2. (4) At a round table, 2015 people are sitting. Every minute, any two people who are friends can swap places. It turns out that, after spending some time, the people at the table can sit in any order. What is the minimum number of pairs of friends among these 2015 people?
Solution: Consider a graph where the vertices are people, and two vertices are connected by an edge if the corresponding two people are friends. To achieve any arbitrary order, we need to be able to swap any two people. For this, we need the corresponding graph to be connected. Moreover, it is not hard to understand that connectivity is sufficient. Therefore, the graph with the minimum number of edges that we are looking for is a tree with 2015 vertices, which means the minimum number of friendship edges is 2014.