Is line graph $ L(C_n) $ of cycle graph $ C_n $ isomorphic to $ C_n $ itself?

Question

I guess the answer is true. How can it be proved if it is true?

Answer 1

A simple undirected graph is a cycle graph iff it is 2-regular and has only one connected component. Now, each edge in a cycle is incident to two other edges, hence $L(C_n)$ is 2-regular. Meanwhile, it is trivial that if $G$ has $c$ connected components with edges, then $L(G)$ has $c$ connected components. Using this fact we get that $L(C_n)$ has one connected component. Therefore, by the initial statement $L(C_n)$ is a cycle graph. Since $C_n$ has $n$ edges, $L(C_n)$ has $n$ vertices and is thus isomorphic to $C_n$.