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, 2024•ElsevierA 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 …
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).
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