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There Does Not Exist a Distance-Regular Graph with Intersection Array \(\{80, 54,12; 1, 6, 60\}\)

  • Jack H. Koolen
  • Quaid Iqbal
  • Jongyook ParkEmail author
  • Masood Ur Rehman
Original Paper
  • 14 Downloads

Abstract

In this paper, we will show that there does not exist a distance-regular graph \(\Gamma \) with intersection array \(\{80, 54,12; 1, 6, 60\}\). We first show that a local graph \(\Delta \) of \(\Gamma \) does not contain a coclique with 5 vertices, and then we prove that the graph \(\Gamma \) is geometric by showing that \(\Delta \) consists of 4 disjoint cliques with 20 vertices. Then we apply a result of Koolen and Bang to the graph \(\Gamma \), and we could obtain that there is no such a distance-regular graph.

Keywords

Distance-regular graphs Geometric distance-regular graphs Delsarte cliques The claw-bound 

Mathematics Subject Classification

05C50 05E30 

Notes

Acknowledgements

Q. Iqbal and M.U. Rehman are supported by Chinese Scholarship Council at USTC, Hefei, China. J.H. Koolen has been partially supported by the National Natural Science Foundation of China (Grant Nos. 11471009 and No. 11671376) and by the Anhui Initiative in Quantum Information Technologies (Grant No. AHY150200). J. Park is supported by Basic Research Program through the National Research Foundation of Korea funded by Ministry of Education (NRF-2017R1D1A1B03032016). This work was (partially) done while J. Park was working at Wonkwang University.

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Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  • Jack H. Koolen
    • 1
    • 2
  • Quaid Iqbal
    • 1
  • Jongyook Park
    • 3
    Email author
  • Masood Ur Rehman
    • 1
  1. 1.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China
  2. 2.Wen-Tsun Wu Key Laboratory of CASHefeiPeople’s Republic of China
  3. 3.Department of MathematicsKyungpook National UniversityDaeguRepublic of Korea

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