On 2-distance primitive graphs of prime valency

Abstract

A vertex transitive non-complete graph is called 2-distance primitive if the stabilizer of a vertex is primitive on both the first and the second step neighbourhoods. In 2019, Jin, Huang and Liu proved that all connected 2-distance primitive graphs of prime valency must belong to some known families of distance-transitive graphs if the socle of the stabilizer of a vertex is not a 2-transitive linear group with some further restrictions. They posed a problem to determine the connected 2-distance primitive graphs of prime valency when the socle of the stabilizer of a vertex is a 2-transitive linear group. In this paper, we make a crucial progress to this problem by proving that either the graph under consideration in this problem is the folded 5-cube or the socle of the stabilizer of a vertex is PSL(2,q) and the graph has girth 4 and contains no 5-cycles. We also give a short proof of the main result of Jin, Huang and Liu (2019), and we find some new graphs that are not in the list of Jin, Huang and Liu (2019).