A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Let 𝑝 be a prime. It was shown by Folkman (J. Combin. Theory 3(1967) 215--232) that a regular edge-transitive graph of order $2p$ or $2p^2$ is necessarily vertex-transitive. In this paper, an extension of his result in the case of cubic graphs is given. It is proved that, every cubic edge-transitive graph of order $8p$ is symmetric, and then all such graphs are classified.

An undirected graph without isolated vertices is said to be semisymmetric if its full automorphism group acts transitively on its edge set but not on its vertex set. In this paper, we inquire the existence of connected semisymmetric cubic graphs of order $16p^2$. It is shown that for every odd prime 𝑝, there exists a semisymmetric cubic graph of order $16p^2$ and its structure is explicitly specified by giving the corresponding voltage rules generating the covering projections.

A graph is called symmetric if its automorphism group acts transitively on its arc set. In this paper, we classify all connected cubic symmetric graphs of order $16p^2$ for each prime 𝑝.

Let 𝐺 be a permutation group on a set 𝛺 with no fixed points in 𝛺 and let 𝑚 be a positive integer. If no element of 𝐺 moves any subset of 𝛺 by more than 𝑚 points (that is, $|\Gamma^g\backslash\Gamma|\leq m$ for every $\Gamma\subseteq\Omega$ and $g\in G$), and also if each 𝐺-orbit has size greater than 2, then the number 𝑡 of 𝐺-orbits in 𝛺 is at most $\frac{1}{2}(3m-1)$. Moreover, the equality holds if and only if 𝐺 is an elementary abelian 3-group.

In this paper, we enumerate the parameter matrices of all perfect 2-colorings of the generalized Petersen graphs $GP(n, 2)$, where $n \geq 5$. We also present some basic results for $GP(n, k)$, where $n \geq 5$ and $k \geq 3$.