Let $P_n$ be a path graph. This is from my textbook: View $P_n$ as the result of folding $C_{2n}$, where the folding has no fixed vertices. An eigenvector of $C_{2n}$ that is constant on the preimages of the folding yields an eigenvector of $P_n$ with the same eigenvalue.
I don't know what does the folding mean and what is the preimage of the folding?
The $\ C_{2n}\ $ regarded as a regular polygon of $\ 2n\ $ sides has $\ n\ $ reflection axes each with no fixed vertex (along with $\ n\ $ reflection lines each with a pair of fixed vertices). Folding apparently means to project the polygon onto the reflection axis forming the path graph $\ P_n\ $ in which each vertex of $\ P_n\ $ comes from a pair of vertices of $\ C_{2n}.\ $ In other words, what this means is that you can construct $\ C_{2n}\ $ by using two copies of $\ P_n\ $ and then join the corresponding end vertices.