Abstract
We study (G, 2)-geodesic-transitive graphs of odd order. We first give a reduction result on this family of graphs: Let N be an intransitive normal subgroup of G. Suppose that such a graph \(\Gamma \) is neither (G, 2)-arc-transitive nor \(\textrm{K}_{m[b]}\) where mb is odd and \(m,b\ge 3\). Then, we show that \(\Gamma \) is a cover of \(\Gamma _N\), G/N is faithful and quasiprimitive on \(V(\Gamma _N)\), \(\Gamma _N\) is \((G/N,s')\)-geodesic-transitive of odd order and girth 3 where \(s'=\min \{2,\textrm{diam}(\Gamma _N)\}\). We next investigate odd order (G, 2)-geodesic-transitive graphs where G acts quasiprimitively on the vertex set and determine all the possible quasiprimitive action types and give examples for them, and we also classify the family of (G, 2)-geodesic-transitive graphs of odd order where G is primitive of product action type on the vertex set. Finally, we find all the odd order 3-geodesic-transitive graphs which are covers of complete graphs.
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Acknowledgements
The project is supported by the NNSF of China (12271524,12061034), NSF of Jiangxi (20224ACB201002, 20212BAB201010) and NSF of Hunan (2022JJ30674).
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Jin, W. Two-geodesic-transitive graphs of odd order. J Algebr Comb 58, 291–305 (2023). https://doi.org/10.1007/s10801-023-01253-3
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DOI: https://doi.org/10.1007/s10801-023-01253-3