{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T14:14:18Z","timestamp":1777644858932,"version":"3.51.4"},"reference-count":0,"publisher":"SAGE Publications","issue":"4","license":[{"start":{"date-parts":[[2017,8,3]],"date-time":"2017-08-03T00:00:00Z","timestamp":1501718400000},"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":["Fundamenta Informaticae"],"published-print":{"date-parts":[[2017,8,3]]},"abstract":"<jats:p>\n                    We continue the study of finite connected edge-bipartite graphs \u0394, with m \u2265 2 vertices (a class of signed graphs), started in [SIAM J. Discrete Math. 27(2013), 827-854] and developed in [Fund. Inform. 139(2015), 249-275, 145(2016), 19-48] by means of the non-symmetric Gram matrix\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\" altimg=\"eq-00001.gif\">\n                      <mml:mrow>\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mover>\n                              <mml:mi>G<\/mml:mi>\n                              <mml:mo>\u2228<\/mml:mo>\n                            <\/mml:mover>\n                          <\/mml:mrow>\n                          <mml:mi>\u0394<\/mml:mi>\n                        <\/mml:msub>\n                        <mml:mo>\u2208<\/mml:mo>\n                        <mml:msub>\n                          <mml:mi>\ud835\udd44<\/mml:mi>\n                          <mml:mi>n<\/mml:mi>\n                        <\/mml:msub>\n                        <mml:mo stretchy=\"false\">(<\/mml:mo>\n                        <mml:mi>\u2124<\/mml:mi>\n                        <mml:mo stretchy=\"false\">)<\/mml:mo>\n                      <\/mml:mrow>\n                    <\/mml:math>\n                    defining \u0394, its symmetric Gram matrix\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\" altimg=\"eq-00002.gif\">\n                      <mml:mrow>\n                        <mml:msub>\n                          <mml:mi>G<\/mml:mi>\n                          <mml:mi>\u0394<\/mml:mi>\n                        <\/mml:msub>\n                        <mml:mo>:<\/mml:mo>\n                        <mml:mo>=<\/mml:mo>\n                        <mml:mfrac>\n                          <mml:mn>1<\/mml:mn>\n                          <mml:mn>2<\/mml:mn>\n                        <\/mml:mfrac>\n                        <mml:mrow>\n                          <mml:mo>[<\/mml:mo>\n                          <mml:mrow>\n                            <mml:mover>\n                              <mml:mrow>\n                                <mml:msub>\n                                  <mml:mi>G<\/mml:mi>\n                                  <mml:mi>\u0394<\/mml:mi>\n                                <\/mml:msub>\n                              <\/mml:mrow>\n                              <mml:mo>\u2228<\/mml:mo>\n                            <\/mml:mover>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mover>\n                              <mml:mrow>\n                                <mml:msubsup>\n                                  <mml:mi>G<\/mml:mi>\n                                  <mml:mi>\u0394<\/mml:mi>\n                                  <mml:mrow>\n                                    <mml:mi>t<\/mml:mi>\n                                    <mml:mi>r<\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:msubsup>\n                              <\/mml:mrow>\n                              <mml:mo>\u2228<\/mml:mo>\n                            <\/mml:mover>\n                          <\/mml:mrow>\n                          <mml:mo>]<\/mml:mo>\n                        <\/mml:mrow>\n                        <mml:mo>\u2208<\/mml:mo>\n                        <mml:msub>\n                          <mml:mi>\ud835\udd44<\/mml:mi>\n                          <mml:mi>n<\/mml:mi>\n                        <\/mml:msub>\n                        <mml:mo stretchy=\"false\">(<\/mml:mo>\n                        <mml:mfrac>\n                          <mml:mn>1<\/mml:mn>\n                          <mml:mn>2<\/mml:mn>\n                        <\/mml:mfrac>\n                        <mml:mi>\u2124<\/mml:mi>\n                        <mml:mo stretchy=\"false\">)<\/mml:mo>\n                      <\/mml:mrow>\n                    <\/mml:math>\n                    , and the Gram quadratic form q\n                    <jats:sub>\u0394<\/jats:sub>\n                    : \u2124\n                    <jats:sup>n<\/jats:sup>\n                    \u2192 \u2124. In the present paper we study connected positive Cox-regular edge-bipartite graphs \u0394, with n \u2265 2 vertices, in the sense that the symmetric Gram matrix G\n                    <jats:sub>\u0394<\/jats:sub>\n                    \u2208 \ud835\udd44\n                    <jats:sub>n<\/jats:sub>\n                    (\u2124) of \u0394 is positive definite. Our aim is to classify such Cox-regular edge-bipartite graphs with at least one loop by means of an inflation algorithm, up to the weak Gram \u2124-congruence \u0394 ~\n                    <jats:sub>\u2124<\/jats:sub>\n                    \u0394\u2032, where \u0394 ~\n                    <jats:sub>\u2124<\/jats:sub>\n                    \u0394\u2032 means that G\n                    <jats:sub>\u0394<\/jats:sub>\n                    \u2032 = B\n                    <jats:sup>tr<\/jats:sup>\n                    \u00b7 G\n                    <jats:sub>\u0394<\/jats:sub>\n                    \u00b7 B, for some B \u2208 \ud835\udd44\n                    <jats:sub>n<\/jats:sub>\n                    (\u2124) such that det B = \u00b11. Our main result of the paper asserts that, given a positive connected Cox-regular edge-bipartite graph \u0394 with n \u2265 2 vertices and with at least one loop there exists a Cox-regular edge-bipartite Dynkin graph \ud835\udc9f\n                    <jats:sub>n<\/jats:sub>\n                    \u2228 {\u212c\n                    <jats:sub>n<\/jats:sub>\n                    , \ud835\udc9e\n                    <jats:sub>n<\/jats:sub>\n                    , \u2131\n                    <jats:sub>4<\/jats:sub>\n                    , \ud835\udca2\n                    <jats:sub>2<\/jats:sub>\n                    } with loops and a suitably chosen sequence\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\" altimg=\"eq-00003.gif\">\n                      <mml:mrow>\n                        <mml:msubsup>\n                          <mml:mrow>\n                            <mml:mstyle mathvariant=\"bold\">\n                              <mml:mtext>t<\/mml:mtext>\n                            <\/mml:mstyle>\n                          <\/mml:mrow>\n                          <mml:mo>\u2022<\/mml:mo>\n                          <mml:mo>\u2212<\/mml:mo>\n                        <\/mml:msubsup>\n                      <\/mml:mrow>\n                    <\/mml:math>\n                    of the inflation operators of one of the types\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\" altimg=\"eq-00004.gif\">\n                      <mml:mrow>\n                        <mml:msup>\n                          <mml:mi>\u0394<\/mml:mi>\n                          <mml:mo>\u2032<\/mml:mo>\n                        <\/mml:msup>\n                        <mml:mo>\u21a6<\/mml:mo>\n                        <mml:msubsup>\n                          <mml:mrow>\n                            <mml:mstyle mathvariant=\"bold\">\n                              <mml:mtext>t<\/mml:mtext>\n                            <\/mml:mstyle>\n                          <\/mml:mrow>\n                          <mml:mi>a<\/mml:mi>\n                          <mml:mo>\u2212<\/mml:mo>\n                        <\/mml:msubsup>\n                        <mml:msup>\n                          <mml:mi>\u0394<\/mml:mi>\n                          <mml:mo>\u2032<\/mml:mo>\n                        <\/mml:msup>\n                      <\/mml:mrow>\n                    <\/mml:math>\n                    and\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\" altimg=\"eq-00005.gif\">\n                      <mml:mrow>\n                        <mml:msup>\n                          <mml:mi>\u0394<\/mml:mi>\n                          <mml:mo>\u2032<\/mml:mo>\n                        <\/mml:msup>\n                        <mml:mo>\u21a6<\/mml:mo>\n                        <mml:msubsup>\n                          <mml:mrow>\n                            <mml:mstyle mathvariant=\"bold\">\n                              <mml:mtext>t<\/mml:mtext>\n                            <\/mml:mstyle>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mi>b<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mo>\u2212<\/mml:mo>\n                        <\/mml:msubsup>\n                        <mml:msup>\n                          <mml:mi>\u0394<\/mml:mi>\n                          <mml:mo>\u2032<\/mml:mo>\n                        <\/mml:msup>\n                      <\/mml:mrow>\n                    <\/mml:math>\n                    such that the composite operator\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\" altimg=\"eq-00006.gif\">\n                      <mml:mrow>\n                        <mml:mi>\u0394<\/mml:mi>\n                        <mml:mo>\u21a6<\/mml:mo>\n                        <mml:msubsup>\n                          <mml:mrow>\n                            <mml:mstyle mathvariant=\"bold\">\n                              <mml:mtext>t<\/mml:mtext>\n                            <\/mml:mstyle>\n                          <\/mml:mrow>\n                          <mml:mi>\u2022<\/mml:mi>\n                          <mml:mo>\u2212<\/mml:mo>\n                        <\/mml:msubsup>\n                        <mml:mi>\u0394<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:math>\n                    reduces \u0394 to the bigraph \ud835\udc9f\n                    <jats:sub>n<\/jats:sub>\n                    such that \u0394 ~\n                    <jats:sub>\u2124<\/jats:sub>\n                    \ud835\udc9f\n                    <jats:sub>n<\/jats:sub>\n                    and the bigraphs \u0394, \ud835\udc9f\n                    <jats:sub>n<\/jats:sub>\n                    have the same number of loops. The algorithm does not change loops and the number of vertices, and computes a matrix B \u2208 \ud835\udd44\n                    <jats:sub>n<\/jats:sub>\n                    (\u2124), with det B = \u00b11, defining the weak Gram \u2124-congruence \u0394 ~\n                    <jats:sub>\u2124<\/jats:sub>\n                    \ud835\udc9f\n                    <jats:sub>n<\/jats:sub>\n                    , that is, satisfying the equation G\n                    <jats:sub>\n                      \ud835\udc9f\n                      <jats:sub>n<\/jats:sub>\n                    <\/jats:sub>\n                    = B\n                    <jats:sup>tr<\/jats:sup>\n                    \u00b7 G\n                    <jats:sub>\u0394<\/jats:sub>\n                    \u00b7 B.\n                  <\/jats:p>","DOI":"10.3233\/fi-2017-1545","type":"journal-article","created":{"date-parts":[[2017,8,4]],"date-time":"2017-08-04T11:35:21Z","timestamp":1501846521000},"page":"367-398","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":6,"title":["Inflation Agorithm for Cox-regular Postive Edge-bipartite Graphs with Loops"],"prefix":"10.1177","volume":"153","author":[{"given":"Bartosz","family":"Makuracki","sequence":"first","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12\/18, 87-100 Toru\u0144, Poland. {bartmak,simson,bzstyler}@mat.umk.pl"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daniel","family":"Simson","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12\/18, 87-100 Toru\u0144, Poland. {bartmak,simson,bzstyler}@mat.umk.pl"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"B\u0142a\u017cej","family":"Zyglarski","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12\/18, 87-100 Toru\u0144, Poland. {bartmak,simson,bzstyler}@mat.umk.pl"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2017,8,3]]},"container-title":["Fundamenta Informaticae"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/FI-2017-1545","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/FI-2017-1545","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T06:30:24Z","timestamp":1777444224000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/FI-2017-1545"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,8,3]]},"references-count":0,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2017,8,3]]}},"alternative-id":["10.3233\/FI-2017-1545"],"URL":"https:\/\/doi.org\/10.3233\/fi-2017-1545","relation":{},"ISSN":["0169-2968","1875-8681"],"issn-type":[{"value":"0169-2968","type":"print"},{"value":"1875-8681","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,8,3]]}}}