Characteristics of Cycle Graph

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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.


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

all its edges are distinct
all its vertices (except for the start and end) are distinct.

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.