# Characteristics of Cycle Graph

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## Theorem

Let $G = \struct {V, E}$ be an (undirected) graph whose order is greater than $2$.

Then $G$ is a cycle graph if and only if:

- $G$ is connected
- every vertex of $G$ is adjacent to $2$ other vertices
- every edge of $G$ is adjacent to $2$ other edges.

## Proof

Recall that a cycle is a closed walk with the properties:

From Cycle Graph is Connected, $G$ is connected.

From Cycle Graph is 2-Regular, $G$ is $2$-regular.

Hence every vertex of $G$ is incident with $2$ edges.

This needs considerable tedious hard slog to complete it.In particular: I hate trying to prove obvious thingsTo discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Finish}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

- 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): Chapter $1$: Mathematical Models: $\S 1.3$: Graphs: Problem $23$