{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T02:37:42Z","timestamp":1769049462672,"version":"3.49.0"},"reference-count":53,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2025,8,6]],"date-time":"2025-08-06T00:00:00Z","timestamp":1754438400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,8,6]],"date-time":"2025-08-06T00:00:00Z","timestamp":1754438400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Graphs and Combinatorics"],"published-print":{"date-parts":[[2025,10]]},"abstract":"Abstract<\/jats:title>\n Let \n \n $$\\Gamma$$<\/jats:tex-math>\n \n \u0393<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> denote a Q<\/jats:italic>-polynomial distance-regular graph with diameter \n \n $$D\\ge 1$$<\/jats:tex-math>\n \n \n D<\/mml:mi>\n \u2265<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. For a vertex x<\/jats:italic> of \n \n $$\\Gamma$$<\/jats:tex-math>\n \n \u0393<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> the corresponding subconstituent algebra \n \n $$T=T(x)$$<\/jats:tex-math>\n \n \n T<\/mml:mi>\n =<\/mml:mo>\n T<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is generated by the adjacency matrix A<\/jats:italic> of \n \n $$\\Gamma$$<\/jats:tex-math>\n \n \u0393<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and the dual adjacency matrix \n \n $$A^*=A^*(x)$$<\/jats:tex-math>\n \n \n \n A<\/mml:mi>\n \u2217<\/mml:mo>\n <\/mml:msup>\n =<\/mml:mo>\n \n A<\/mml:mi>\n \u2217<\/mml:mo>\n <\/mml:msup>\n \n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> of \n \n $$\\Gamma$$<\/jats:tex-math>\n \n \u0393<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> with respect to x<\/jats:italic>. We introduce a T<\/jats:italic>-module \n \n $${\\mathcal {N}} = {\\mathcal {N}}(x)$$<\/jats:tex-math>\n \n \n N<\/mml:mi>\n =<\/mml:mo>\n N<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> called the nucleus of \n \n $$\\Gamma$$<\/jats:tex-math>\n \n \u0393<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> with respect to x<\/jats:italic>. We describe \n \n $${\\mathcal {N}}$$<\/jats:tex-math>\n \n N<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> from various points of view. We show that all the irreducible T<\/jats:italic>-submodules of \n \n $${\\mathcal {N}}$$<\/jats:tex-math>\n \n N<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> are thin. Under the assumption that \n \n $$\\Gamma$$<\/jats:tex-math>\n \n \u0393<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is a nonbipartite dual polar graph, we give an explicit basis for \n \n $${\\mathcal {N}}$$<\/jats:tex-math>\n \n N<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and the action of \n \n $$A, A^*$$<\/jats:tex-math>\n \n \n A<\/mml:mi>\n ,<\/mml:mo>\n \n A<\/mml:mi>\n \u2217<\/mml:mo>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> on this basis. The basis is in bijection with the set of elements for the projective geometry \n \n $$L_D(q)$$<\/jats:tex-math>\n \n \n \n L<\/mml:mi>\n D<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n q<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, where GF<\/jats:italic>(q<\/jats:italic>) is the finite field used to define \n \n $$\\Gamma$$<\/jats:tex-math>\n \n \u0393<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00373-025-02960-3","type":"journal-article","created":{"date-parts":[[2025,8,6]],"date-time":"2025-08-06T06:26:42Z","timestamp":1754461602000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Nucleus of a Q-Polynomial Distance-Regular Graph"],"prefix":"10.1007","volume":"41","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0942-5489","authenticated-orcid":false,"given":"Paul","family":"Terwilliger","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,8,6]]},"reference":[{"key":"2960_CR1","volume-title":"Geometric Algebra","author":"E Artin","year":"1957","unstructured":"Artin, E.: Geometric Algebra. Interscience, New York (1957)"},{"key":"2960_CR2","doi-asserted-by":"publisher","DOI":"10.1515\/9783110630251","volume-title":"Algebraic Combinatorics","author":"E Bannai","year":"2021","unstructured":"Bannai, E., Bannai, E., Ito, T., Tanaka, R.: Algebraic Combinatorics, vol. 5. Walter de Gruyter GmbH & Co KG (2021). https:\/\/doi.org\/10.1515\/9783110630251"},{"key":"2960_CR3","volume-title":"Algebraic Combinatorics. I. Association Schemes","author":"E Bannai","year":"1984","unstructured":"Bannai, E., Ito, T.: Algebraic Combinatorics. I. Association Schemes. Benjamin\/Cummings, Menlo Park, CA (1984)"},{"issue":"12","key":"2960_CR4","doi-asserted-by":"crossref","first-page":"113169","DOI":"10.1016\/j.disc.2022.113169","volume":"345","author":"P Bernard","year":"2022","unstructured":"Bernard, P., Cramp\u00e9, N., Vinet, L.: The terwilliger algebra of symplectic dual polar graphs, the subspace lattices and $$U_q(\\mathfrak{sl} _2)$$. Discrete Math. 345(12), 113169 (2022). arXiv:2108.13819","journal-title":"Discrete Math."},{"key":"2960_CR5","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511608704","volume-title":"Algebraic Graph Theory","author":"NL Biggs","year":"1974","unstructured":"Biggs, N.L.: Algebraic Graph Theory, vol. 67. Cambridge University Press, London (1974)"},{"key":"2960_CR6","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-74341-2","volume-title":"Distance Regular-Graphs","author":"AE Brouwer","year":"1989","unstructured":"Brouwer, A.E., Cohen, A., Neumaier, A.: Distance Regular-Graphs. Springer-Verlag, Berlin (1989)"},{"key":"2960_CR7","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1016\/S0097-3165(03)00006-2","volume":"102","author":"A Brouwer","year":"2003","unstructured":"Brouwer, A., Godsil, C., Koolen, J., Martin, W.: Width and dual width of subsets in polynomial association schemes. J. Combin. Theory Ser. A 102, 255\u2013271 (2003)","journal-title":"J. Combin. Theory Ser. A"},{"key":"2960_CR8","volume-title":"Projective and Polar Spaces, Volume 13 of QMW Maths Notes","author":"PJ Cameron","year":"1992","unstructured":"Cameron, P.J.: Projective and Polar Spaces, Volume 13 of QMW Maths Notes. Queen Mary and Westfield College, University of London, London (1992)"},{"key":"2960_CR9","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1016\/S0012-365X(98)00196-4","volume":"196","author":"JS Caughman","year":"1999","unstructured":"Caughman, J.S.: The Terwilliger algebras of bipartite $$P$$- and $$Q$$-polynomial schemes. Discrete Math. 196, 65\u201395 (1999)","journal-title":"Discrete Math."},{"key":"2960_CR10","doi-asserted-by":"crossref","first-page":"1573","DOI":"10.1016\/j.laa.2010.06.005","volume":"433","author":"D Cerzo","year":"2010","unstructured":"Cerzo, D.: Structure of thin irreducible modules of a $$Q$$-polynomial distance-regular graph. Linear Algebra Appl. 433, 1573\u20131613 (2010). arXiv:1003.5368","journal-title":"Linear Algebra Appl."},{"key":"2960_CR11","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/S0012-365X(97)00226-4","volume":"187","author":"B Curtin","year":"1998","unstructured":"Curtin, B.: 2-homogeneous bipartite distance-regular graphs. Discrete Math. 187, 39\u201370 (1998)","journal-title":"Discrete Math."},{"key":"2960_CR12","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1007\/s003730050049","volume":"15","author":"B Curtin","year":"1999","unstructured":"Curtin, B.: Bipartite distance-regular graphs I. Graphs Combin. 15, 143\u2013158 (1999)","journal-title":"Graphs Combin."},{"key":"2960_CR13","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1006\/jctb.2000.2002","volume":"81","author":"B Curtin","year":"2001","unstructured":"Curtin, B.: The Terwilliger algebra of a 2-homogeneous bipartite distance-regular graph. J. Combin. Theory Ser. B 81, 125\u2013141 (2001)","journal-title":"J. Combin. Theory Ser. B"},{"key":"2960_CR14","doi-asserted-by":"crossref","unstructured":"van Dam, E.R., Koolen, J.H., Tanaka, H.: Distance-regular graphs. Electron. J. Combin. DS22 (2016). arXiv:1410.6294","DOI":"10.37236\/4925"},{"key":"2960_CR15","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1007\/s00373-008-0814-8","volume":"24","author":"A Hiraki","year":"2008","unstructured":"Hiraki, A.: Strongly closed subgraphs in a distance-regular graph with $$c_2>1$$. Graphs Combin. 24, 537\u2013550 (2008)","journal-title":"Graphs Combin."},{"key":"2960_CR16","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1007\/s00373-011-1064-8","volume":"28","author":"A Hiraki","year":"2012","unstructured":"Hiraki, A.: A characterization of the Hamming graphs and the dual polar graphs by completely regular subgraphs. Graphs Combin. 28, 449\u2013467 (2012)","journal-title":"Graphs Combin."},{"key":"2960_CR17","first-page":"707","volume":"6","author":"S Ghosh","year":"2023","unstructured":"Ghosh, S., Srinivasan, M.: A $$q$$-analog of the adjacency matrix of the $$n$$-cube. Algebr. Comb. 6, 707\u2013725 (2023). arXiv:2204.05540","journal-title":"Algebr. Comb."},{"key":"2960_CR18","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1006\/eujc.2000.0514","volume":"23","author":"JT Go","year":"2002","unstructured":"Go, J.T.: The Terwilliger algebra of the hypercube. European J. Combin. 23, 399\u2013429 (2002)","journal-title":"European J. Combin."},{"key":"2960_CR19","first-page":"205","volume":"32","author":"T Ito","year":"2019","unstructured":"Ito, T.: TD-pairs and the $$q$$-Onsager algebra. Su-hak 32, 205\u2013232 (2019)","journal-title":"Su-hak"},{"key":"2960_CR20","doi-asserted-by":"crossref","first-page":"1857","DOI":"10.1016\/j.laa.2011.03.032","volume":"435","author":"T Ito","year":"2011","unstructured":"Ito, T., Nomura, K., Terwilliger, P.: A classification of sharp tridiagonal pairs. Linear Algebra Appl. 435, 1857\u20131884 (2011). arXiv:1001.1812","journal-title":"Linear Algebra Appl."},{"key":"2960_CR21","doi-asserted-by":"crossref","unstructured":"Ito, T., Tanabe, K., Terwilliger, P.: Some algebra related to $$P$$- and $$Q$$-polynomial association schemes. Codes and Association Schemes (Piscataway NJ, 1999), pp. 167\u2013192, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 56, Amer. Math. Soc., Providence RI (2001); arXiv:math.CO\/0406556","DOI":"10.1090\/dimacs\/056\/14"},{"key":"2960_CR22","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/s11139-006-0242-4","volume":"13","author":"T Ito","year":"2007","unstructured":"Ito, T., Terwilliger, P.: Tridiagonal pairs and the quantum affine algebra $$U_q({\\widehat{sl}}_2)$$. Ramanujan J. 13, 39\u201362 (2007). arXiv:math.QA\/0310042","journal-title":"Ramanujan J."},{"key":"2960_CR23","doi-asserted-by":"crossref","first-page":"68","DOI":"10.1016\/j.jalgebra.2009.04.008","volume":"322","author":"T Ito","year":"2009","unstructured":"Ito, T., Terwilliger, P.: Tridiagonal pairs of $$q$$-Racah type. J. Algebra 322, 68\u201393 (2009). arXiv:0807.0271","journal-title":"J. Algebra"},{"key":"2960_CR24","doi-asserted-by":"crossref","first-page":"81","DOI":"10.2206\/kyushujm.64.81","volume":"64","author":"T Ito","year":"2010","unstructured":"Ito, T., Terwilliger, P.: The augmented tridiagonal algebra. Kyushu J. Math. 64, 81\u2013144 (2010). arXiv:0904.2889","journal-title":"Kyushu J. Math."},{"key":"2960_CR25","doi-asserted-by":"crossref","first-page":"1828","DOI":"10.1016\/j.disc.2010.01.004","volume":"310","author":"J Kim","year":"2010","unstructured":"Kim, J.: A duality between pairs of split decompositions for a $$Q$$-polynomial distance-regular graph. Discrete Math. 310, 1828\u20131834 (2010). arXiv:0705.0167","journal-title":"Discrete Math."},{"key":"2960_CR26","doi-asserted-by":"crossref","first-page":"53","DOI":"10.4067\/S0719-06462010000200005","volume":"12","author":"F Levstein","year":"2010","unstructured":"Levstein, F., Maldonado, C.: Generalized quadrangles and subconstituent algebra. Cubo 12, 53\u201375 (2010)","journal-title":"Cubo"},{"key":"2960_CR27","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1007\/s00373-018-1915-7","volume":"34","author":"S Mamart","year":"2018","unstructured":"Mamart, S.: A group commutator involving the last distance matrix and dual distance matrix of a $$Q$$-polynomial distance-regular graph: the Hamming graph case. Graphs Combin. 34, 803\u2013817 (2018). arXiv:1801.05494","journal-title":"Graphs Combin."},{"key":"2960_CR28","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1016\/j.laa.2007.01.028","volume":"424","author":"K Nomura","year":"2007","unstructured":"Nomura, K., Terwilliger, P.: The split decomposition of a tridiagonal pair. Linear Algebra Appl. 424, 339\u2013345 (2007). arXiv:math\/0612460","journal-title":"Linear Algebra Appl."},{"key":"2960_CR29","doi-asserted-by":"crossref","first-page":"1083","DOI":"10.1016\/j.laa.2007.09.002","volume":"428","author":"K Nomura","year":"2008","unstructured":"Nomura, K., Terwilliger, P.: The switching element for a leonard pair. Linear Algebra Appl. 428, 1083\u20131108 (2008). arXiv:math\/0608623","journal-title":"Linear Algebra Appl."},{"key":"2960_CR30","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1016\/j.laa.2008.02.006","volume":"429","author":"K Nomura","year":"2008","unstructured":"Nomura, K., Terwilliger, P.: Sharp tridiagonal pairs. Linear Algebra Appl. 429, 79\u201399 (2008). arXiv:0712.3665","journal-title":"Linear Algebra Appl."},{"key":"2960_CR31","doi-asserted-by":"crossref","first-page":"1647","DOI":"10.1016\/j.laa.2008.04.042","volume":"429","author":"K Nomura","year":"2008","unstructured":"Nomura, K., Terwilliger, P.: The structure of a tridiagonal pair. Linear Algebra Appl. 429, 1647\u20131662 (2008). arXiv:0802.1096","journal-title":"Linear Algebra Appl."},{"key":"2960_CR32","doi-asserted-by":"crossref","first-page":"1073","DOI":"10.1006\/eujc.2002.0607","volume":"23","author":"A Pascasio","year":"2002","unstructured":"Pascasio, A.: On the multiplicities of the primitive idempotents of a $$Q$$-polynomial distance-regular graph. European J. Combin. 23, 1073\u20131078 (2002)","journal-title":"European J. Combin."},{"key":"2960_CR33","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/0097-3165(85)90052-4","volume":"40","author":"D Stanton","year":"1985","unstructured":"Stanton, D.: Harmonics on posets. J. Combin. Theory Ser. A 40, 136\u2013149 (1985)","journal-title":"J. Combin. Theory Ser. A"},{"key":"2960_CR34","doi-asserted-by":"crossref","first-page":"P34","DOI":"10.37236\/2389","volume":"19","author":"M Srinivasan","year":"2012","unstructured":"Srinivasan, M.: A positive combinatorial formula for the complexity of the $$q$$-analog of the $$n$$-cube. Electron. J. Combin. 19, P34 (2012). arXiv:1111.1799","journal-title":"Electron. J. Combin."},{"key":"2960_CR35","doi-asserted-by":"crossref","first-page":"903","DOI":"10.1016\/j.jcta.2005.08.006","volume":"113","author":"H Tanaka","year":"2006","unstructured":"Tanaka, H.: Classification of subsets with minimal width and dual width in grassmann, bilinear forms and dual polar graphs. J. Combin. Theory Ser. A 113, 903\u2013910 (2006)","journal-title":"J. Combin. Theory Ser. A"},{"key":"2960_CR36","doi-asserted-by":"crossref","first-page":"167, 32","DOI":"10.37236\/654","volume":"18","author":"H Tanaka","year":"2011","unstructured":"Tanaka, H.: Vertex subsets with minimal width and dual width in $$Q$$-polynomial distance-regular graphs. Electron. J. Combin. 18, 167, 32 (2011). arXiv:1011.2000","journal-title":"Electron. J. Combin."},{"key":"2960_CR37","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/978-1-4613-8994-1_15","volume-title":"The incidence algebra of a uniform poset. Coding Theory and Design Theory: Part I Coding Theory","author":"P Terwilliger","year":"1990","unstructured":"Terwilliger, P.: The incidence algebra of a uniform poset. Coding Theory and Design Theory: Part I Coding Theory, pp. 193\u2013212. Springer, New York (1990)"},{"key":"2960_CR38","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1023\/A:1022494701663","volume":"1","author":"P Terwilliger","year":"1992","unstructured":"Terwilliger, P.: The subconstituent algebra of an association scheme I. J. Algebraic Combin. 1, 363\u2013388 (1992)","journal-title":"J. Algebraic Combin."},{"key":"2960_CR39","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1023\/A:1022480715311","volume":"2","author":"P Terwilliger","year":"1993","unstructured":"Terwilliger, P.: The subconstituent algebra of an association scheme II. J. Algebraic Combin. 2, 73\u2013103 (1993)","journal-title":"J. Algebraic Combin."},{"key":"2960_CR40","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1023\/A:1022415825656","volume":"2","author":"P Terwilliger","year":"1993","unstructured":"Terwilliger, P.: The subconstituent algebra of an association scheme III. J. Algebraic Combin. 2, 177\u2013210 (1993)","journal-title":"J. Algebraic Combin."},{"key":"2960_CR41","doi-asserted-by":"crossref","first-page":"323","DOI":"10.2969\/aspm\/02410323","volume-title":"Progress in Algebraic Combinatorics. Advanced Studies in Pure Mathematics","author":"P Terwilliger","year":"1996","unstructured":"Terwilliger, P.: Quantum matroids. In: Bannai, E., Munemasa, A. (eds.) Progress in Algebraic Combinatorics. Advanced Studies in Pure Mathematics, vol. 24, pp. 323\u2013441. Mathematical Society of Japan, Tokyo (1996)"},{"key":"2960_CR42","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1016\/S0024-3795(01)00242-7","volume":"330","author":"P Terwilliger","year":"2001","unstructured":"Terwilliger, P.: Two linear transformations each tridiagonal with respect to an eigenbasis of the other. Linear Algebra Appl. 330, 149\u2013203 (2001). arXiv:math.RA\/0406555","journal-title":"Linear Algebra Appl."},{"key":"2960_CR43","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1016\/S0377-0427(02)00600-3","volume":"153","author":"P Terwilliger","year":"2003","unstructured":"Terwilliger, P.: Introduction to leonard pairs. J. Comput. Appl. Math. 153, 463\u2013475 (2003)","journal-title":"J. Comput. Appl. Math."},{"key":"2960_CR44","doi-asserted-by":"crossref","first-page":"263","DOI":"10.1007\/s00373-004-0594-8","volume":"21","author":"P Terwilliger","year":"2005","unstructured":"Terwilliger, P.: The displacement and split decompositions for a $$Q$$-polynomial distance-regular graph. Graphs Combin. 21, 263\u2013276 (2005). arXiv:math.CO\/0306142","journal-title":"Graphs Combin."},{"key":"2960_CR45","doi-asserted-by":"crossref","first-page":"1687","DOI":"10.1007\/s00373-021-02357-y","volume":"37","author":"P Terwilliger","year":"2021","unstructured":"Terwilliger, P.: Notes on the leonard system classification. Graphs Combin. 37, 1687\u20131748 (2021). arXiv:2003.09668","journal-title":"Graphs Combin."},{"key":"2960_CR46","first-page":"430","volume":"487","author":"P Terwilliger","year":"2024","unstructured":"Terwilliger, P.: Distance-regular graphs, the subconstituent algebra, and the $$Q$$-polynomial property. London Math. Soc. Lecture Note Ser. 487, 430\u2013491 (2024). arXiv:2207.07747","journal-title":"London Math. Soc. Lecture Note Ser."},{"key":"2960_CR47","doi-asserted-by":"crossref","first-page":"105724","DOI":"10.1016\/j.jcta.2022.105724","volume":"196","author":"P Terwilliger","year":"2023","unstructured":"Terwilliger, P.: Tridiagonal pairs, alternating elements, and distance-regular graphs. J. Combin. Theory Ser. A 196, 105724 (2023). arXiv:2207.07741","journal-title":"J. Combin. Theory Ser. A"},{"issue":"4","key":"2960_CR48","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/s00373-023-02661-9","volume":"39","author":"P Terwilliger","year":"2023","unstructured":"Terwilliger, P.: A $$Q$$-polynomial structure associated with the projective geometry $$L_N(q)$$. Graphs Combin. 39(4), 63 (2023). arXiv:2208.13098","journal-title":"Graphs Combin."},{"issue":"2","key":"2960_CR49","doi-asserted-by":"crossref","first-page":"114321","DOI":"10.1016\/j.disc.2024.114321","volume":"348","author":"P Terwilliger","year":"2025","unstructured":"Terwilliger, P.: Projective geometries, $$Q$$-polynomial structures, and quantum groups. Discret. Math. 348(2), 114321 (2025). arXiv:2407.14964","journal-title":"Discret. Math."},{"key":"2960_CR50","doi-asserted-by":"crossref","first-page":"411","DOI":"10.1142\/S0219498804000940","volume":"3","author":"P Terwilliger","year":"2004","unstructured":"Terwilliger, P., Vidunas, R.: Leonard pairs and the Askey-Wilson relations. J. Algebra Appl. 3, 411\u2013426 (2004). arXiv:math\/0305356","journal-title":"J. Algebra Appl."},{"key":"2960_CR51","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1006\/eujc.1995.0083","volume":"18","author":"CW Weng","year":"1997","unstructured":"Weng, C.W.: $$D$$-bounded distance-regular graphs. European J. Combin. 18, 211\u2013229 (1997)","journal-title":"European J. Combin."},{"key":"2960_CR52","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1007\/s003730050031","volume":"14","author":"CW Weng","year":"1998","unstructured":"Weng, C.W.: Weak-geodetically closed subgraphs in distance-regular graphs. Graphs Combin. 14, 275\u2013304 (1998)","journal-title":"Graphs Combin."},{"key":"2960_CR53","doi-asserted-by":"crossref","first-page":"443","DOI":"10.1016\/j.laa.2012.08.016","volume":"438","author":"C Worawannotai","year":"2013","unstructured":"Worawannotai, C.: Dual polar graphs, the quantum algebra $$U_q(\\mathfrak{sl} _2)$$, and Leonard systems of dual $$q$$-krawtchouk type. Linear Algebra Appl. 438, 443\u2013497 (2013). arXiv:1205.2144","journal-title":"Linear Algebra Appl."}],"container-title":["Graphs and Combinatorics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00373-025-02960-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00373-025-02960-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00373-025-02960-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T05:11:28Z","timestamp":1760505088000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00373-025-02960-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,6]]},"references-count":53,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["2960"],"URL":"https:\/\/doi.org\/10.1007\/s00373-025-02960-3","relation":{},"ISSN":["0911-0119","1435-5914"],"issn-type":[{"value":"0911-0119","type":"print"},{"value":"1435-5914","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,6]]},"assertion":[{"value":"30 August 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 July 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 August 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The author declares that no funds, grants, or other support were received during the preparation of this manuscript.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Funding"}},{"value":"The author has no relevant financial or non-financial interests to disclose.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"94"}}