{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T06:25:46Z","timestamp":1777703146726,"version":"3.51.4"},"reference-count":10,"publisher":"SAGE Publications","issue":"6","license":[{"start":{"date-parts":[[2019,5,9]],"date-time":"2019-05-09T00:00:00Z","timestamp":1557360000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Journal of Intelligent &amp; Fuzzy Systems"],"published-print":{"date-parts":[[2019,6,11]]},"abstract":"<jats:p>\n                    Let\n                    <jats:italic>\u0394<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>\u0393<\/jats:italic>\n                    <\/jats:sub>\n                    \u00a0(\n                    <jats:italic>G<\/jats:italic>\n                    ) be Gallai simplicial complex defined as a generalization of Gallai graph\n                    <jats:italic>\u0393<\/jats:italic>\n                    \u00a0(\n                    <jats:italic>G<\/jats:italic>\n                    ) such that\n                    <jats:italic>G<\/jats:italic>\n                    is a finite simple graph. The Betti numbers are topological objects which were proved to be invariants by Poincar\u00e9. In this note, we give formula for Betti numbers of Gallai simplicial complex associated to prism graph by coding in MATLAB. We prove that a simple graph\n                    <jats:italic>G<\/jats:italic>\n                    having\n                    <jats:italic>n<\/jats:italic>\n                    vertices of the form\n                    <jats:italic>n<\/jats:italic>\n                    \u00a0=\u00a03\n                    <jats:italic>l<\/jats:italic>\n                    \u00a0+\u00a02 or 3\n                    <jats:italic>l<\/jats:italic>\n                    \u00a0+\u00a03 is\n                    <jats:italic>f<\/jats:italic>\n                    -Gallai graph for\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi>G<\/mml:mi>\n                        <mml:mo>=<\/mml:mo>\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mstyle mathvariant=\"double-struck\">\n                              <mml:mi>\ud835\udd4a<\/mml:mi>\n                            <\/mml:mstyle>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mn>4<\/mml:mn>\n                            <mml:mi>l<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    (respectively\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi>G<\/mml:mi>\n                        <mml:mo>=<\/mml:mo>\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mstyle mathvariant=\"double-struck\">\n                              <mml:mi>\ud835\udd4a<\/mml:mi>\n                            <\/mml:mstyle>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mn>4<\/mml:mn>\n                            <mml:mi>l<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    ) a graph consisting of star graphs\n                    <jats:italic>S<\/jats:italic>\n                    <jats:sub>\n                      2\n                      <jats:italic>l<\/jats:italic>\n                    <\/jats:sub>\n                    and\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:msubsup>\n                          <mml:mi>S<\/mml:mi>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>l<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mo>\u2032<\/mml:mo>\n                        <\/mml:msubsup>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    (respectively\n                    <jats:italic>S<\/jats:italic>\n                    <jats:sub>\n                      2\n                      <jats:italic>l<\/jats:italic>\n                    <\/jats:sub>\n                    and\n                    <jats:italic>S<\/jats:italic>\n                    <jats:sub>\n                      2\n                      <jats:italic>l<\/jats:italic>\n                      +1\n                    <\/jats:sub>\n                    ) with\n                    <jats:italic>l<\/jats:italic>\n                    common vertices such that\n                    <jats:italic>l<\/jats:italic>\n                    \u00a0\u2265\u00a02.\n                  <\/jats:p>","DOI":"10.3233\/jifs-181478","type":"journal-article","created":{"date-parts":[[2019,5,10]],"date-time":"2019-05-10T10:43:01Z","timestamp":1557484981000},"page":"5645-5651","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":1,"title":["Gallai simplicial complexes"],"prefix":"10.1177","volume":"36","author":[{"given":"Shahid","family":"Muhmood","sequence":"first","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road, Lahore, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Imran","family":"Ahmed","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road, Lahore, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Adnan","family":"Liaquat","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Defence Road, Off Raiwind Road, Lahore, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2019,5,9]]},"reference":[{"key":"e_1_3_2_2_2","doi-asserted-by":"publisher","DOI":"10.1142\/S1005386712000788"},{"key":"e_1_3_2_3_2","unstructured":"I. 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