Friendship Graph for Five People

Is it possible for 5 people to each be friends with exactly 2 other people?

This problem explores the idea of creating a graph representation of a social network, where each person (or vertex) has exactly two friends (or edges). From an abstract point of view, this concept is related to cycles in graph theory. Specifically, each person being connected to exactly two others is equivalent to constructing a cycle graph, where all vertices form a single cycle. The key challenge is determining if such a cycle can be constructed with the given number of vertices, which, in this problem, is five.

To solve this, one must understand that a cycle graph of 'n' vertices is possible as long as each vertex is connected in such a way that forms a continuous loop without any breaks. For an odd number of vertices like five, it is indeed possible to create one cycle. Therefore, constructing this graph involves connecting these five vertices sequentially and then closing the loop to ensure each vertex has exactly two connections. This basic but fundamental insight connects to larger discussions in graph theory about Eulerian cycles, connectivity, and even social network models as graph-based representations. Understanding such problems aids in developing both theoretical knowledge of graph properties and practical skills in modeling real-world networks using discrete mathematics concepts.