The smallest eigenvalues of Hamming graphs, Johnson graphs and other distance-regular graphs with classical parameters

We prove a conjecture by Van Dam and Sotirov on the smallest eigenvalue of (distance-$j$) Hamming graphs and a conjecture by Karloff on the smallest eigenvalue of (distance-$j$) Johnson graphs. More generally, we study the smallest eigenvalue and the second largest eigenvalue in absolute value of the graphs of the relations of classical $P$- and $Q$-polynomial association schemes.

[1]  D. Stanton.  Some q-Krawtchouk Polynomials on Chevalley Groups , 1980 .

[2]  Jörg Eisfeld,et al.  The Eigenspaces of the Bose-Mesner-Algebras of the Association Schemes Corresponding to Projective Spaces and Polar Spaces , 1999, Des. Codes Cryptogr..

[3]  Simon Litsyn,et al.  On Integral Zeros of Krawtchouk Polynomials , 1996, J. Comb. Theory, Ser. A.

[4]  Vasek Chvátal,et al.  The tail of the hypergeometric distribution , 1979, Discret. Math..

[5]  Ronald L. Graham,et al.  The Radon transform on $Z^k_2$. , 1985 .

[6]  Dennis Stanton,et al.  A Partially Ordered Set and q-Krawtchouk Polynomials , 1981, J. Comb. Theory, Ser. A.

[7]  A. Neumaier,et al.  Distance Regular Graphs , 1989 .

[8]  Zoltán Füredi,et al.  Forbidding Just One Intersection , 1985, J. Comb. Theory, Ser. A.

[9]  Jean-Marie Goethals,et al.  Alternating Bilinear Forms over GF(q) , 1975, J. Comb. Theory A.

[10]  P. Delsarte.  AN ALGEBRAIC APPROACH TO THE ASSOCIATION SCHEMES OF CODING THEORY , 2011 .

[11]  Philippe Delsarte,et al.  Bilinear Forms over a Finite Field, with Applications to Coding Theory , 1978, J. Comb. Theory A.

[12]  G. M. Saha,et al.  Reduction of the number of associate classes of hypercubic association schemes , 1978 .

[13]  R. Sotirov,et al.  New bounds for the max-k-cut and chromatic number of a graph , 2015, 1503.06595.

[14]  Subhash Khot,et al.  Pseudorandom Sets in Grassmann Graph Have Near-Perfect Expansion , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[15]  Klaus Metsch,et al.  Large {0, 1, ..., t}-cliques in dual polar graphs , 2018, J. Comb. Theory, Ser. A.

[16]  Noga Alon,et al.  Bipartite Subgraphs and the Smallest Eigenvalue , 2000, Combinatorics, Probability and Computing.

[17]  Kai-Uwe Schmidt,et al.  Hermitian rank distance codes , 2017, Des. Codes Cryptogr..

[18]  Philippe Delsarte,et al.  Association Schemes and t-Designs in Regular Semilattices , 1976, J. Comb. Theory A.

[19]  de Bmm Benne Weger,et al.  On integral zeroes of binary Krawtchouk polynomials , 1999 .

[20]  Ronald L. Graham,et al.  THE RADON TRANSFORM ON Z , 1985 .

[21]  Ilya Dumer,et al.  Spherically Punctured Biorthogonal Codes , 2013, IEEE Transactions on Information Theory.

[22]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[23]  F. Vanhove.  Incidence geometry from an algebraic graph theory point of view , 2011 .

[24]  Roelof J. Stroeker,et al.  On a quartic diophantine equation , 1996 .

[25]  Philippe Delsarte,et al.  Properties and Applications of the Recurrence $F( {i + 1,k + 1,n + 1} ) = q^{k + 1} F( {i,k + 1,n} ) - q^k F( {i,k,n} )$ , 1976 .

[26]  Guillaume Hanrot,et al.  Résolution effective d'équations diophantiennes : algorithmes et applications , 1997 .

[27]  Ilya Dumer,et al.  Spherically Punctured Biorthogonal Codes , 2013, IEEE Trans. Inf. Theory.

[28]  Willem H. Haemers,et al.  Spectra of Graphs , 2011 .

[29]  Frank R. Kschischang,et al.  A Rank-Metric Approach to Error Control in Random Network Coding , 2007, IEEE Transactions on Information Theory.

[30]  Dennis Stanton,et al.  Zeros of generalized Krawtchouk polynomials , 1990 .

[31]  Laurent Habsieger,et al.  More zeros of krawtchouk polynomials , 1993, Graphs Comb..

[32]  Howard J. Karloff,et al.  How good is the Goemans-Williamson MAX CUT algorithm? , 1996, STOC '96.

[33]  M. Skala.  Hypergeometric tail inequalities: ending the insanity , 2013, 1311.5939.

[34]  Klaus Metsch,et al.  On a Characterization of Bilinear Forms Graphs , 1999, Eur. J. Comb..