Friendship Graphs with Specific Degrees

Is it possible for 5 people to each be friends with exactly 3 other people?

This problem is a wonderful introduction to the concept of graph theory, specifically focusing on how graphs can be used to model social relationships like friendships. The question is essentially asking if it's possible to construct a graph where nodes, representing people, have a degree of exactly 3. This relates to the concept of the 'degree of a vertex,' which is the number of edges incident to the vertex. In terms of social relationships, an edge represents a friendship.

One high-level strategy to approach this problem is to consider the handshaking lemma, which states that the sum of the degrees of all vertices in a graph is twice the number of edges. For this particular problem, with 5 people and each having 3 friends, you need to verify if such a configuration is possible. Analyzing if a simple cycle or other types of graph structures meet these conditions could provide insight.

Another important concept to understand is the idea of 'regular graphs,' where each vertex has the same degree—in this case, 3. Constructing regular graphs and testing small configurations is an excellent way to gain intuition. Concepts like parity and constraints on the number of odd degree vertices could also help deduce the possibility of the requested configuration.