On 2-distance primitive graphs of prime valency

JJ Huang, YQ Feng, YS Kwon, JX Zhou, FG Yin - Discrete Mathematics, 2024 - Elsevier
JJ Huang, YQ Feng, YS Kwon, JX Zhou, FG Yin
Discrete Mathematics, 2024Elsevier
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 …
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).
Elsevier
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