The classification of endpoint-transitive graphs with geodesic condition
Abstract
We introduce endpoint k-path-transitive graphs, in which the automorphism group fixing two prescribed vertices pointwise acts transitively on the paths of length k joining them. We study the geodesic case, where k equals the distance between these vertices. We identify the union of these geodesics with the Hasse graph of a finite bounded graded poset and reduce it to proper blocks by complete cuts andmatching compression. Assume that the induced group K acts faithfully and 2-homogeneously on the first distance layer. We first prove that every nontrivial normal subgroup of K is transitive on every internal layer if and only if soc(K) is. Under this normal-basic condition, we classify the proper blocks in the affine case and the equal-width proper blocks with a 2-transitive internal layer in the almost-simple case. The nontrivial design interfaces are Paley or affine symplectic designs in the affine case, and projective designs, the 2-(11, 5, 2) design, the HigmanSims design, or their complements in the almost-simple case. The proper blocks have reduced rank at most four, whereas almost-simple proper blocks can have arbitrarily large reduced rank without the equal-width condition. We also give a stabilizer factorization criterion for assembling blocks while preserving endpoint-geodesic transitivity.
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BibTeX
@misc{wang2026classification,
title = {{The classification of endpoint-transitive graphs with geodesic condition}},
author = {Ximin Wang and Zheng Huang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36622},
eprint = {2609.36622},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36622},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The classification of endpoint-transitive graphs with geodesic condition
AU - Ximin Wang
AU - Zheng Huang
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.36622
UR - https://arxiv.org/abs/2609.36622
AB - We introduce endpoint k-path-transitive graphs, in which the automorphism group fixing two prescribed vertices pointwise acts transitively on the paths of length k joining them. We study the geodesic case, where k equals the distance between these vertices. We identify the union of these geodesics with the Hasse graph of a finite bounded graded poset and reduce it to proper blocks by complete cuts andmatching compression. Assume that the induced group K acts faithfully and 2-homogeneously on the first distance layer. We first prove that every nontrivial normal subgroup of K is transitive on every internal layer if and only if soc(K) is. Under this normal-basic condition, we classify the proper blocks in the affine case and the equal-width proper blocks with a 2-transitive internal layer in the almost-simple case. The nontrivial design interfaces are Paley or affine symplectic designs in the affine case, and projective designs, the 2-(11, 5, 2) design, the HigmanSims design, or their complements in the almost-simple case. The proper blocks have reduced rank at most four, whereas almost-simple proper blocks can have arbitrarily large reduced rank without the equal-width condition. We also give a stabilizer factorization criterion for assembling blocks while preserving endpoint-geodesic transitivity.
ER -