{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T14:53:14Z","timestamp":1753887194693,"version":"3.41.2"},"reference-count":11,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2023,1,1]],"date-time":"2023-01-01T00:00:00Z","timestamp":1672531200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,5,16]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The Collatz-Sinogowitz irregularity index is the oldest known numerical measure of graph irregularity. For a simple and connected graph <jats:inline-formula>\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_math-2022-0572_eq_001.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mi>G<\/m:mi>\n                        <\/m:math>\n                        <jats:tex-math>G<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> of order <jats:inline-formula>\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_math-2022-0572_eq_002.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mi>n<\/m:mi>\n                        <\/m:math>\n                        <jats:tex-math>n<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> and size <jats:inline-formula>\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_math-2022-0572_eq_003.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mi>m<\/m:mi>\n                        <\/m:math>\n                        <jats:tex-math>m<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, it is defined as <jats:inline-formula>\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_math-2022-0572_eq_004.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mspace width=\"0.1em\"\/>\n                           <m:mtext>CS<\/m:mtext>\n                           <m:mspace width=\"0.1em\"\/>\n                           <m:mrow>\n                              <m:mo>(<\/m:mo>\n                              <m:mrow>\n                                 <m:mi>G<\/m:mi>\n                              <\/m:mrow>\n                              <m:mo>)<\/m:mo>\n                           <\/m:mrow>\n                           <m:mo>=<\/m:mo>\n                           <m:msub>\n                              <m:mrow>\n                                 <m:mi>\u03bb<\/m:mi>\n                              <\/m:mrow>\n                              <m:mrow>\n                                 <m:mn>1<\/m:mn>\n                              <\/m:mrow>\n                           <\/m:msub>\n                           <m:mo>\u2212<\/m:mo>\n                           <m:mn>2<\/m:mn>\n                           <m:mi>m<\/m:mi>\n                           <m:mspace width=\"0.1em\"\/>\n                           <m:mtext>\/<\/m:mtext>\n                           <m:mspace width=\"0.1em\"\/>\n                           <m:mi>n<\/m:mi>\n                           <m:mo>,<\/m:mo>\n                        <\/m:math>\n                        <jats:tex-math>\\hspace{0.1em}\\text{CS}\\hspace{0.1em}\\left(G)={\\lambda }_{1}-2m\\hspace{0.1em}\\text{\/}\\hspace{0.1em}n,<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> where <jats:inline-formula>\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_math-2022-0572_eq_005.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mrow>\n                                 <m:mi>\u03bb<\/m:mi>\n                              <\/m:mrow>\n                              <m:mrow>\n                                 <m:mn>1<\/m:mn>\n                              <\/m:mrow>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:tex-math>{\\lambda }_{1}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> is the largest eigenvalue of the adjacency matrix of <jats:inline-formula>\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_math-2022-0572_eq_006.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mi>G<\/m:mi>\n                        <\/m:math>\n                        <jats:tex-math>G<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, and <jats:inline-formula>\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_math-2022-0572_eq_007.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mn>2<\/m:mn>\n                           <m:mi>m<\/m:mi>\n                           <m:mspace width=\"0.1em\"\/>\n                           <m:mtext>\/<\/m:mtext>\n                           <m:mspace width=\"0.1em\"\/>\n                           <m:mi>n<\/m:mi>\n                        <\/m:math>\n                        <jats:tex-math>2m\\hspace{0.1em}\\text{\/}\\hspace{0.1em}n<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> is the average vertex degree of <jats:inline-formula>\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_math-2022-0572_eq_008.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mi>G<\/m:mi>\n                        <\/m:math>\n                        <jats:tex-math>G<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>. Here, the Collatz-Sinogowitz inverse irregularity problem is studied. For every integer <jats:inline-formula>\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_math-2022-0572_eq_009.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mi>i<\/m:mi>\n                           <m:mo>\u2265<\/m:mo>\n                           <m:mn>0<\/m:mn>\n                        <\/m:math>\n                        <jats:tex-math>i\\ge 0<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, it is shown that there exists a graph <jats:inline-formula>\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_math-2022-0572_eq_010.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mi>G<\/m:mi>\n                        <\/m:math>\n                        <jats:tex-math>G<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> such that <jats:inline-formula>\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_math-2022-0572_eq_011.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mspace width=\"0.1em\"\/>\n                           <m:mtext>CS<\/m:mtext>\n                           <m:mspace width=\"0.1em\"\/>\n                           <m:mrow>\n                              <m:mo>(<\/m:mo>\n                              <m:mrow>\n                                 <m:mi>G<\/m:mi>\n                              <\/m:mrow>\n                              <m:mo>)<\/m:mo>\n                           <\/m:mrow>\n                           <m:mo>=<\/m:mo>\n                           <m:mi>i<\/m:mi>\n                        <\/m:math>\n                        <jats:tex-math>\\hspace{0.1em}\\text{CS}\\hspace{0.1em}\\left(G)=i<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>. Also, for every interval <jats:inline-formula>\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_math-2022-0572_eq_012.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mrow>\n                                 <m:mi>I<\/m:mi>\n                              <\/m:mrow>\n                              <m:mrow>\n                                 <m:mi>i<\/m:mi>\n                              <\/m:mrow>\n                           <\/m:msub>\n                           <m:mo>=<\/m:mo>\n                           <m:mrow>\n                              <m:mo>(<\/m:mo>\n                              <m:mrow>\n                                 <m:mi>i<\/m:mi>\n                                 <m:mo>,<\/m:mo>\n                                 <m:mi>i<\/m:mi>\n                                 <m:mo>+<\/m:mo>\n                                 <m:mn>1<\/m:mn>\n                              <\/m:mrow>\n                              <m:mo>)<\/m:mo>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>{I}_{i}=\\left(i,i+1)<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, it is shown that there are infinitely many graphs whose Collatz-Sinogowitz irregularity lies in <jats:inline-formula>\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_math-2022-0572_eq_013.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msub>\n                              <m:mrow>\n                                 <m:mi>I<\/m:mi>\n                              <\/m:mrow>\n                              <m:mrow>\n                                 <m:mi>i<\/m:mi>\n                              <\/m:mrow>\n                           <\/m:msub>\n                        <\/m:math>\n                        <jats:tex-math>{I}_{i}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1515\/math-2022-0572","type":"journal-article","created":{"date-parts":[[2023,5,16]],"date-time":"2023-05-16T10:37:54Z","timestamp":1684233474000},"source":"Crossref","is-referenced-by-count":0,"title":["On the inverse Collatz-Sinogowitz irregularity problem"],"prefix":"10.1515","volume":"21","author":[{"given":"Abdullah","family":"Alazemi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Kuwait University , Safat 13060 , Kuwait"}]},{"given":"Milica","family":"An\u0111eli\u0107","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Kuwait University , Safat 13060 , Kuwait"}]},{"given":"Darko","family":"Dimitrov","sequence":"additional","affiliation":[{"name":"Faculty of Information Studies , 8000 Novo Mesto , Slovenia"}]}],"member":"374","published-online":{"date-parts":[[2023,5,16]]},"reference":[{"key":"2023090109330527195_j_math-2022-0572_ref_001","doi-asserted-by":"crossref","unstructured":"L. Collatz and U. Sinogowitz, Spektren endlicher Graphen, Abh. Math. Semin. Univ. Hambg. 21 (1957), 63\u201377.","DOI":"10.1007\/BF02941924"},{"key":"2023090109330527195_j_math-2022-0572_ref_002","doi-asserted-by":"crossref","unstructured":"F. K. Bell, A note on the irregularity of graphs, Linear Algebra Appl. 161 (1992), 45\u201354.","DOI":"10.1016\/0024-3795(92)90004-T"},{"key":"2023090109330527195_j_math-2022-0572_ref_003","unstructured":"M. O. Albertson, The irregularity of a graph, Ars Combin. 46 (1997), 219\u2013225."},{"key":"2023090109330527195_j_math-2022-0572_ref_004","unstructured":"I. Gutman, M. Togan, A. Yurttas, A. S. Cevik, and I. N. Cangul, Inverse problem for sigma index, MATCH Commun. Math. Comput. Chem. 79 (2018), 491\u2013508."},{"key":"2023090109330527195_j_math-2022-0572_ref_005","doi-asserted-by":"crossref","unstructured":"D. Dimitrov and D. Stevanovi\u0107, On the \u03c3t-irregularity and the inverse irregularity problem, Appl. Math. 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Guo, Some upper bounds on the spectral radius of a graph, Linear Algebra Appl. 601 (2020), 101\u2013112.","DOI":"10.1016\/j.laa.2020.04.024"},{"key":"2023090109330527195_j_math-2022-0572_ref_009","doi-asserted-by":"crossref","unstructured":"A. E. Brouwer and W. H. Haemers, Spectra of Graphs, Springer, New York, 2012.","DOI":"10.1007\/978-1-4614-1939-6"},{"key":"2023090109330527195_j_math-2022-0572_ref_010","doi-asserted-by":"crossref","unstructured":"F. Belardo, E. M. Li Marzi, and S. K. Simi\u0107, Some results on the index of unicyclic graphs, Linear Algebra Appl. 416 (2006), 1048\u20131059.","DOI":"10.1016\/j.laa.2006.01.008"},{"key":"2023090109330527195_j_math-2022-0572_ref_011","unstructured":"M. Lepovi\u0107 and I. Gutman, Some spectral properties of starlike trees, Bull. Cl. Sci. Math. Nat. Sci. Math. 26 (2001), 107\u2013113."}],"container-title":["Open Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/math-2022-0572\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/math-2022-0572\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,1]],"date-time":"2023-09-01T09:38:17Z","timestamp":1693561097000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/math-2022-0572\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,1]]},"references-count":11,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2023,7,10]]},"published-print":{"date-parts":[[2023,7,10]]}},"alternative-id":["10.1515\/math-2022-0572"],"URL":"https:\/\/doi.org\/10.1515\/math-2022-0572","relation":{},"ISSN":["2391-5455"],"issn-type":[{"type":"electronic","value":"2391-5455"}],"subject":[],"published":{"date-parts":[[2023,1,1]]},"article-number":"20220572"}}