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Distance-Regular Graphs Distance-regular graphs Edwin R. van Dam Jack H. Koolen Department of Econometrics and O.R. School of Mathematical Sciences Tilburg University University of Science and Technology of China The Netherlands and [email protected] Wu Wen-Tsun Key Laboratory of Mathematics of CAS Hefei, Anhui, 230026, China [email protected] Hajime Tanaka Research Center for Pure and Applied Mathematics Graduate School of Information Sciences Tohoku University Sendai 980-8579, Japan [email protected] Submitted: Dec 19, 2014; Accepted: Apr 7, 2016 1st edition, Apr 15, 2016 Mathematics Subject Classifications: 05E30, 05Cxx, 05Exx 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. Keywords: Distance-regular graph; survey; association scheme; P -polynomial; Q- polynomial; geometric the electronic journal of combinatorics (2016), #DS22 1 Contents 1 Introduction6 2 An introduction to distance-regular graphs7 2.1 Definition....................................8 2.2 A few examples.................................8 2.2.1 The complete graph..........................8 2.2.2 The polygons..............................9 2.2.3 The Petersen graph and other Odd graphs..............9 2.3 Which graphs are determined by their intersection array?..........9 2.4 Some combinatorial conditions for the intersection array.......... 10 2.5 The spectrum of eigenvalues and multiplicities................ 12 2.6 Association schemes.............................. 15 2.7 The Q-polynomial property.......................... 17 2.8 Delsarte cliques and geometric graphs..................... 18 2.9 Imprimitivity.................................. 18 2.10 Distance-transitive graphs........................... 20 3 Examples 20 3.1 The classical families with unbounded diameter............... 20 3.1.1 Classical parameters.......................... 22 3.1.2 Other families with unbounded diameter............... 23 3.2 New constructions............................... 24 3.2.1 The twisted Grassmann graphs.................... 24 3.2.2 Brouwer-Pasechnik and Kasami graphs................ 25 3.2.3 De Caen, Mathon, and Moorhouse's Preparata graphs and crooked graphs.................................. 25 3.2.4 Soicher graphs and Meixner graphs.................. 27 3.2.5 The Koolen-Riebeek graph and the Moscow-Soicher graph..... 27 4 More background 27 4.1 Miscellaneous definitions............................ 27 4.2 A few comments on the eigenspaces...................... 29 4.3 The Terwilliger algebra............................. 29 4.4 Equitable partitions and completely regular codes.............. 31 4.4.1 Interlacing, the quotient matrix, and the quotient graph...... 31 4.4.2 Completely regular codes........................ 32 4.5 Distance-biregular graphs and weakly geometric graphs........... 33 4.6 Homogeneity.................................. 35 4.7 Designs..................................... 35 the electronic journal of combinatorics (2016), #DS22 2 5 Q-polynomial distance-regular graphs 36 5.1 The graphs with classical parameters with b = 1............... 36 5.2 Recent results on graphs with classical parameters.............. 37 5.3 Imprimitive graphs with classical parameters and partition graphs..... 38 5.4 Characterizations of the Q-polynomial property............... 38 5.5 Classification results.............................. 40 5.5.1 Case p 6= ±2............................... 40 5.5.2 Case p = 2, r 6= 0, r? 6= 0........................ 41 5.5.3 Case p = 2, r = 0, r? 6= 0........................ 41 5.5.4 Case p = 2, r 6= 0, r? = 0........................ 41 5.5.5 Case p = 2, r = r? = 0......................... 41 5.5.6 Case p = −2............................... 41 5.5.7 Bipartite graphs............................. 41 5.5.8 Antipodal graphs............................ 41 5.5.9 Almost bipartite graphs........................ 42 5.5.10 Almost Q-bipartite graphs....................... 42 5.6 The Terwilliger algebras of Q-polynomial distance-regular graphs..... 42 5.7 Further results on Q-polynomial distance-regular graphs.......... 44 5.7.1 Antipodal covers............................ 44 5.7.2 Distance-regular graphs with multiple Q-polynomial orderings... 44 5.7.3 Bounds for the girth.......................... 45 5.7.4 The Erd}os-Ko-Radotheorem...................... 45 5.7.5 Unimodality of the multiplicities.................... 45 5.7.6 Posets associated with Q-polynomial distance-regular graphs.... 45 5.8 Tridiagonal systems............................... 47 6 The Terwilliger algebra and combinatorics 49 6.1 Homogeneity and tight distance-regular graphs................ 49 6.1.1 Tight distance-regular graphs..................... 49 6.1.2 The CAB condition and 1-homogeneous distance-regular graphs.. 51 6.1.3 More results on homogeneity...................... 51 6.2 Thin modules.................................. 51 6.3 Vanishing Krein parameters.......................... 53 6.3.1 Triple intersection numbers...................... 53 6.3.2 Hadamard products of two primitive idempotents.......... 54 6.4 Relaxations of homogeneity.......................... 55 7 Growth of intersection numbers and bounds on the diameter 57 7.1 The Ivanov bound............................... 57 7.2 Distance-regular graphs of order (s; t)..................... 58 7.3 A bound for distance-regular graphs with c2 > 2............... 58 7.4 The Pyber Bound................................ 59 the electronic journal of combinatorics (2016), #DS22 3 8 The Bannai-Ito conjecture 59 8.1 Proof of the Bannai-Ito conjecture....................... 59 8.2 Extensions of the Bannai-Ito conjecture.................... 61 8.3 The distance-regular graphs with small valency............... 61 9 Geometric distance-regular graphs 62 9.1 Metsch's characterizations........................... 62 9.2 Characterization of Doubled Odd and Doubled Grassmann graphs..... 63 9.3 Bounds on claws................................ 63 9.4 Sufficient conditions............................... 65 9.5 Distance-regular graphs with a fixed smallest eigenvalue........... 65 9.6 Regular near polygons............................. 66 10 Spectral characterizations 68 10.1 Distance-regularity from the spectrum.................... 68 10.2 The p-rank.................................... 70 10.3 Spectral excess theorem............................ 70 10.4 Almost distance-regular graphs........................ 72 11 Subgraphs 73 11.1 Strongly closed subgraphs........................... 73 11.2 Bipartite closed subgraphs........................... 75 11.3 Maximal cliques................................. 75 11.4 Convex subgraphs................................ 76 11.5 Designs..................................... 76 11.6 The Terwilliger algebra with respect to a code................ 77 12 Completely regular codes 78 12.1 Parameters................................... 79 12.2 Leonard completely regular codes....................... 79 12.3 Completely regular codes in the Hamming graphs.............. 79 12.3.1 Completely transitive codes and generalizations........... 80 12.3.2 Arithmetic completely regular codes................. 81 12.4 Completely regular codes in other distance-regular graphs.......... 82 13 More combinatorial properties 83 13.1 Distance-regular graphs with a relatively small number of vertices..... 83 13.2 Distance-regular graphs with multiple P -polynomial orderings....... 85 13.3 Characterizing antipodality and the height.................. 85 13.4 Bounds on kD for primitive distance-regular graphs............. 86 13.5 Terwilliger graphs and existence of quadrangles............... 86 13.6 Connectivity and the second eigenvalue.................... 87 13.6.1 Connectivity and matchings...................... 87 13.6.2 The second largest eigenvalue..................... 87 the electronic journal of combinatorics (2016), #DS22 4 13.6.3 The standard sequence......................... 88 13.7 The distance-D graph............................. 88 14 Multiplicities 89 14.1 Terwilliger's tree bound............................ 89 14.2 Godsil's bound................................. 91 14.3 The distance-regular graphs with a small multiplicity............ 92 14.4 Integrality of multiplicities........................... 92 15 Applications 93 15.1 Combinatorial optimization.......................... 93 15.2 Random walks, diffusion models, and quantum walks............ 94 15.2.1 Diffusion models and stock portfolios................. 94 15.2.2 Chip-firing and the abelian sandpile model.............. 95 15.2.3 Biggs' conjecture on resistance and potential............. 95 15.2.4 Quantum walks............................. 96 15.3 Miscellaneous applications........................... 97 16 Miscellaneous 97 16.1 Distance-transitive graphs........................... 97 16.2 The metric dimension.............................. 98 16.3 The chromatic number............................. 98 16.4 Cores....................................... 99 16.5 Modular representations............................ 99 16.6 Asymptotic spectral analysis.......................... 100 16.7 Spin models................................... 100 16.8 Cometric association schemes......................... 101 17 Tables 102 17.1 Diameter 3 and primitive............................ 102 17.1.1 Uniqueness..............................
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