Introduction to Graph Theory and its Applications

Foundations of Graph Theory: Understand Structures That Drive Modern Systems

Unlock the mathematical language behind networks, systems, and complex relationships with Introduction to Graph Theory and Its Applications. This foundational course explores how graph structures—networks of nodes and connections—are used to solve problems in computer science, engineering, biology, social sciences, machine learning, and more. Through a balance of theory and application, you will dive into the building blocks of graph theory, from trees, bipartite graphs, and Eulerian cycles to graph coloring, planarity, and shortest path algorithms. You'll learn how graphs represent data structures, model real-world systems, and support algorithmic solutions for challenges in routing, scheduling, genome mapping, and neural networks.

Ideal for students and professionals in mathematics, computer science, data science, and related fields, this course offers both the conceptual tools and the practical skills to approach modern networked systems with confidence.

Key Topics

• Introduction to Graphs: Definitions, terminology, and graph classifications.
• Graph Representations: Adjacency lists, adjacency matrices, incidence matrices.
• Eulerian and Hamiltonian Graphs: Paths, cycles, and real-world applications.
• Trees and Forests: Spanning trees, bridges, and properties of acyclic graphs.
• Connectivity and Components: Paths, articulation points, and graph traversal.
• Planar Graphs and Graph Drawing: Euler’s formula and platonic solids.
• Graph Coloring: Chromatic number, coloring algorithms, and scheduling problems.
• Bipartite Graphs and Matching: Characterizations, Hall’s Theorem, maximal matchings.
• Applications of Graph Theory: Network design, bioinformatics, machine learning, transportation systems, and more.

Course Benefits

• Build a Strong Mathematical Foundation: Gain a clear understanding of the fundamental elements of graphs and how to compute essential graph parameters.
• Master Graph Representation Techniques: Learn to represent graphs using adjacency and incidence matrices—key tools for computer-based graph analysis.
• Analyze Graph Structure and Connectivity: Develop the ability to assess connectivity, identify components, and explore critical elements such as bridges and articulation points.
• Understand Key Graph Types: Explore the properties and applications of bipartite graphs, trees, Eulerian and Hamiltonian graphs, and their roles in solving complex problems.
• Solve Practical Problems Using Graph Coloring: Apply graph coloring methods to real-world scheduling, allocation, and resource management challenges.
• Visualize and Classify Planar Graphs: Understand planarity and learn to recognize and work with planar graphs in various systems and designs.
• Explore Matchings and Network Optimization: Define and identify matchings in graphs, a core concept in network flows, resource allocation, and optimization.
• Apply Graph Theory Across Disciplines: Discover how graph theory is used in diverse fields—from social networks and biology to transportation, chemistry, and machine learning.
• Translate Theory into Practice: Gain the confidence to model real-world systems using graphs and select appropriate algorithms and analytical methods to solve them.

Course Details and Next Steps

• Course typically offered: Online during Fall and Spring quarter
• Prerequisites: Basic knowledge in linear algebra, and familiarity with mathematical proofs and counting are recommended
• Next steps: Upon completion, consider enrolling in other Applied Mathematics coursework for continued learning
• Contact: For more information about this course, please contact unex-techdata@ucsd.edu