Today’s entry is a Theorem of the Day:

The Friendship Theorem:

In a finite graph in which any two distinct vertices share exactly one common neighbour, some vertex is adjacent to all other vertices.

In other words, if every pair of people at a party shares exactly one mutual friend at the party then some guest is a friend of everybody present. Maybe this will come in handy at Christmas parties? For more information, and other ways in which graph theory can help you in social situations, see the full listing at Theorem of the Day: the Friendship Theorem.

This is part of the Aperiodical Advent Calendar. We’ll be posting a new surprise for you each morning until Christmas!