DOI QR코드

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AN UPPER BOUND ON THE CHEEGER CONSTANT OF A DISTANCE-REGULAR GRAPH

  • Kim, Gil Chun (Department of Mathematics Dong-A University) ;
  • Lee, Yoonjin (Department of Mathematics Ewha Womans University)
  • 투고 : 2016.02.22
  • 발행 : 2017.03.31

초록

We present an upper bound on the Cheeger constant of a distance-regular graph. Recently, the authors found an upper bound on the Cheeger constant of distance-regular graph under a certain restriction in their previous work. Our new bound in the current paper is much better than the previous bound, and it is a general bound with no restriction. We point out that our bound is explicitly computable by using the valencies and the intersection matrix of a distance-regular graph. As a major tool, we use the discrete Green's function, which is defined as the inverse of ${\beta}$-Laplacian for some positive real number ${\beta}$. We present some examples of distance-regular graphs, where we compute our upper bound on their Cheeger constants.

과제정보

연구 과제 주관 기관 : National Research Foundation of Korea (NRF)

참고문헌

  1. E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, Menlo Park, 1984.
  2. N. Biggs, Algebraic Graph Theory, Cambridge Tracts in Mathematics, No. 67. Cam-bridge University Press, London, 1974.
  3. A. Brouwer and W. Haemers, Eigenvalues and perfect matchings, Linear Algebra Appl. 395 (2005), 155-162. https://doi.org/10.1016/j.laa.2004.08.014
  4. A. Brouwer and J. H. Koolen, The vertex-connectivity of a distance regular-graph, Eu-ropean J. Combin. 30 (2009), no. 3, 668-673. https://doi.org/10.1016/j.ejc.2008.07.006
  5. F. Chung, PageRank and random walks on graphs, Fete of Combinatorics and Computer Science (G. O. H. Katona, A. Schrijver and T. Szonyi, Eds.), pp. 43-62, Springer, Berlin, 2010.
  6. F. Chung, PageRank as a discrete Green's function, Geometry and Analysis. No. 1, 285-302, Adv. Lect. Math. (ALM), 17, Int. Press, Somerville, MA, 2011.
  7. F. Chung and S.-T. Yau, Covering, heat kernels and spanning tree, Electron. J. Combin. 6 (1999), Research Paper 12, 21 pp.
  8. P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl. 10 (1973), 97 pp.
  9. J. Dodziuk and W. S. Kendall, Combinatorial Laplacians and isoperimetric inequality, From local times to global geometry, control and physics (Coventry, 1984/85), 68-74, Pitman Res. Notes Math. Ser., 150, Longman Sci. Tech., Harlow, 1986.
  10. G. C. Kim and Y. Lee, Explicit expression of the Krawtchouk polynomial via a discrete Green's function, J. Korean Math. Soc. 50 (2013), no. 3, 509-527. https://doi.org/10.4134/JKMS.2013.50.3.509
  11. G. C. Kim and Y. Lee, A Cheeger inequality of a distance regular graph using Green's function, Dis-crete Math. 313 (2013), no. 20, 2337-2347. https://doi.org/10.1016/j.disc.2013.06.012
  12. G. C. Kim and Y. Lee, Corrigendum to "A Cheeger inequality of a distance-regular graph using Green's function" [Discrete Mathematics 313 (2013), no. 20, 2337-2347], Discrete Math. 338 (2015), no. 9, 1621-1623. https://doi.org/10.1016/j.disc.2015.04.010
  13. J. H. Koolen, J. Park, and H. Yu, An inequality involving the second largest and smallest eigenvalue of a distance regular graph, Linear Algebra Appl. 434 (2011), no. 12, 2404-2412. https://doi.org/10.1016/j.laa.2010.12.032
  14. G. Oshikiri, Cheeger constant and connectivity of graphs, Interdiscip. Inform. Sci. 8 (2002), no. 2, 147-150. https://doi.org/10.4036/iis.2002.147
  15. J. Tan, On cheeger inequalities of a graph, Discrete Math. 269 (2003), no. 1-3, 315-323. https://doi.org/10.1016/S0012-365X(03)00127-4
  16. P. Terwilliger, An inequality involving the local eigenvalues of a distance-regular graph, J. Algebraic Combin. 19 (2004), no. 2, 143-172. https://doi.org/10.1023/B:JACO.0000023004.62272.8c