{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T02:50:53Z","timestamp":1747191053448,"version":"3.40.5"},"reference-count":9,"publisher":"Wiley","license":[{"start":{"date-parts":[[2022,11,25]],"date-time":"2022-11-25T00:00:00Z","timestamp":1669334400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2022,11,25]]},"abstract":"<jats:p>The permanent is important invariants of a graph with some applications in physics. If <jats:inline-formula>\n                     <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\">\n                        <a:mi>G<\/a:mi>\n                     <\/a:math>\n                  <\/jats:inline-formula> is a graph with adjacency matrix <jats:inline-formula>\n                     <c:math xmlns:c=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\">\n                        <c:mi>A<\/c:mi>\n                        <c:mo>=<\/c:mo>\n                        <c:mfenced open=\"[\" close=\"]\" separators=\"|\">\n                           <c:mrow>\n                              <c:msub>\n                                 <c:mrow>\n                                    <c:mi>a<\/c:mi>\n                                 <\/c:mrow>\n                                 <c:mrow>\n                                    <c:mi>i<\/c:mi>\n                                    <c:mi>j<\/c:mi>\n                                 <\/c:mrow>\n                              <\/c:msub>\n                           <\/c:mrow>\n                        <\/c:mfenced>\n                     <\/c:math>\n                  <\/jats:inline-formula>, then the permanent of <jats:inline-formula>\n                     <h:math xmlns:h=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\">\n                        <h:mi>A<\/h:mi>\n                     <\/h:math>\n                  <\/jats:inline-formula> is defined as <jats:inline-formula>\n                     <j:math xmlns:j=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\">\n                        <j:mtext>perm<\/j:mtext>\n                        <j:mfenced open=\"(\" close=\")\" separators=\"|\">\n                           <j:mrow>\n                              <j:mi>A<\/j:mi>\n                           <\/j:mrow>\n                        <\/j:mfenced>\n                        <j:mo>=<\/j:mo>\n                        <j:mstyle displaystyle=\"true\">\n                           <j:msub>\n                              <j:mrow>\n                                 <j:mo stretchy=\"false\">\u2211<\/j:mo>\n                              <\/j:mrow>\n                              <j:mrow>\n                                 <j:mi>\u03c3<\/j:mi>\n                                 <j:mo>\u2208<\/j:mo>\n                                 <j:msub>\n                                    <j:mrow>\n                                       <j:mi>S<\/j:mi>\n                                    <\/j:mrow>\n                                    <j:mrow>\n                                       <j:mi>n<\/j:mi>\n                                    <\/j:mrow>\n                                 <\/j:msub>\n                              <\/j:mrow>\n                           <\/j:msub>\n                           <j:mrow>\n                              <j:mstyle displaystyle=\"true\">\n                                 <j:msubsup>\n                                    <j:mo stretchy=\"false\">\u220f<\/j:mo>\n                                    <j:mrow>\n                                       <j:mi>i<\/j:mi>\n                                       <j:mo>=<\/j:mo>\n                                       <j:mn>1<\/j:mn>\n                                    <\/j:mrow>\n                                    <j:mi>n<\/j:mi>\n                                 <\/j:msubsup>\n                                 <j:mrow>\n                                    <j:msub>\n                                       <j:mrow>\n                                          <j:mi>a<\/j:mi>\n                                       <\/j:mrow>\n                                       <j:mrow>\n                                          <j:mi>i<\/j:mi>\n                                          <j:mi>\u03c3<\/j:mi>\n                                          <j:mfenced open=\"(\" close=\")\" separators=\"|\">\n                                             <j:mrow>\n                                                <j:mi>i<\/j:mi>\n                                             <\/j:mrow>\n                                          <\/j:mfenced>\n                                       <\/j:mrow>\n                                    <\/j:msub>\n                                 <\/j:mrow>\n                              <\/j:mstyle>\n                           <\/j:mrow>\n                        <\/j:mstyle>\n                     <\/j:math>\n                  <\/jats:inline-formula>, where <jats:inline-formula>\n                     <v:math xmlns:v=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\">\n                        <v:msub>\n                           <v:mrow>\n                              <v:mi>S<\/v:mi>\n                           <\/v:mrow>\n                           <v:mrow>\n                              <v:mi>n<\/v:mi>\n                           <\/v:mrow>\n                        <\/v:msub>\n                     <\/v:math>\n                  <\/jats:inline-formula> denotes the symmetric group on <jats:inline-formula>\n                     <x:math xmlns:x=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\">\n                        <x:mi>n<\/x:mi>\n                     <\/x:math>\n                  <\/jats:inline-formula> symbols. In this paper, the general form of the adjacency matrices of hexagonal and armchair chains will be computed. As a consequence of our work, it is proved that if <jats:inline-formula>\n                     <z:math xmlns:z=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\">\n                        <z:mi>G<\/z:mi>\n                        <z:mfenced open=\"[\" close=\"]\" separators=\"|\">\n                           <z:mrow>\n                              <z:mi>k<\/z:mi>\n                           <\/z:mrow>\n                        <\/z:mfenced>\n                     <\/z:math>\n                  <\/jats:inline-formula> and <jats:inline-formula>\n                     <eb:math xmlns:eb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\">\n                        <eb:mi>H<\/eb:mi>\n                        <eb:mfenced open=\"[\" close=\"]\" separators=\"|\">\n                           <eb:mrow>\n                              <eb:mi>k<\/eb:mi>\n                           <\/eb:mrow>\n                        <\/eb:mfenced>\n                     <\/eb:math>\n                  <\/jats:inline-formula> denote the hexagonal and armchair chains, respectively, then <jats:inline-formula>\n                     <jb:math xmlns:jb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\">\n                        <jb:mtext>perm<\/jb:mtext>\n                        <jb:mfenced open=\"(\" close=\")\" separators=\"|\">\n                           <jb:mrow>\n                              <jb:mi>A<\/jb:mi>\n                              <jb:mfenced open=\"(\" close=\")\" separators=\"|\">\n                                 <jb:mrow>\n                                    <jb:mi>G<\/jb:mi>\n                                    <jb:mfenced open=\"[\" close=\"]\" separators=\"|\">\n                                       <jb:mrow>\n                                          <jb:mn>1<\/jb:mn>\n                                       <\/jb:mrow>\n                                    <\/jb:mfenced>\n                                 <\/jb:mrow>\n                              <\/jb:mfenced>\n                           <\/jb:mrow>\n                        <\/jb:mfenced>\n                        <jb:mo>=<\/jb:mo>\n                        <jb:mn>4<\/jb:mn>\n                     <\/jb:math>\n                  <\/jats:inline-formula>, <jats:inline-formula>\n                     <ub:math xmlns:ub=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\">\n                        <ub:mtext>perm<\/ub:mtext>\n                        <ub:mfenced open=\"(\" close=\")\" separators=\"|\">\n                           <ub:mrow>\n                              <ub:mi>A<\/ub:mi>\n                              <ub:mfenced open=\"(\" close=\")\" separators=\"|\">\n                                 <ub:mrow>\n                                    <ub:mi>G<\/ub:mi>\n                                    <ub:mfenced open=\"[\" close=\"]\" separators=\"|\">\n                                       <ub:mrow>\n                                          <ub:mi>k<\/ub:mi>\n                                       <\/ub:mrow>\n                                    <\/ub:mfenced>\n                                 <\/ub:mrow>\n                              <\/ub:mfenced>\n                           <\/ub:mrow>\n                        <\/ub:mfenced>\n                        <ub:mo>=<\/ub:mo>\n                        <ub:msup>\n                           <ub:mrow>\n                              <ub:mfenced open=\"(\" close=\")\" separators=\"|\">\n                                 <ub:mrow>\n                                    <ub:mi>k<\/ub:mi>\n                                    <ub:mo>+<\/ub:mo>\n                                    <ub:mn>1<\/ub:mn>\n                                 <\/ub:mrow>\n                              <\/ub:mfenced>\n                           <\/ub:mrow>\n                           <ub:mrow>\n                              <ub:mn>2<\/ub:mn>\n                           <\/ub:mrow>\n                        <\/ub:msup>\n                     <\/ub:math>\n                  <\/jats:inline-formula>, <jats:inline-formula>\n                     <ic:math xmlns:ic=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\">\n                        <ic:mi>k<\/ic:mi>\n                        <ic:mo>\u2265<\/ic:mo>\n                        <ic:mn>2<\/ic:mn>\n                     <\/ic:math>\n                  <\/jats:inline-formula>, and <jats:inline-formula>\n                     <kc:math xmlns:kc=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\">\n                        <kc:mtext>perm<\/kc:mtext>\n                        <kc:mfenced open=\"(\" close=\")\" separators=\"|\">\n                           <kc:mrow>\n                              <kc:mi>A<\/kc:mi>\n                              <kc:mfenced open=\"(\" close=\")\" separators=\"|\">\n                                 <kc:mrow>\n                                    <kc:mi>H<\/kc:mi>\n                                    <kc:mfenced open=\"[\" close=\"]\" separators=\"|\">\n                                       <kc:mrow>\n                                          <kc:mi>k<\/kc:mi>\n                                       <\/kc:mrow>\n                                    <\/kc:mfenced>\n                                 <\/kc:mrow>\n                              <\/kc:mfenced>\n                           <\/kc:mrow>\n                        <\/kc:mfenced>\n                        <kc:mo>=<\/kc:mo>\n                        <kc:msup>\n                           <kc:mrow>\n                              <kc:mn>4<\/kc:mn>\n                           <\/kc:mrow>\n                           <kc:mrow>\n                              <kc:mi>k<\/kc:mi>\n                           <\/kc:mrow>\n                        <\/kc:msup>\n                     <\/kc:math>\n                  <\/jats:inline-formula> with <jats:inline-formula>\n                     <vc:math xmlns:vc=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\">\n                        <vc:mi>k<\/vc:mi>\n                        <vc:mo>\u2265<\/vc:mo>\n                        <vc:mn>1<\/vc:mn>\n                     <\/vc:math>\n                  <\/jats:inline-formula>. One question about the permanent of a hexagonal zig-zag chain is also presented.<\/jats:p>","DOI":"10.1155\/2022\/7786922","type":"journal-article","created":{"date-parts":[[2022,11,25]],"date-time":"2022-11-25T19:46:33Z","timestamp":1669405593000},"page":"1-6","source":"Crossref","is-referenced-by-count":0,"title":["Permanents of Hexagonal and Armchair Chains"],"prefix":"10.1155","volume":"2022","author":[{"given":"O.","family":"Nekooei","sequence":"first","affiliation":[{"name":"Department of Mathematics, Tafresh University, Tafresh 39518-79611, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8916-7733","authenticated-orcid":true,"given":"H.","family":"Barzegar","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Tafresh University, Tafresh 39518-79611, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"A. R.","family":"Ashrafi","sequence":"additional","affiliation":[{"name":"Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-53153, Iran"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"key":"1","volume-title":"Algebraic Graph Theory","author":"N. Biggs","year":"1993","edition":"2nd"},{"volume-title":"Matching Theory","year":"1986","author":"L. Lovasz","key":"2"},{"key":"3","first-page":"17","article-title":"Permanents of 3X3 invertible matrices modulo n","volume":"22","author":"A. Bohra","year":"2022","journal-title":"Integers"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1016\/j.mcm.2007.12.001"},{"key":"5","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-00892-8","volume-title":"Kekul\u00e9 Structures in Benzenoid Hydrocarbons","author":"S. Cyvin","year":"1988"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1155\/2021\/9939469"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1007\/bf02579161"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1080\/0025570x.2001.11953063"},{"volume-title":"Mathematica, Version 13.1","year":"2022","author":"Wolfram ResearchInc","key":"9"}],"container-title":["International Journal of Mathematics and Mathematical Sciences"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2022\/7786922.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2022\/7786922.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/ijmms\/2022\/7786922.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,25]],"date-time":"2022-11-25T19:46:39Z","timestamp":1669405599000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/ijmms\/2022\/7786922\/"}},"subtitle":[],"editor":[{"given":"Anwar","family":"Saleh Alwardi","sequence":"additional","affiliation":[],"role":[{"role":"editor","vocabulary":"crossref"}]}],"short-title":[],"issued":{"date-parts":[[2022,11,25]]},"references-count":9,"alternative-id":["7786922","7786922"],"URL":"https:\/\/doi.org\/10.1155\/2022\/7786922","relation":{},"ISSN":["1687-0425","0161-1712"],"issn-type":[{"type":"electronic","value":"1687-0425"},{"type":"print","value":"0161-1712"}],"subject":[],"published":{"date-parts":[[2022,11,25]]}}}