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Cores and Independence Numbers of Grassmann Graphs

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Abstract

A graph G is a core if every endomorphism of G is an automorphism. Let \(J_q(n,m)\) be the Grassmann graph with parameters q, m, n. We prove that many Grassmann graphs are cores, and both \(J_2(2k,2)\) and \(J_q(2^k,2)\) are not cores. We also obtain the independence number of \(J_q(n,2)\). In further to study cores and coding theory, it is important to estimate the upper bound of the independence number of \(J_q(n,m)\). Using a vertex-transitive subgraph of \(J_q(n,m)\), we obtain upper bounds on the independence number of \(J_q(n,m)\), which are also an improvement of bounds for the size of constant dimension codes in a 2011 paper of Etzion and Vardy.

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Acknowledgements

The authors are grateful to the referees for many useful comments and suggestions. Projects 11371072, 11501036 supported by NSFC. Supported by the Fundamental Research Funds for the Central University of China, Youth Scholar Program of Beijing Normal University (2014NT31) and China Postdoctoral Science Foundation (2015M570958).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, 410004, China

    Li-Ping Huang

  2. Sch. Math. Sci. and Lab. Math. Com. Sys., Beijing Normal University, Beijing, 100875, China

    Benjian Lv

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  1. Li-Ping Huang
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  2. Benjian Lv
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Correspondence to Li-Ping Huang.

Additional information

Projects 11371072, 11501036 supported by National Natural Science Foundation of China.

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Huang, LP., Lv, B. Cores and Independence Numbers of Grassmann Graphs. Graphs and Combinatorics 33, 1607–1620 (2017). https://doi.org/10.1007/s00373-017-1858-4

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  • DOI: https://doi.org/10.1007/s00373-017-1858-4

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