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Vertex-transitive Diameter Two Graphs

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Abstract

We investigate the family of vertex-transitive graphs with diameter 2. Let Γ be such a graph. Suppose that its automorphism group is transitive on the set of ordered non-adjacent vertex pairs. Then either Γ is distance-transitive or Γ has girth at most 4. Moreover, if Γ has valency 2, then Γ ≅ C4 or C5; and for any integer n ≥ 3, there exist such graphs Γ of valency n such that its automorphism group is not transitive on the set of arcs. Also, we determine this family of graphs of valency less than 5. Finally, the family of diameter 2 circulants is characterized.

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Acknowledgments

The authors are grateful to the anonymous referees for valuable suggestions and comments.

Funding

This paper is supported by the National Natural Science Foundation of China (12061034,12071484,11661039), Natural Science Foundation of Jiangxi (20212BAB201010,GJJ190273,20192ACBL21007,2018ACB21001), Natural Science Foundation of Hunan (2020JJ4675) and China Postdoctoral Science Foundation (2019T120563).

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Authors and Affiliations

  1. School of Statistics, Jiangxi University of Finance and Economics, Nanchang, 330013, China

    Wei ** & Li Tan

  2. School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, 410075, China

    Wei **

  3. Research Center of Applied Statistics, Jiangxi University of Finance and Economics, Nanchang, 330013, China

    Li Tan

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  1. Wei **
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  2. Li Tan
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Correspondence to Wei **.

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**, W., Tan, L. Vertex-transitive Diameter Two Graphs. Acta Math. Appl. Sin. Engl. Ser. 38, 209–222 (2022). https://doi.org/10.1007/s10255-022-1058-8

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  • DOI: https://doi.org/10.1007/s10255-022-1058-8

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