{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,4,12]],"date-time":"2025-04-12T22:44:47Z","timestamp":1744497887158,"version":"3.28.0"},"reference-count":71,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2024,11,13]],"date-time":"2024-11-13T00:00:00Z","timestamp":1731456000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,11,13]],"date-time":"2024-11-13T00:00:00Z","timestamp":1731456000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Supercomput"],"published-print":{"date-parts":[[2025,1]]},"DOI":"10.1007\/s11227-024-06633-5","type":"journal-article","created":{"date-parts":[[2024,11,13]],"date-time":"2024-11-13T23:34:49Z","timestamp":1731540889000},"update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Bandwidth of WK-recursive networks and its sparse matrix computation"],"prefix":"10.1007","volume":"81","author":[{"given":"R.","family":"Nathiya","sequence":"first","affiliation":[]},{"given":"D.","family":"Meiyappan","sequence":"additional","affiliation":[]},{"given":"Savari","family":"Prabhu","sequence":"additional","affiliation":[]},{"given":"Sudeep","family":"Stephen","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,11,13]]},"reference":[{"issue":"8","key":"6633_CR1","doi-asserted-by":"publisher","first-page":"441","DOI":"10.1145\/366707.367535","volume":"6","author":"DH Thurnau","year":"1963","unstructured":"Thurnau DH (1963) Algorithm 195-bandsolve. Commun. ACM 6(8):441","journal-title":"Commun. ACM"},{"key":"6633_CR2","doi-asserted-by":"crossref","unstructured":"Cuthill E, McKee J (1969) Reducing the bandwidth of sparse symmetric matrices, In: Proceedings of the 24th national conference 157\u2013172","DOI":"10.1145\/800195.805928"},{"issue":"3","key":"6633_CR3","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1002\/nme.1620060306","volume":"6","author":"RJ Collins","year":"1973","unstructured":"Collins RJ (1973) Bandwidth reduction by automatic renumbering. Int J Numer Meth Eng 6(3):345\u2013356","journal-title":"Int J Numer Meth Eng"},{"key":"6633_CR4","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1016\/j.engappai.2016.02.008","volume":"52","author":"X Lai","year":"2016","unstructured":"Lai X, Hao JK, L\u00fc Z, Glover F (2016) A learning-based path relinking algorithm for the bandwidth coloring problem. Eng Appl Artif Intell 52:81\u201391","journal-title":"Eng Appl Artif Intell"},{"issue":"3","key":"6633_CR5","doi-asserted-by":"publisher","first-page":"1126","DOI":"10.1016\/j.ejor.2022.09.004","volume":"306","author":"S Cavero","year":"2023","unstructured":"Cavero S, Pardo EG, Duarte A (2023) Efficient iterated greedy for the two-dimensional bandwidth minimization problem. Eur J Oper Res 306(3):1126\u20131139","journal-title":"Eur J Oper Res"},{"issue":"2","key":"6633_CR6","doi-asserted-by":"publisher","first-page":"75","DOI":"10.1002\/(SICI)1097-0118(199906)31:2<75::AID-JGT1>3.0.CO;2-S","volume":"31","author":"YL Lai","year":"1999","unstructured":"Lai YL, Williams K (1999) A survey of solved problems and applications on bandwidth, edgesum, and profile of graphs. J Graph Theory 31(2):75\u201394","journal-title":"J Graph Theory"},{"issue":"1","key":"6633_CR7","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1137\/0608002","volume":"8","author":"FR Chung","year":"1987","unstructured":"Chung FR, Leighton FT, Rosenberg AL (1987) Embedding graphs in books: A layout problem with applications to VLSI design. SIAM J Algeb Discrete Methods 8(1):33\u201358","journal-title":"SIAM J Algeb Discrete Methods"},{"issue":"10","key":"6633_CR8","doi-asserted-by":"publisher","first-page":"967","DOI":"10.1016\/j.sysarc.2008.04.005","volume":"54","author":"MH Farahabady","year":"2008","unstructured":"Farahabady MH, Imani N, Sarbazi-Azad H (2008) Some topological and combinatorial properties of WK-recursive mesh and WK-pyramid interconnection networks. J Syst Architect 54(10):967\u2013976","journal-title":"J Syst Architect"},{"issue":"2","key":"6633_CR9","doi-asserted-by":"publisher","first-page":"361","DOI":"10.1016\/0045-7949(96)00028-4","volume":"61","author":"MA Ajiz","year":"1996","unstructured":"Ajiz MA, Hamed MM (1996) Manzur and Hamed (MH) algorithm for resequencing nodes in the undirected graphs to minimize the storage by reducing the bandwidth of sparse symmetric matrices. Comput Struct 61(2):361\u2013374","journal-title":"Comput Struct"},{"key":"6633_CR10","doi-asserted-by":"publisher","first-page":"108","DOI":"10.1016\/j.compeleceng.2022.108488","volume":"105","author":"T Li","year":"2023","unstructured":"Li T, Shen L (2023) A sparse matrix-vector multiplication accelerator based on high-bandwidth memory. Comput Electr Eng 105:108\u2013488","journal-title":"Comput Electr Eng"},{"issue":"1","key":"6633_CR11","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1137\/0112012","volume":"12","author":"LH Harper","year":"1964","unstructured":"Harper LH (1964) Optimal assignments of numbers to vertices. J Soc Ind Appl Math 12(1):131\u2013135","journal-title":"J Soc Ind Appl Math"},{"issue":"3","key":"6633_CR12","doi-asserted-by":"publisher","first-page":"263","DOI":"10.1007\/BF02280884","volume":"16","author":"CH Papadimitriou","year":"1976","unstructured":"Papadimitriou CH (1976) The NP-completeness of the bandwidth minimization problem. Computing 16(3):263\u2013270","journal-title":"Computing"},{"issue":"3","key":"6633_CR13","doi-asserted-by":"publisher","first-page":"477","DOI":"10.1137\/0134037","volume":"34","author":"MR Garey","year":"1978","unstructured":"Garey MR, Graham RL, Johnson DS, Knuth DE (1978) Complexity results for bandwidth minimization. SIAM J Appl Math 34(3):477\u2013495","journal-title":"SIAM J Appl Math"},{"issue":"4","key":"6633_CR14","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1137\/0607057","volume":"7","author":"B Monien","year":"1986","unstructured":"Monien B (1986) The bandwidth minimization problem for caterpillars with hair length 3 is NP-complete. SIAM J Algeb Discrete Methods 7(4):505\u2013512","journal-title":"SIAM J Algeb Discrete Methods"},{"issue":"3","key":"6633_CR15","doi-asserted-by":"publisher","first-page":"567","DOI":"10.1016\/S0304-3975(03)00238-X","volume":"307","author":"D Muradian","year":"2003","unstructured":"Muradian D (2003) The bandwidth minimization problem for cyclic caterpillars with hair length 1 is NP-complete. Theoret Comput Sci 307(3):567\u2013572","journal-title":"Theoret Comput Sci"},{"issue":"3","key":"6633_CR16","doi-asserted-by":"publisher","first-page":"266","DOI":"10.1016\/j.tcs.2006.07.015","volume":"363","author":"M Lin","year":"2006","unstructured":"Lin M, Lin Z, Xu J (2006) Graph bandwidth of weighted caterpillars. Theoret Comput Sci 363(3):266\u2013277","journal-title":"Theoret Comput Sci"},{"issue":"4","key":"6633_CR17","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1137\/0602041","volume":"2","author":"SF Assmann","year":"1981","unstructured":"Assmann SF, Peck GW, Sys\u0142o MM, Zak J (1981) The bandwidth of caterpillars with hairs of length 1 and 2. SIAM J Algeb Discrete Methods 2(4):387\u2013393","journal-title":"SIAM J Algeb Discrete Methods"},{"issue":"1","key":"6633_CR18","doi-asserted-by":"publisher","first-page":"190","DOI":"10.1007\/s00453-007-9002-0","volume":"55","author":"U Feige","year":"2009","unstructured":"Feige U, Talwar K (2009) Approximating the bandwidth of caterpillars. Algorithmica 55(1):190\u2013204","journal-title":"Algorithmica"},{"issue":"2","key":"6633_CR19","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1016\/0020-0190(93)90206-O","volume":"46","author":"F Makedon","year":"1993","unstructured":"Makedon F, Sheinwald D, Wolfsthal Y (1993) A simple linear-time algorithm for the recognition of bandwidth-2 biconnected graphs. Inf Process Lett 46(2):103\u2013107","journal-title":"Inf Process Lett"},{"issue":"2","key":"6633_CR20","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1016\/S0304-3975(00)00068-2","volume":"249","author":"S Zhou","year":"2000","unstructured":"Zhou S (2000) Bounding the bandwidths for graphs. Theoret Comput Sci 249(2):357\u2013368","journal-title":"Theoret Comput Sci"},{"key":"6633_CR21","doi-asserted-by":"publisher","first-page":"491","DOI":"10.1016\/S0304-3975(03)00052-5","volume":"301","author":"LH Harper","year":"2003","unstructured":"Harper LH (2003) On the bandwidth of a Hamming graph. Theoret Comput Sci 301:491\u2013498","journal-title":"Theoret Comput Sci"},{"key":"6633_CR22","doi-asserted-by":"publisher","first-page":"488","DOI":"10.1016\/j.tcs.2008.07.029","volume":"407","author":"J Balogh","year":"2008","unstructured":"Balogh J, Bezrukov SL, Harper LH, Seress A (2008) On the bandwidth of 3-dimensional Hamming graphs. Theoret Comput Sci 407:488\u2013495","journal-title":"Theoret Comput Sci"},{"issue":"2","key":"6633_CR23","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1016\/0012-365X(92)90268-K","volume":"103","author":"GW Peck","year":"1992","unstructured":"Peck GW, Shastri A (1992) Bandwidth of theta graphs with short paths. Discret Math 103(2):177\u2013187","journal-title":"Discret Math"},{"issue":"4","key":"6633_CR24","doi-asserted-by":"publisher","first-page":"533","DOI":"10.1016\/j.jda.2008.11.001","volume":"7","author":"P Heggernes","year":"2009","unstructured":"Heggernes P, Kratsch D, Meister D (2009) Bandwidth of bipartite permutation graphs in polynomial time. J Discrete Algo 7(4):533\u2013544","journal-title":"J Discrete Algo"},{"key":"6633_CR25","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1016\/S0012-365X(98)00009-0","volume":"189","author":"MM Sys\u0142o","year":"1998","unstructured":"Sys\u0142o MM, Weaver ML, West DB (1998) Bandwidth and density for block graphs. Discret Math 189:163\u2013176","journal-title":"Discret Math"},{"key":"6633_CR26","doi-asserted-by":"publisher","first-page":"3701","DOI":"10.1016\/j.tcs.2010.06.018","volume":"411","author":"M Cygan","year":"2010","unstructured":"Cygan M, Pilipczuk M (2010) Exact and approximate bandwidth. Theoret Comput Sci 411:3701\u20133713","journal-title":"Theoret Comput Sci"},{"issue":"11","key":"6633_CR27","doi-asserted-by":"publisher","first-page":"469","DOI":"10.1016\/j.ipl.2010.04.013","volume":"110","author":"L Lin","year":"2010","unstructured":"Lin L, Lin Y (2010) Two models of two-dimensional bandwidth problems. Inf Process Lett 110(11):469\u2013473","journal-title":"Inf Process Lett"},{"key":"6633_CR28","doi-asserted-by":"publisher","first-page":"44","DOI":"10.1016\/j.tcs.2013.05.038","volume":"500","author":"Y Bartal","year":"2013","unstructured":"Bartal Y, Carroll DE, Meyerson A, Neiman O (2013) Bandwidth and low dimensional embedding. Theoret Comput Sci 500:44\u201356","journal-title":"Theoret Comput Sci"},{"issue":"2","key":"6633_CR29","doi-asserted-by":"publisher","first-page":"140","DOI":"10.1016\/0890-5401(87)90028-9","volume":"74","author":"D Kratsch","year":"1987","unstructured":"Kratsch D (1987) Finding the minimum bandwidth of an interval graph. Inf Comput 74(2):140\u2013158","journal-title":"Inf Comput"},{"issue":"2","key":"6633_CR30","doi-asserted-by":"publisher","first-page":"218","DOI":"10.1016\/0890-5401(91)90045-4","volume":"95","author":"R Mahesh","year":"1991","unstructured":"Mahesh R, Rangan CP, Srinivasan A (1991) On finding the minimum bandwidth of interval graphs. Inf Comput 95(2):218\u2013224","journal-title":"Inf Comput"},{"key":"6633_CR31","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1016\/0012-365X(93)E0219-T","volume":"142","author":"L Smithline","year":"1995","unstructured":"Smithline L (1995) Bandwidth of the complete $$k$$-ary tree. Discret Math 142:203\u2013212","journal-title":"Discret Math"},{"key":"6633_CR32","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1016\/0012-365X(89)90083-6","volume":"75","author":"FR Chung","year":"1989","unstructured":"Chung FR, Seymour PD (1989) Graphs with small bandwidth and cutwidth. Discret Math 75:113\u2013119","journal-title":"Discret Math"},{"issue":"6","key":"6633_CR33","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1016\/S0020-0190(98)00173-2","volume":"68","author":"T Kloks","year":"1998","unstructured":"Kloks T, Kratsch D, M\u00fcller H (1998) Bandwidth of chain graphs. Inf Process Lett 68(6):313\u2013315","journal-title":"Inf Process Lett"},{"key":"6633_CR34","first-page":"281","volume":"66","author":"MM Sys\u0142o","year":"1982","unstructured":"Sys\u0142o MM, \u017bak J (1982) The bandwidth problem: Critical subgraphs and the solution for caterpillars. Ann Discrete Math 66:281\u2013286","journal-title":"Ann Discrete Math"},{"issue":"1","key":"6633_CR35","doi-asserted-by":"publisher","first-page":"24","DOI":"10.1006\/jagm.2000.1118","volume":"40","author":"A Gupta","year":"2001","unstructured":"Gupta A (2001) Improved bandwidth approximation for trees and chordal graphs. J Algo 40(1):24\u201336","journal-title":"J Algo"},{"issue":"50","key":"6633_CR36","doi-asserted-by":"publisher","first-page":"7001","DOI":"10.1016\/j.tcs.2011.09.011","volume":"412","author":"P Golovach","year":"2011","unstructured":"Golovach P, Heggernes P, Kratsch D, Lokshtanov D, Meister D, Saurabh S (2011) Bandwidth on AT-free graphs. Theoret Comput Sci 412(50):7001\u20137008","journal-title":"Theoret Comput Sci"},{"key":"6633_CR37","doi-asserted-by":"publisher","first-page":"238","DOI":"10.1016\/j.tcs.2007.03.012","volume":"377","author":"YL Lai","year":"2007","unstructured":"Lai YL, Tian CS (2007) An extremal graph with given bandwidth. Theoret Comput Sci 377:238\u2013242","journal-title":"Theoret Comput Sci"},{"issue":"4","key":"6633_CR38","doi-asserted-by":"publisher","first-page":"752","DOI":"10.1016\/j.jctb.2007.11.005","volume":"98","author":"J B\u00f6ttcher","year":"2008","unstructured":"B\u00f6ttcher J, Schacht M, Taraz A (2008) Spanning 3-colourable subgraphs of small bandwidth in dense graphs. J Comb Theory Ser B 98(4):752\u2013777","journal-title":"J Comb Theory Ser B"},{"issue":"4","key":"6633_CR39","doi-asserted-by":"publisher","first-page":"679","DOI":"10.1007\/s10878-012-9548-x","volume":"27","author":"GJ Chang","year":"2014","unstructured":"Chang GJ, Chia ML, Kuo D, Lin JY, Yan JH (2014) Bandwidth sums of block graphs and cacti. J Comb Optim 27(4):679\u2013687","journal-title":"J Comb Optim"},{"key":"6633_CR40","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1016\/j.comnet.2015.03.012","volume":"83","author":"D Wang","year":"2015","unstructured":"Wang D (2015) Bandwidth-efficiency-oriented topology optimization for integrated switching systems based on circulant graphs. Comput Netw 83:199\u2013216","journal-title":"Comput Netw"},{"key":"6633_CR41","doi-asserted-by":"publisher","first-page":"84","DOI":"10.1016\/j.dam.2017.04.008","volume":"227","author":"T Jiang","year":"2017","unstructured":"Jiang T, Miller Z, Yager D (2017) On the bandwidth of the Kneser graph. Discret Appl Math 227:84\u201394","journal-title":"Discret Appl Math"},{"key":"6633_CR42","doi-asserted-by":"crossref","unstructured":"Staden K, Treglown A (2020) The bandwidth theorem for locally dense graphs, In Forum of Mathematics Sigma 8(42) Cambridge University Press","DOI":"10.1017\/fms.2020.39"},{"issue":"5","key":"6633_CR43","doi-asserted-by":"publisher","first-page":"1217","DOI":"10.1016\/j.ejc.2009.10.010","volume":"31","author":"J B\u00f6ttcher","year":"2010","unstructured":"B\u00f6ttcher J, Pruessmann KP, Taraz A, W\u00fcrfl A (2010) Bandwidth, expansion, treewidth, separators and universality for bounded-degree graphs. Eur J Comb 31(5):1217\u20131227","journal-title":"Eur J Comb"},{"issue":"7","key":"6633_CR44","doi-asserted-by":"publisher","first-page":"7988","DOI":"10.1007\/s10489-022-03802-y","volume":"53","author":"B Koohestani","year":"2023","unstructured":"Koohestani B (2023) On the solution of the graph bandwidth problem by means of search methods. Appl Intell 53(7):7988\u20138004","journal-title":"Appl Intell"},{"issue":"7","key":"6633_CR45","doi-asserted-by":"publisher","first-page":"112","DOI":"10.1016\/j.disc.2021.112427","volume":"344","author":"C You","year":"2021","unstructured":"You C, Lin Q, Chen X (2021) Ramsey numbers of large books and bipartite graphs with small bandwidth. Discret Math 344(7):112\u2013427","journal-title":"Discret Math"},{"issue":"3","key":"6633_CR46","doi-asserted-by":"publisher","first-page":"191","DOI":"10.1016\/0012-365X(89)90319-1","volume":"76","author":"RD Dutton","year":"1989","unstructured":"Dutton RD, Brigham RC (1989) On the size of graphs of a given bandwidth. Discret Math 76(3):191\u2013195","journal-title":"Discret Math"},{"issue":"2","key":"6633_CR47","doi-asserted-by":"publisher","first-page":"178","DOI":"10.1016\/0890-5401(92)90012-5","volume":"100","author":"R Board","year":"1992","unstructured":"Board R (1992) The online graph bandwidth problem. Inf Comput 100(2):178\u2013201","journal-title":"Inf Comput"},{"issue":"3","key":"6633_CR48","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1002\/jgt.3190060302","volume":"6","author":"PZ Chinn","year":"1982","unstructured":"Chinn PZ, Chv\u00e1talov\u00e1 J, Dewdney AK, Gibbs NE (1982) The bandwidth problem for graphs and matrices: A survey. J Graph Theory 6(3):223\u2013254","journal-title":"J Graph Theory"},{"issue":"1\u20133","key":"6633_CR49","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1016\/0012-365X(96)00205-1","volume":"150","author":"K Ando","year":"1996","unstructured":"Ando K, Kaneko A, Gervacio S (1996) The bandwidth of a tree with $$k$$ leaves is at most $$\\lceil \\frac{k}{2}\\rceil $$. Discret Math 150(1\u20133):403\u2013406","journal-title":"Discret Math"},{"issue":"3","key":"6633_CR50","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1016\/0167-739X(88)90007-6","volume":"4","author":"G Della Vecchia","year":"1988","unstructured":"Della Vecchia G, Sanges C (1988) A recursively scalable network VLSI implementation. Futur Gener Comput Syst 4(3):235\u2013243","journal-title":"Futur Gener Comput Syst"},{"issue":"4","key":"6633_CR51","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1016\/0898-1221(88)90213-1","volume":"15","author":"F Harary","year":"1988","unstructured":"Harary F, Hayes JP, Wu HJ (1988) A survey of the theory of hypercube graphs. Comp Math Appl 15(4):277\u2013289","journal-title":"Comp Math Appl"},{"issue":"1\u20132","key":"6633_CR52","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1016\/S0020-0255(98)10045-2","volume":"113","author":"CP Chang","year":"1999","unstructured":"Chang CP, Wang JN, Hsu LH (1999) Topological properties of twisted cube. Inf Sci 113(1\u20132):147\u2013167","journal-title":"Inf Sci"},{"issue":"2","key":"6633_CR53","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1002\/net.10033","volume":"40","author":"SA Choudum","year":"2002","unstructured":"Choudum SA, Sunitha V (2002) Augmented cubes. Networks 40(2):71\u201384","journal-title":"Networks"},{"issue":"3","key":"6633_CR54","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1016\/S0020-0190(97)00013-6","volume":"61","author":"I Fri\u0161","year":"1997","unstructured":"Fri\u0161 I, Havel I, Liebl P (1997) The diameter of the cube-connected cycles. Inf Process Lett 61(3):157\u2013160","journal-title":"Inf Process Lett"},{"key":"6633_CR55","first-page":"187","volume":"79","author":"M Livingston","year":"1990","unstructured":"Livingston M, Stout QF (1990) Perfect dominating sets, University of Michigan, In. Congr Numer 79:187\u2013203","journal-title":"Congr Numer"},{"issue":"1","key":"6633_CR56","doi-asserted-by":"publisher","first-page":"632","DOI":"10.1016\/j.akcej.2018.06.005","volume":"17","author":"P Ragukumar","year":"2020","unstructured":"Ragukumar P, Sethuraman G (2020) Binomial trees are graceful. AKCE Int J Graphs Comb 17(1):632\u2013636","journal-title":"AKCE Int J Graphs Comb"},{"key":"6633_CR57","doi-asserted-by":"publisher","first-page":"15783","DOI":"10.1007\/s11227-024-06053-5","volume":"80","author":"S Prabhu","year":"2024","unstructured":"Prabhu S, Manimozhi V, Davoodi A, Guirao JLG (2024) Fault-tolerant basis of generalized fat trees and perfect binary tree derived architectures. J Supercomput 80:15783\u201315798","journal-title":"J Supercomput"},{"key":"6633_CR58","first-page":"128037","volume":"452","author":"M Arulperumjothi","year":"2023","unstructured":"Arulperumjothi M, Klav\u017ear S, Prabhu S (2023) Redefining fractal cubic networks and determining their metric dimension and fault-tolerant metric dimension. Appl Math Comput 452:128037","journal-title":"Appl Math Comput"},{"key":"6633_CR59","first-page":"126897","volume":"420","author":"S Prabhu","year":"2022","unstructured":"Prabhu S, Manimozhi V, Arulperumjothi M, Klav\u017ear S (2022) Twin vertices in fault-tolerant metric sets and fault-tolerant metric dimension of multistage interconnection networks. Appl Math Comput 420:126897","journal-title":"Appl Math Comput"},{"issue":"3","key":"6633_CR60","doi-asserted-by":"publisher","first-page":"468","DOI":"10.1006\/jpdc.1994.1158","volume":"23","author":"DR Duh","year":"1994","unstructured":"Duh DR, Chen GH (1994) Topological properties of WK-recursive networks. J Parallel Distrib Comput 23(3):468\u2013474","journal-title":"J Parallel Distrib Comput"},{"key":"6633_CR61","unstructured":"INMOS Limited (1988) C. O. R. P. O. R. A. T. E. Transputer reference manual, Prentice Hall International (UK) Ltd."},{"issue":"9","key":"6633_CR62","doi-asserted-by":"publisher","first-page":"853","DOI":"10.1109\/TPDS.2005.109","volume":"16","author":"JS Fu","year":"2005","unstructured":"Fu JS (2005) Hamiltonicity of the WK-recursive network with and without faulty nodes. IEEE Trans Parallel Distrib Syst 16(9):853\u2013865","journal-title":"IEEE Trans Parallel Distrib Syst"},{"issue":"24","key":"6633_CR63","doi-asserted-by":"publisher","first-page":"5611","DOI":"10.1016\/j.ins.2007.04.013","volume":"177","author":"JF Fang","year":"2007","unstructured":"Fang JF, Wang YR, Huang HL (2007) The $$m$$-pancycle-connectivity of a WK-recursive network. Inf Sci 177(24):5611\u20135619","journal-title":"Inf Sci"},{"issue":"4","key":"6633_CR64","doi-asserted-by":"publisher","first-page":"367","DOI":"10.1109\/71.588608","volume":"8","author":"MY Su","year":"1997","unstructured":"Su MY, Chen GH, Duh DR (1997) A shortest path routing algorithm for incomplete WK-recursive networks. IEEE Trans Parallel Distrib Syst 8(4):367\u2013379","journal-title":"IEEE Trans Parallel Distrib Syst"},{"issue":"2","key":"6633_CR65","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1007\/s006070050057","volume":"63","author":"L Verdoscia","year":"1999","unstructured":"Verdoscia L, Vaccaro R (1999) An adaptive routing algorithm for WK-recursive topologies. Computing 63(2):171\u2013184","journal-title":"Computing"},{"key":"6633_CR66","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1016\/j.tcs.2022.01.039","volume":"910","author":"L You","year":"2022","unstructured":"You L, Han Y, Fan J, Ye D (2022) Structure fault tolerance of WK-recursive networks. Theoret Comput Sci 910:113\u2013120","journal-title":"Theoret Comput Sci"},{"key":"6633_CR67","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1016\/j.tcs.2017.12.023","volume":"713","author":"L You","year":"2018","unstructured":"You L, Fan J, Han Y (2018) Super spanning connectivity on WK-recursive networks. Theoret Comput Sci 713:42\u201355","journal-title":"Theoret Comput Sci"},{"key":"6633_CR68","doi-asserted-by":"publisher","first-page":"236","DOI":"10.1016\/j.ins.2014.02.087","volume":"271","author":"TY Ho","year":"2014","unstructured":"Ho TY, Lin CK, Tan JJ, Hsu LH (2014) Fault-tolerant Hamiltonian connectivity of the WK-recursive networks. Inf Sci 271:236\u2013245","journal-title":"Inf Sci"},{"issue":"7","key":"6633_CR69","doi-asserted-by":"publisher","first-page":"9879","DOI":"10.1007\/s11227-021-04296-0","volume":"78","author":"FH Wang","year":"2022","unstructured":"Wang FH, Wang SI (2022) Independent spanning trees on WK-recursive networks and WK-recursive pyramids. J Supercomput 78(7):9879\u20139906","journal-title":"J Supercomput"},{"issue":"7","key":"6633_CR70","doi-asserted-by":"publisher","first-page":"944","DOI":"10.3390\/math12070944","volume":"12","author":"FH Wang","year":"2024","unstructured":"Wang FH, Hsu CJ (2024) Rainbow Connection Numbers of WK-Recursive Networks and WK-Recursive Pyramids. Mathematics 12(7):944","journal-title":"Mathematics"},{"issue":"3","key":"6633_CR71","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1016\/0012-365X(75)90039-4","volume":"11","author":"J Chv\u00e1talov\u00e1","year":"1975","unstructured":"Chv\u00e1talov\u00e1 J (1975) Optimal labelling of a product of two paths. Discret Math 11(3):249\u2013253","journal-title":"Discret Math"}],"container-title":["The Journal of Supercomputing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11227-024-06633-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11227-024-06633-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11227-024-06633-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,14]],"date-time":"2024-11-14T00:07:45Z","timestamp":1731542865000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11227-024-06633-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11,13]]},"references-count":71,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2025,1]]}},"alternative-id":["6633"],"URL":"https:\/\/doi.org\/10.1007\/s11227-024-06633-5","relation":{},"ISSN":["0920-8542","1573-0484"],"issn-type":[{"type":"print","value":"0920-8542"},{"type":"electronic","value":"1573-0484"}],"subject":[],"published":{"date-parts":[[2024,11,13]]},"assertion":[{"value":"19 October 2024","order":1,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 November 2024","order":2,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that there is no conflict of interest regarding the publication of this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"175"}}