Classifying cubic symmetric graphs of order $18 p^2$ - Alaeiyan, Mehdi; Hosseinipoor, Mohammad Kazem; Akbarizadeh, Masoumeh (Armenian Journal of Mathematics, 2022)
A $s$-arc in a graph is an ordered $(s+1)$-tuple $(v_{0}, v_{1}, …, v_{s-1}, v_{s})$ of vertices such that $v_{i-1}$ is adjacent to $v_{i}$ for $1≤ i ≤ s$ and $v_{i-1} eq v_{i+1}$ for $1≤ i < s$. A graph $X$ is called $s$-regular if its automorphism group acts regularly on the set of its $s$-arcs. In this paper, we classify all connected cubic $s$-regular graphs of order $18p^2$ for each $s≥1$ and each prime $p$.