Distance-regular graphs

Abstract

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.

156 pages

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• The existence of designs
• New upper bounds for kissing numbers from semidefinite programming
• Quantum Random Walks Hit Exponentially Faster
• Separation cut-offs for birth and death chains
• Perfect state transfer on distance-regular graphs and association schemes
• Commutative association schemes
• Experimental study of energy-minimizing point configurations on spheres
• On almost distance-regular graphs
• Two linear transformations each tridiagonal with respect to an eigenbasis of the other; an algebraic approach to the Askey scheme of orthogonal polynomials
• Resolving sets for Johnson and Kneser graphs
• Matrix Algebras and Semidefinite Programming Techniques for Codes
• Perfect state transfer over distance-regular spin networks
• On semidefinite programming relaxations of the traveling salesman problem
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• A new family of tight sets in ${\mathcal{Q}}^{+}(5,q)$
• Some algebra related to $P$-and $Q$-polynomial association schemes
• Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems
• Investigation of Continuous-Time Quantum Walk Via Modules of Bose-Mesner and Terwilliger Algebras
• Quantum central limit theorem for continuous-time quantum walks on odd graphs in quantum probability theory
• New proofs of the Assmus-Mattson theorem based on the Terwilliger algebra
• The Drinfel'd polynomial of a tridiagonal pair
• Complex Hadamard Matrices, Instantaneous Uniform Mixing and Cubes
• Neighbour transitivity on codes in Hamming graphs
• The metric dimension of small distance-regular and strongly regular graphs
• A "Continuous" Limit of the Complementary Bannai-Ito Polynomials: Chihara Polynomials
• The Erdős-Ko-Rado theorem for twisted Grassmann graphs
• Dual polar graphs, the quantum algebra U_q(sl_2), and Leonard systems of dual q-Krawtchouk type
• On a conjecture of Brouwer involving the connectivity of strongly regular graphs
• A bilinear form relating two Leonard systems
• Point configurations that are asymmetric yet balanced
• Arithmetic completely regular codes
• Nonexistence of exceptional imprimitive Q-polynomial association schemes with six classes
• Geometric aspects of 2-walk-regular graphs
• Four-Weight Spin Models and Jones Pairs
• On bounding the bandwidth of graphs with symmetry
• Optimal Embeddings of Distance Regular Graphs into Euclidean Spaces
• On perturbations of almost distance-regular graphs
• Dual concepts of almost distance-regularity and the spectral excess theorem
• Some Spectral and Quasi-Spectral Characterizations of Distance-Regular Graphs
• The Laplacian spectral excess theorem for distance-regular graphs
• Classification of a family of completely transitive codes
• Large connected strongly regular graphs are Hamiltonian
• On the metric dimension of imprimitive distance-regular graphs
• Hypercubes, Leonard triples and the anticommutator spin algebra
• Distance-regular graph with large a1 or c2
• A note on the span of Hadamard products of vectors
• There are only finitely many distance-regular graphs with valency k at least three, fixed ratio k2/k and large diameter
• The endomorphism of Grassmann graphs
• Distance-regular graphs with valency k having smallest eigenvalue at most -k/2
• New characterisations of the Nordstrom-Robinson codes
• The uniqueness of a distance-regular graph with intersection array {32,27,8,1;1,4,27,32} and related results
• Comparison of two techniques for proving nonexistence of strongly regular graphs
• Cameron-Liebler line classes with parameter $x=\frac{q^2-1}{2}$
• Completely Transitive Designs
• New parameters of subsets in polynomial schemes
• A characterization of the graphs of bilinear $(d\times d)$-forms over ${\mathbb{F}}_2$
• On a Certain Subalgebra of $U_q({\widehat{\mathfrak{sl}}}_2)$ Related to the Degenerate $q$-Onsager Algebra
• Twice $Q$-polynomial distance-regular graphs of diameter 4
• A note on Cameron - Liebler line classes in PG(n,4)
• A spectral characterization of strongly distance-regular graphs with diameter four
• The Erdős-Ko-Rado basis for a Leonard system
• Two characterizations of crooked functions
• On a Koolen -- Park inequality and Terwilliger graphs
• The displacement and split decompositions for a $Q$-polynomial distance-regular graph
• Recursive calculation of effective resistances in distance-regular networks based on Bose-Mesner algebra and Christoffel-Darboux identity
• On the local spectra of the subconstituents of a vertex set and completely pseudo-regular codes
• An odd characterization of the generalized odd graphs

Cited by in corpus (44)

• Eigenvalues of Cayley graphs
• Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems
• Spatial Search on Johnson Graphs by Continuous-Time Quantum Walk
• On relative $t$-designs in polynomial association schemes
• Max-cut and extendability of matchings in distance-regular graphs
• Strongly walk-regular graphs
• Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials
• Arithmetic completely regular codes
• Entanglement of Free Fermions on Hadamard Graphs
• Distance-regular Cayley graphs with least eigenvalue $-2$
• Geometric aspects of 2-walk-regular graphs
• A compact presentation for the alternating central extension of the positive part of $U_q({\widehat{\mathfrak{sl}}}_2)$
• Nonsymmetric Askey-Wilson polynomials and $Q$-polynomial distance-regular graphs
• Godsil-McKay switching and twisted Grassmann graphs
• A characterization of Johnson and Hamming graphs and proof of Babai's conjecture
• Some Spectral and Quasi-Spectral Characterizations of Distance-Regular Graphs
• The Terwilliger algebra of the twisted Grassmann graph: the thin case
• Multimarked Spatial Search by Continuous-Time Quantum Walk
• On the metric dimension of imprimitive distance-regular graphs
• Remarks on pseudo-vertex-transitive graphs with small diameter
• Distance-regular graphs with valency k having smallest eigenvalue at most -k/2
• On the spectrum and linear programming bound for hypergraphs
• Scaling Limits for the Gibbs States on Distance-Regular Graphs with Classical Parameters
• The uniqueness of a distance-regular graph with intersection array {32,27,8,1;1,4,27,32} and related results
• Thick weakly distance-regular digraphs
• Grassmann graphs, degenerate DAHA, and non-symmetric dual $q$-Hahn polynomials
• Characterizing graphs with fully positive semidefinite $Q$-matrices
• A characterization of the graphs of bilinear $(d\times d)$-forms over ${\mathbb{F}}_2$
• Equivalent characterizations of the spectra of graphs and applications to measures of distance-regularity
• A spectral characterization of the s-clique extension of the square grid graphs
• Tight relative $t$-designs on two shells in hypercubes, and Hahn and Hermite polynomials
• Twice $Q$-polynomial distance-regular graphs of diameter 4
• On the coset graph construction of distance-regular graphs
• Nonassociativity of the Norton Algebras of some distance regular graphs
• On the Cheeger constant for distance-regular graphs
• On the spectrum of two families of non-distance-regular graphs
• The super-connectivity of Johnson graphs
• A characterization of 2-neighborhood degree list of diameter 2 graphs
• A spectral characterization of the $s$-clique extension of the triangular graphs
• Primitive weakly distance-regular circulant digraphs
• On completely regular and completely transitive codes derived from Hamming codes
• Speeding up finite-time consensus via minimal polynomial of a weighted graph - a numerical approach
• Spectra of strongly Deza graphs
• On the automorphism groups of distance-regular graphs and rank-4 primitive coherent configurations