- Volume 54 Issue 2
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
연구 과제 주관 기관 : National Research Foundation of Korea (NRF)
- E. Bannai and T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, Menlo Park, 1984.
- N. Biggs, Algebraic Graph Theory, Cambridge Tracts in Mathematics, No. 67. Cam-bridge University Press, London, 1974.
- 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
- 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
- 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.
- 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.
- F. Chung and S.-T. Yau, Covering, heat kernels and spanning tree, Electron. J. Combin. 6 (1999), Research Paper 12, 21 pp.
- P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl. 10 (1973), 97 pp.
- 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.
- 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
- 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
- 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
- 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
- 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
- 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
- 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