What do your friendships, molecules and road maps have in common?

Friendships, molecules and road maps can all be drawn as the same thing: dots joined by lines. Mathematicians call such a drawing a graph, and graph theory is the study of these structures, which model pairwise relations between objects. The dots are vertices, the lines are edges. In a friendship graph, people are vertices and an edge means they know each other. In a molecule, atoms are vertices and chemical bonds are edges. In a road map, junctions are vertices and roads are edges, each carrying a length or a travel time, which is exactly how satnavs work out your route.

The field began with a puzzle. In 1736 Leonhard Euler asked whether you could cross the seven bridges of Königsberg exactly once each, and his paper is regarded as the first in graph theory. The word graph itself came later, in 1878, when James Joseph Sylvester borrowed the idea from chemists' drawings of molecules.

Graphs also measure how close people are. Paul Erdős, a mathematician who published at least 1,525 papers and travelled with everything he owned in two suitcases, inspired the Erdős number: his co-authors have number 1, their co-authors 2, and so on, the length of the shortest path to him in the graph of collaborations. Albert Einstein's is 2. Most mathematicians with one sit around 5.