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Inverse Problems in Graph Theory: Nets

  • A. A. Makhnev
  • M. P. Golubyatnikov
  • Wenbin Guo
Article
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Abstract

Let \(\varGamma \) be a distance-regular graph of diameter 3 with strong regular graph \(\varGamma _3\). The determination of the parameters \(\varGamma _3\) over the intersection array of the graph \(\varGamma \) is a direct problem. Finding an intersection array of the graph \(\varGamma \) with respect to the parameters \(\varGamma _3\) is an inverse problem. Previously, inverse problems were solved for \(\varGamma _3\) by Makhnev and Nirova. In this paper, we study the intersection arrays of distance-regular graph \(\varGamma \) of diameter 3, for which the graph \({\bar{\varGamma }}_3\) is a pseudo-geometric graph of the net \(PG_{m}(n, m)\). New infinite series of admissible intersection arrays for these graphs are found. We also investigate the automorphisms of distance-regular graph with the intersection array \(\{20,16,5; 1,1,16 \}\).

Keywords

Distance-regular graph Pseudo-geometric graph Strong regular graph 

Mathematics Subject Classification

05C25 05E30 

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Copyright information

© School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • A. A. Makhnev
    • 1
  • M. P. Golubyatnikov
    • 2
  • Wenbin Guo
    • 3
  1. 1.Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of SciencesYekaterinburgRussia
  2. 2.Ural Federal University named after the First President of Russia B. N. YeltsinYekaterinburgRussia
  3. 3.University of Science and Technology of ChinaHefeiChina

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