What is the difference between cycles and paths in graph theory?

What is the difference between cycles and paths in graph theory?

In graph theory, a path is a sequence of edges that connects a sequence of vertices without revisiting any vertex. A cycle, on the other hand, is a path that starts and ends at the same vertex, forming a closed loop.

Can you give me an example of a cycle?

Sure! In a triangle graph with vertices A, B, and C, the sequence A -> B -> C -> A forms a cycle.

What about a path? Can you show me an example?

Absolutely! In the same triangle graph, the sequence A -> B -> C is a path, as it connects the vertices without returning to A.

Are there any specific properties that distinguish cycles from paths?

Yes, one key property is that cycles must have at least three vertices and cannot repeat any edges, while paths can have any number of vertices and do not revisit any vertex.