Strip away everything from a map except the connections — which cities link to which, never mind distance or direction — and what remains is a graph: a set of dots joined by lines. Graph theory is the study of these networks of relationships, and it turns out that almost anything worth understanding is secretly a graph: road maps, the internet, friendships, molecules, family trees, the states of a puzzle.
Born from a puzzle about the bridges of Königsberg, graph theory is now the backbone of
computer science — routing packets, ranking web pages, scheduling exams, compiling code. Its
charm is that the objects are so simple (just dots and lines) yet the questions run from
childishly easy to among the hardest open problems in mathematics. It is where deep theory and
real-world
One thread runs through everything here. Once you record only what is connected to what, a torrent of questions becomes precise: Can I get from here to there? Is there a route that uses every road exactly once? How few colours label a map with no two neighbours alike? Can this network be drawn without crossings? Each answer depends only on the pattern of connections — the graph — not on the messy details you threw away.
This course moves in three stages, each building on the last.
We begin with the object itself — just dots and lines — and the surprising amount of vocabulary and insight that a single picture of connections already carries.