Journal of Algebraic Combinatorics

, Volume 38, Issue 1, pp 191–195 | Cite as

On the connectedness of the complement of a ball in distance-regular graphs

Article

Abstract

An important property of strongly regular graphs is that the second subconstituent of any primitive strongly regular graph is always connected. Brouwer asked to what extent this statement can be generalized to distance-regular graphs. In this paper, we show that if γ is any vertex of a distance-regular graph Γ and t is the index where the standard sequence corresponding to the second largest eigenvalue of Γ changes sign, then the subgraph induced by the vertices at distance at least t from γ, is connected.

Keywords

Distance-regular graph Strongly regular graph Subconstituent Connectivity Eigenvalue Standard sequence 

Notes

Acknowledgements

The authors are grateful to Andries Brouwer for many comments and suggestions that have greatly improved the original manuscript. Sebastian M. Cioabă thanks C.L. Toma for interesting conversations. This work was partially supported by a grant from the Simons Foundation (#209309 to Sebastian M. Cioabă). Jack H. Koolen was partially supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant number 2010-0028061). The authors thank the organizers (Edwin van Dam and Willem Haemers) and the participants of the GAC5 conference for creating a wonderful research atmosphere in which part of this work was done and the referees for useful comments and suggestions.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
  2. 2.Department of MathematicsPOSTECHPohangSouth Korea

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