Classifying cubic symmetric graphs of order 18 p2

A ss-arc in a graph is an ordered (s+1)(s+1)-tuple (v0,v1,⋯,vs−1,vs)(v0,v1,⋯,vs−1,vs) of vertices such that vi−1vi−1 is adjacent to vivi for 1≤i≤s1≤i≤s and vi−1≠vi+1vi−1≠vi+1 for 1≤i<s1≤i<s. A graph XX is called ss-regular if its automorphism group acts regularly on the set of its ss-arcs. In this paper, we classify all connected cubic ss-regular graphs of order 18p218p2 for each s≥1s≥1 and each prime pp.