structured journey from the fundamental concepts of Graph Theory to classical algorithms, AI pathfinding, network flows, social network analysis, community detection, link prediction, and graph mining.

By connecting mathematical graph structures with real-world AI applications, the book helps readers understand how graphs can be used to model relationships, search complex spaces, analyze networks, optimize routes, detect communities, and support intelligent decision-making.

VOL-1 establishes the foundation for advanced graph algorithms, graph-based Machine Learning, Graph Neural Networks, graph embeddings, and modern neural approaches explored in subsequent volumes.