24.1.8.2. Cycle Graph

Then there is another special simple and directed graph called as the cycle graph denoted as C_n. Here the vertex set will be consisting of n nodes and you will have E = {(v_1, v_2), (v_2, v_3), …, (v_{n-1}, v_n), (v_n, v_1)}. Now since the graph is simple, the cycle graph is defined only when the number of vertices is V ≥ 3.

A cycle graph is a type of graph that consists of a circular arrangement of vertices and edges connecting them. It is denoted as Cₙ, where 'n' represents the number of vertices. The edges connect each vertex to the next one in the sequence, and the last vertex connects back to the first, forming a cycle. For a graph to be considered a cycle, it must have at least three vertices because having only two vertices would not allow a proper cycle without violating the simple graph definition, which allows only one edge between any two vertices.

You can think of a cycle graph like a roundabout or a circular track where you have multiple exits. Imagine a running track where there are three checkpoints: one, two, and three. Runners can go from checkpoint one to two, two to three, and then back from three to one, forming a continuous loop. Just like in a cycle graph, you cannot define such a loop with only two checkpoints.

Because if I try to define a cycle graph between involving just two nodes, then as per the definition the edge set will be the following. You have an edge between 1 and 2 and again you have an edge between 2 to 1 that will be the definition of the edge set as per this general definition.

To clarify, if a cycle graph were to consist of just two nodes (let's say A and B), you could connect A to B and B back to A. This leads to a situation where both edges are effectively going in opposite directions and do not form a true 'cycle'. A cycle requires a sequence of connections that can loop back on itself without redundancy. Therefore, at least three distinct vertices are necessary to create a legitimate cycle.

Imagine trying to create a one-way street that goes in a circle—it's impossible with only two stop signs! You need a minimum of three points, like a triangle, to make a proper loop. Similarly, in a cycle graph, you need at least three vertices to set up a complete connection that returns to the starting point.

So, if I consider a cycle graph with three nodes then that will be something like this you have v_1, v_2, v_3 and then you have an edge back from 3 to 1.

Cycle graphs can be easily illustrated. For instance, with three vertices: v₁, v₂, and v₃, you would draw edges connecting v₁ to v₂, v₂ to v₃, and finally v₃ back to v₁. This forms a triangle shape, visually depicting the cycle. The properties notable to cycle graphs include that they are always connected and every vertex has a degree of 2, meaning each vertex connects to exactly two edges. Additionally, they are undirected, signifying that the edges have no direction.

Think of it like a musical chairs game where each person (vertex) is connected to the next one and they must keep moving in a circle. Each person has two neighbors—one on each side, illustrating the cycle's property where everyone connects back at the starting point, completing the circuit.

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