Introduction to Graph Theory

Go to Course: https://www.coursera.org/learn/graphs

Introduction

## Course Review: Introduction to Graph Theory If you’ve ever marveled at the intricacies of networks or pondered how devices communicate in the vast web of the internet, then the “Introduction to Graph Theory” course on Coursera is a must-take for you. This course perfectly blends the elegance of visual representation in mathematics with practical applications that extend into various fields, from engineering to biology. Below, I detail my experience and recommendation for this standout offering. ### Overview Graph Theory is a captivating and multidimensional subject that appeals to both the analytical mind and the creative spirit. This course invites educators, students, and curious minds alike to immerse themselves in a realm where mathematics meets real-world applications. Whether you're puzzled by GPS systems navigating roads, or intrigued by how integrated circuits are designed, this course lays the groundwork for understanding these complex yet fascinating systems through graphs. ### Course Structure and Syllabus The course is split into several comprehensive weeks, each focusing on crucial aspects of graph theory: 1. **What is a Graph?** The course kicks off with a fundamental exploration of graphs. Here, you'll learn what constitutes a graph, its applications, and why it’s a fundamental concept in mathematical modeling. Engaging interactive puzzles make this learning stage both challenging and enjoyable, providing insight into the pervasive presence of graphs in daily life. 2. **Cycles** Moving on to cycles in graphs, you will delve into connected components and apply them to practical programming problems, such as the Guarini puzzle and project dependency graphs. A standout moment is the discussion of Eulerian vs. Hamiltonian cycles, which is not only intellectually stimulating but also relevant to genome assembly, showcasing the course's connection to current biological research. 3. **Graph Classes** This week focuses on key graph classes: trees, bipartite graphs, and planar graphs. Through engaging lessons, you’ll uncover algorithms for connecting cities cost-effectively and analyze job matching scenarios. The practical problems and interactive components keep learners intrigued and motivated. 4. **Graph Parameters** Here, the course shifts to exploring important graph parameters such as colorings. You’ll discover historical context, like the famous four-color theorem in political map coloring, and delve into Ramsey's Theorem. Such theorems not only intrigue but also arm you with the tools to prove substantial results in the field. 5. **Flows and Matchings** The final segment highlights the practical applications of graph theory through flow and matching algorithms. The course provides insights on optimizing traffic flow and pairing systems in various industries. The application of mathematical concepts to real-life scenarios, like student placements and organ transplants, underscores the importance of graph theory in modern problem-solving. ### User Experience The course effectively balances theoretical knowledge and practical application, making complex concepts accessible through clear explanations and real-world examples. Its interactive structure encourages deeper engagement, while the quizzes and puzzles reinforce learning and retention. ### Recommendation I wholeheartedly recommend the “Introduction to Graph Theory” course for anyone interested in mathematics, computer science, or engineering. Whether you are a beginner or have some prior knowledge of mathematical concepts, this course will serve as an excellent resource. The real-world applications highlighted throughout the syllabus ensure that you not only learn but also understand the significance and utility of graph theory in our everyday lives. The course structure and engaging content make learning enjoyable and effective. Plus, with Coursera's flexible online format, you can navigate the modules at your own pace. Embark on this fascinating journey of graph theory; you won't be disappointed! In conclusion, if you have a thirst for knowledge and a curiosity about the interconnected world around you, signing up for “Introduction to Graph Theory” is an investment in your intellectual growth. Join this enlightening journey and unlock the door to a world of mathematical wonder!

Syllabus

What is a Graph?

What are graphs? What do we need them for? This week we'll see that a graph is a simple pictorial way to represent almost any relations between objects. We'll see that we use graph applications daily! We'll learn what graphs are, when and how to use them, how to draw graphs, and we'll also see the most important graph classes. We start off with two interactive puzzles. While they may be hard, they demonstrate the power of graph theory very well! If you don't find these puzzles easy, please see the videos and reading materials after them.

We’ll consider connected components of a graph and how they can be used to implement a simple program for solving the Guarini puzzle and for proving optimality of a certain protocol. We’ll see how to find a valid ordering of a to-do list or project dependency graph. Finally, we’ll figure out the dramatic difference between seemingly similar Eulerian cycles and Hamiltonian cycles, and we’ll see how they are used in genome assembly!

This week we will study three main graph classes: trees, bipartite graphs, and planar graphs. We'll define minimum spanning trees, and then develop an algorithm which finds the cheapest way to connect arbitrary cities. We'll study matchings in bipartite graphs, and see when a set of jobs can be filled by applicants. We'll also learn what planar graphs are, and see when subway stations can be connected without intersections. Stay tuned for more interactive puzzles!

We'll focus on the graph parameters and related problems. First, we'll define graph colorings, and see why political maps can be colored in just four colors. Then we will see how cliques and independent sets are related in graphs. Using these notions, we'll prove Ramsey Theorem which states that in a large system, complete disorder is impossible! Finally, we'll study vertex covers, and learn how to find the minimum number of computers which control all network connections.

This week we'll develop an algorithm that finds the maximum amount of water which can be routed in a given water supply network. This algorithm is also used in practice for optimization of road traffic and airline scheduling. We'll see how flows in networks are related to matchings in bipartite graphs. We'll then develop an algorithm which finds stable matchings in bipartite graphs. This algorithm solves the problem of matching students with schools, doctors with hospitals, and organ donors with patients. By the end of this week, we'll implement an algorithm which won the Nobel Prize in Economics!