<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE article PUBLIC "-//NLM//DTD JATS (Z39.96) Journal Publishing DTD v1.3 20210610//EN" "JATS-journalpublishing1-3.dtd">
<article article-type="research-article" dtd-version="1.3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xml:lang="ru"><front><journal-meta><journal-id journal-id-type="publisher-id">sibsutis</journal-id><journal-title-group><journal-title xml:lang="ru">Вестник СибГУТИ</journal-title><trans-title-group xml:lang="en"><trans-title>The Herald of the Siberian State University of Telecommunications and Information Science</trans-title></trans-title-group></journal-title-group><issn pub-type="ppub">1998-6920</issn><publisher><publisher-name>СибГУТИ</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="doi">10.55648/1998-6920-2024-18-4-52-61</article-id><article-id custom-type="elpub" pub-id-type="custom">sibsutis-897</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group><subj-group subj-group-type="section-heading" xml:lang="ru"><subject>Статьи</subject></subj-group></article-categories><title-group><article-title>Результаты по целочисленным графам</article-title><trans-title-group xml:lang="en"><trans-title>Results of Search About Integer Graph</trans-title></trans-title-group></title-group><contrib-group><contrib contrib-type="author" corresp="yes"><contrib-id contrib-id-type="orcid">https://orcid.org/0000-0001-7929-1899</contrib-id><name-alternatives><name name-style="eastern" xml:lang="ru"><surname>Овчаренко</surname><given-names>А. Ю.</given-names></name><name name-style="western" xml:lang="en"><surname>Ovcharenko</surname><given-names>A. Yu.</given-names></name></name-alternatives><bio xml:lang="ru"><p>Овчаренко Алёна Юрьевна - к.т.н., доцент кафедры высшей математики, </p><p>630102, Новосибирск, ул. Кирова, д. 86.</p></bio><bio xml:lang="en"><p>Alena Y. Ovcharenko - PhD in Engineering, Assoc. Professor of the Department of Higher Mathematics,</p><p>86, Kirov St., Novosibirsk, 630102.</p></bio><email xlink:type="simple">shmatova_aaa@mail.ru</email><xref ref-type="aff" rid="aff-1"/></contrib></contrib-group><aff-alternatives id="aff-1"><aff xml:lang="ru"><institution>Сибирский государственный университет телекоммуникаций и информатики (СибГУТИ)</institution><country>Россия</country></aff><aff xml:lang="en"><institution>Siberian State University of Telecommunications and Information Science (SibSUTIS)</institution><country>Russian Federation</country></aff></aff-alternatives><pub-date pub-type="collection"><year>2024</year></pub-date><pub-date pub-type="epub"><day>12</day><month>05</month><year>2024</year></pub-date><volume>18</volume><issue>4</issue><issue-title>Вестник СибГУТИ</issue-title><fpage>52</fpage><lpage>61</lpage><permissions><copyright-statement>Copyright &amp;#x00A9; Овчаренко А.Ю., 2024</copyright-statement><copyright-year>2024</copyright-year><copyright-holder xml:lang="ru">Овчаренко А.Ю.</copyright-holder><copyright-holder xml:lang="en">Ovcharenko A.Y.</copyright-holder><license xml:lang="ru" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>Данная работа распространяется под лицензией Creative Commons Attribution 4.0.</license-p></license><license xml:lang="en" license-type="creative-commons-attribution" xlink:href="https://creativecommons.org/licenses/by/4.0/" xlink:type="simple"><license-p>This work is licensed under a Creative Commons Attribution 4.0 License.</license-p></license></permissions><self-uri xlink:href="https://vestnik.sibsutis.ru/jour/article/view/897">https://vestnik.sibsutis.ru/jour/article/view/897</self-uri><abstract><p>В статье представлен обзор результатов, связанных с поиском целочисленных графов Кэли и смежные вопросы. </p></abstract><trans-abstract xml:lang="en"><p>The article provides an overview of the results related to the search for integer graphs and related issues.</p></trans-abstract><kwd-group xml:lang="ru"><kwd>целочисленные графы</kwd><kwd>спектр графа</kwd><kwd>регулярные графы</kwd><kwd>нерегулярные графы</kwd><kwd>4-регулярные графы</kwd></kwd-group><kwd-group xml:lang="en"><kwd>integer graphs</kwd><kwd>spectrum of graph</kwd><kwd>regular graphs</kwd><kwd>non-regular graphs</kwd><kwd>4-regular graphs</kwd></kwd-group></article-meta></front><back><ref-list><title>References</title><ref id="cit1"><label>1</label><citation-alternatives><mixed-citation xml:lang="ru">Harary F, Schwenk A. J. Which graphs have integral spectra? Graphs and Combinatorics (Bari R. and Harary F., eds.), Springer-Verlag, Berlin (1974), P. 45—51.</mixed-citation><mixed-citation xml:lang="en">Harary F, Schwenk A. J. Which graphs have integral spectra? Graphs and Combinatorics (R. Bari and F. Harary, eds.), Springer-Verlag, Berlin (1974), pp. 45—51.</mixed-citation></citation-alternatives></ref><ref id="cit2"><label>2</label><citation-alternatives><mixed-citation xml:lang="ru">Akers S., Krishnamurthy B. A group theoretic model for symmetric interconnection networks // IEEE Transactions on Computers, 1989. P. 555 - 566.</mixed-citation><mixed-citation xml:lang="en">Akers S., Krishnamurthy B. A group theoretic model for symmetric interconnection networks. IEEE Transactions on Computers, 1989. pp. 555 - 566.</mixed-citation></citation-alternatives></ref><ref id="cit3"><label>3</label><citation-alternatives><mixed-citation xml:lang="ru">Hwang K., Jacobs M., Earl E. Swartzlander (Jr.). Proceedings of the 1986 International Conference on Parallel Processing // IEEE Computer Society Press, 1986, P 1051.</mixed-citation><mixed-citation xml:lang="en">Hwang K., Jacobs M., Earl E. Swartzlander (Jr.). Proceedings of the 1986 International Conference on Parallel Processing. IEEE Computer Society Press, 1986, pp. 1051.</mixed-citation></citation-alternatives></ref><ref id="cit4"><label>4</label><citation-alternatives><mixed-citation xml:lang="ru">Annexstein, F., Baumslag, M., Rosenberg, A.L. Group action graphs and parallel architectures. SIAM J. Comput., 19, 1990. P. 544–569.</mixed-citation><mixed-citation xml:lang="en">Annexstein, F., Baumslag, M., Rosenberg, A. L. Group action graphs and parallel architectures. SIAM J. Comput., 19, 1990. pp. 544–569.</mixed-citation></citation-alternatives></ref><ref id="cit5"><label>5</label><citation-alternatives><mixed-citation xml:lang="ru">Heydemann, M. Cayley graphs and interconnection networks. In: G. Hahn and G. Sabidussi (Eds), Graph Symmetry: Algebraic Methods and Applications. 1997. P. 167–224.</mixed-citation><mixed-citation xml:lang="en">Heydemann, M. Cayley graphs and interconnection networks. In: G. Hahn and G. Sabidussi (Eds), Graph Symmetry: Algebraic Methods and Applications. 1997. pp. 167–224.</mixed-citation></citation-alternatives></ref><ref id="cit6"><label>6</label><citation-alternatives><mixed-citation xml:lang="ru">Xiao W., Parhami B. Some mathematical properties of Cayley digraphs with applications to interconnection network design. International Journal of Computer Mathematics, 2005. P. 521–528.</mixed-citation><mixed-citation xml:lang="en">Xiao W., Parhami B. Some mathematical properties of Cayley digraphs with applications to interconnection network design. International Journal of Computer Mathematics, 2005. pp. 521–528.</mixed-citation></citation-alternatives></ref><ref id="cit7"><label>7</label><citation-alternatives><mixed-citation xml:lang="ru">Lavault C. Interconnection Networks: Graph- and Group Theoretic Modelling. 12th International Conference on Control Systems and Computer Science, 1999. P. 207-214.</mixed-citation><mixed-citation xml:lang="en">Lavault C. Interconnection Networks: Graph- and Group Theoretic Modelling. 12th International Conference on Control Systems and Computer Science, 1999. pp. 207-214.</mixed-citation></citation-alternatives></ref><ref id="cit8"><label>8</label><citation-alternatives><mixed-citation xml:lang="ru">Godsil C. State transfer on graphs. Discrete Mathematics, 312(1): P. 129 – 147, 2012.</mixed-citation><mixed-citation xml:lang="en">Godsil C. State transfer on graphs. Discrete Mathematics, 312(1):129 – 147, 2012.</mixed-citation></citation-alternatives></ref><ref id="cit9"><label>9</label><citation-alternatives><mixed-citation xml:lang="ru">Ekert A., Christandl M., Datta N., Landahl A. J. Perfect state transfer in quantum spin networks. Phys. Rev. Lett., 92:187902, May 2004.</mixed-citation><mixed-citation xml:lang="en">Ekert A., Christandl M., Datta N., Landahl A. J. Perfect state transfer in quantum spin networks. Phys. Rev. Lett., 92:187902, May 2004.</mixed-citation></citation-alternatives></ref><ref id="cit10"><label>10</label><citation-alternatives><mixed-citation xml:lang="ru">Овчаренко А. Ю. Спектры и диаметры графов Кэли некоторых конечных групп // Вестник СибГУТИ. 2018. №3. С. 45—61.</mixed-citation><mixed-citation xml:lang="en">Ovcharenko A. Y. Spektry i diametry grafov Keli nekotorykh konechnykh grupp [Spectra and diameters of Cayley graphs of some finite groups]. Vestnik SibGUTI, 2018, no. 3, pp. 45-61.</mixed-citation></citation-alternatives></ref><ref id="cit11"><label>11</label><citation-alternatives><mixed-citation xml:lang="ru">Balinska K., Cvetkovic D., Radosavljevic Z., Simic S., Stevanovic D. A survey on integral graphs, Publ ETF, 2002, № 13, P. 1-24.</mixed-citation><mixed-citation xml:lang="en">Balinska K., Cvetkovic D., Radosavljevic Z., Simic S., Stevanovic D. A survey on integral graphs. Publ ETF, 2002, no. 13, pp. 1-24.</mixed-citation></citation-alternatives></ref><ref id="cit12"><label>12</label><citation-alternatives><mixed-citation xml:lang="ru">Cvetkovic D., Doob M., Sachs H. Spectra of graphs – Theory and application. Deutscher Verlag der Wissenschaften – Academic Press, Berlin–New York, 1980; secondedition 1982; third edition, Johann Ambrosius Barth Verlag, Heidelberg–Leipzig, 1995.</mixed-citation><mixed-citation xml:lang="en">Cvetkovic D., Doob M., Sachs H. Spectra of graphs – Theory and application. Deutscher Verlag der Wissenschaften – Academic Press, Berlin–New York, 1980; secondedition 1982; third edition, Johann Ambrosius Barth Verlag, Heidelberg–Leipzig, 1995.</mixed-citation></citation-alternatives></ref><ref id="cit13"><label>13</label><citation-alternatives><mixed-citation xml:lang="ru">Bussemaker F. C., Cvetkoviс D. There are exactly 13 connected, cubic, integral graphs. Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. Fiz., Nos. 544–576 (1976), P. 43–48.</mixed-citation><mixed-citation xml:lang="en">Bussemaker F. C., Cvetkoviс D. There are exactly 13 connected, cubic, integral graphs. Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. Fiz., Nos. 544–576 (1976), pp. 43–48.</mixed-citation></citation-alternatives></ref><ref id="cit14"><label>14</label><citation-alternatives><mixed-citation xml:lang="ru">Schwenk A. J. Exactly thirteen connected cubic graphs have integral spectra. Proceedings of the International Graph Theory Conference at Kalamazoo, May 1976, (Y. Alavi and D. Lick, eds.) Springer-Verlag.</mixed-citation><mixed-citation xml:lang="en">Schwenk A. J. Exactly thirteen connected cubic graphs have integral spectra. Proceedings of the International Graph Theory Conference at Kalamazoo, May 1976, (Y. Alavi and D. Lick, eds.) Springer-Verlag.</mixed-citation></citation-alternatives></ref><ref id="cit15"><label>15</label><citation-alternatives><mixed-citation xml:lang="ru">Roitman M. An infinite family of integral graphs, Discrete Math. 52 (1984), №2-3, P. 313— 315.</mixed-citation><mixed-citation xml:lang="en">Roitman M. An infinite family of integral graphs. Discrete Math. 52 (1984), no.2-3, pp. 313—315.</mixed-citation></citation-alternatives></ref><ref id="cit16"><label>16</label><citation-alternatives><mixed-citation xml:lang="ru">Wang L.G., Li X.L., Hoede C. Integral complete r-partite Graphs, Discrete Math. 283 (2004), № 1-3, P. 231-241.</mixed-citation><mixed-citation xml:lang="en">Wang L. G., Li X. L., Hoede C. Integral complete r-partite Graphs. Discrete Math. 283 (2004), no. 13, 231-241.</mixed-citation></citation-alternatives></ref><ref id="cit17"><label>17</label><citation-alternatives><mixed-citation xml:lang="ru">Wang L. A survey of results on integral trees and integral graphs, Department of Applied Mathematics, Faculty of EEMCS, University of Twente The Netherlands, Memorandum № 1763 (2005), P. 1—22.</mixed-citation><mixed-citation xml:lang="en">Wang L. A survey of results on integral trees and integral graphs. Department of Applied Mathematics, Faculty of EEMCS, University of Twente The Netherlands, Memorandum No. 1763 (2005), pp. 1—22.</mixed-citation></citation-alternatives></ref><ref id="cit18"><label>18</label><citation-alternatives><mixed-citation xml:lang="ru">Balinska K. T., Cvetkovic D., Lepovic M., Simic S. K. There are exactly 150 connected integral graphs up to 10 vertices. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 10 (1999), P. 95-105.</mixed-citation><mixed-citation xml:lang="en">Balinska K. T., Cvetkovic D., Lepovic M., Simic S. K. There are exactly 150 connected integral graphs up to 10 vertices. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 10 (1999), 95-105.</mixed-citation></citation-alternatives></ref><ref id="cit19"><label>19</label><citation-alternatives><mixed-citation xml:lang="ru">Balinska K. T., Kupczyk M., Simic S. K., Zwierzynski K. T. On generating all integral graphs on 11 vertices. Computer Science Center Report № 469, Technical University of Poznan, (1999/2000), P. 1–53.</mixed-citation><mixed-citation xml:lang="en">Balinska K. T., Kupczyk M., Simic S. K., Zwierzynski K. T. On generating all integral graphs on 11 vertices. Computer Science Center Report No. 469, Technical University of Poznan, (1999/2000), pp. 1–53.</mixed-citation></citation-alternatives></ref><ref id="cit20"><label>20</label><citation-alternatives><mixed-citation xml:lang="ru">Balinska K. T., Kupczyk M., Simic S. K., Zwierzynski K. T. On generating all integral graphs on 12 vertices. Computer Science Center Report № 482, Technical University of Poznan, (2001), P. 1–36.</mixed-citation><mixed-citation xml:lang="en">Balinska K. T., Kupczyk M., Simic S. K., Zwierzynski K. T. On generating all integral graphs on 12 vertices. Computer Science Center Report No. 482, Technical University of Poznan, (2001), pp. 1– 36.</mixed-citation></citation-alternatives></ref><ref id="cit21"><label>21</label><citation-alternatives><mixed-citation xml:lang="ru">Cvetkovic D., Simic S., Stevanovic D. 4-Regular integral graphs, Univ. Beograd. Publ. Elektr. Fak, Ser. Mat. 9 (1998), P. 89–102.</mixed-citation><mixed-citation xml:lang="en">Cvetkovic D., Simic S., Stevanovic D. 4-Regular integral graphs, Univ. Beograd. Publ. Elektr. Fak, Ser. Mat. 9 (1998), pp. 89–102.</mixed-citation></citation-alternatives></ref><ref id="cit22"><label>22</label><citation-alternatives><mixed-citation xml:lang="ru">Stevanovic D. Nonexistence of some 4-regular integral graphs, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 10 (1999), P. 81-86.</mixed-citation><mixed-citation xml:lang="en">Stevanovic D. Nonexistence of some 4-regular integral graphs, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 10 (1999), pp. 81-86.</mixed-citation></citation-alternatives></ref><ref id="cit23"><label>23</label><citation-alternatives><mixed-citation xml:lang="ru">Stevanovic D. 4-Regular integral graphs avoiding ±3 in the spectrum, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 14 (2003), P. 99-110.</mixed-citation><mixed-citation xml:lang="en">Stevanovic D. 4-Regular integral graphs avoiding ±3 in the spectrum, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. 14 (2003), pp. 99-110.</mixed-citation></citation-alternatives></ref><ref id="cit24"><label>24</label><citation-alternatives><mixed-citation xml:lang="ru">Stevanovic D., de Abreu N.M.M., de Freitas M.A.A., Del-Vecchio R. Walks and Regular Integral Graphs, Linear Algebra Appl. 423 (2007), P. 119–135.</mixed-citation><mixed-citation xml:lang="en">Stevanovic D., de Abreu N. M. M., de Freitas M. A. A., Del-Vecchio R. Walks and Regular Integral Graphs. Linear Algebra Appl. 423 (2007), pp. 119–135.</mixed-citation></citation-alternatives></ref><ref id="cit25"><label>25</label><citation-alternatives><mixed-citation xml:lang="ru">Stevanovic D. Some compositions of graphs and integral graphs (in Serbian), Ph.D. thesis, University of Nis, 2000.</mixed-citation><mixed-citation xml:lang="en">Stevanovic D. Some compositions of graphs and integral graphs (in Serbian), Ph.D. thesis, University of Nis, 2000.</mixed-citation></citation-alternatives></ref><ref id="cit26"><label>26</label><citation-alternatives><mixed-citation xml:lang="ru">Minchenko M., Wanless I. M. Quartic integral Cayley graphs. // Ars Mathematica Contemporanea. – V. 8. – 2015. P. 381–408.</mixed-citation><mixed-citation xml:lang="en">Minchenko M., Wanless I. M. Quartic integral Cayley graphs. Ars Mathematica Contemporanea. – Vol. 8. – 2015. pp. 381–408.</mixed-citation></citation-alternatives></ref><ref id="cit27"><label>27</label><citation-alternatives><mixed-citation xml:lang="ru">Balinska K. T., Simic S. K. The nonregular, bipartite, integral graphs with maximum degree 4. Part I: Basic properties, Graph theory (Kazimierz Dolny, 1997), Discrete Math. 236 (2001), №1-3, P. 13-24.</mixed-citation><mixed-citation xml:lang="en">Balinska K. T., Simic S. K. The nonregular, bipartite, integral graphs with maximum degree 4. Part I: Basic properties, Graph theory (Kazimierz Dolny, 1997), Discrete Math. 236 (2001), no.1-3, pp. 1324.</mixed-citation></citation-alternatives></ref><ref id="cit28"><label>28</label><citation-alternatives><mixed-citation xml:lang="ru">Balinska K. T., Simic S. K. Some remarks on integral graphs with maximum degree four. XIV Conference on Applied Mathematics (Palic, 2000). Novi Sad J. Math. 31 (2001), № 1, P. 19- 25.</mixed-citation><mixed-citation xml:lang="en">Balinska K. T., Simic S. K. Some remarks on integral graphs with maximum degree four. XIV Conference on Applied Mathematics (Palic, 2000). Novi Sad J. Math. 31 (2001), no.1, 19- 25. pp. 13 24.</mixed-citation></citation-alternatives></ref><ref id="cit29"><label>29</label><citation-alternatives><mixed-citation xml:lang="ru">Balinska K. T., Simic S. K. Zwierzynski K. T. Which non-regular bipartite integral graphs with maximum degree four do not have ±1 as eigenvalues? Discrete Math. 286 (2004), № 12, P. 15- 24.</mixed-citation><mixed-citation xml:lang="en">Balinska K. T., Simic S. K. Zwierzynski K. T. Which non-regular bipartite integral graphs with maximum degree four do not have ±1 as eigenvalues? Discrete Math. 286 (2004), no.1-2, pp. 15- 24.</mixed-citation></citation-alternatives></ref><ref id="cit30"><label>30</label><citation-alternatives><mixed-citation xml:lang="ru">Stevanovic D. Some compositions of graphs and integral graphs (in Serbian), Ph.D. thesis, University of Nis, 2000.</mixed-citation><mixed-citation xml:lang="en">Stevanovic D. Some compositions of graphs and integral graphs (in Serbian), Ph.D. thesis, University of Nis, 2000.</mixed-citation></citation-alternatives></ref><ref id="cit31"><label>31</label><citation-alternatives><mixed-citation xml:lang="ru">Ghasemi M. Integral pentavalent Cayley graphs on abelian or dihedral groups. Proceedings of the Indian Academy of Sciences: Mathematical Sciences. – 2017. – V. 127, № 2. – P. 219-224</mixed-citation><mixed-citation xml:lang="en">Ghasemi M. Integral pentavalent Cayley graphs on abelian or dihedral groups. Proceedings of the Indian Academy of Sciences: Mathematical Sciences. – 2017. – Vol. 127, no. 2. – pp. 219-224.</mixed-citation></citation-alternatives></ref><ref id="cit32"><label>32</label><citation-alternatives><mixed-citation xml:lang="ru">Konstantinova E. V., Lytkina D.V. Integral Cayley Graphs over Finite Groups // Algebra Colloquium. – 2020. – V. 27, № 1. – P. 131-136.</mixed-citation><mixed-citation xml:lang="en">Konstantinova E. V., Lytkina D. V. Integral Cayley Graphs over Finite Groups. Algebra Colloquium. – 2020. – Vol. 27, no. 1. – pp. 131-136.</mixed-citation></citation-alternatives></ref><ref id="cit33"><label>33</label><citation-alternatives><mixed-citation xml:lang="ru">Goryainov S., Zhao D., Konstantinova E. V., Li H. Integral graphs obtained by dual Seidel switching. Linear Algebra and its Applications. – 2020. – V. 604. – P. 476-489.</mixed-citation><mixed-citation xml:lang="en">Goryainov S., Zhao D., Konstantinova E. V., Li H. Integral graphs obtained by dual Seidel switching. Linear Algebra and its Applications. – 2020. – Vol. 604. – pp. 476-489.</mixed-citation></citation-alternatives></ref><ref id="cit34"><label>34</label><citation-alternatives><mixed-citation xml:lang="ru">Lu. Lu, Huang Q., Huang X. Integral Cayley graphs over dihedral groups. Journal of Algebraic Combinatorics. – 2018. – V. 47, № 4. – P. 585-601.</mixed-citation><mixed-citation xml:lang="en">Lu. Lu, Huang Q., Huang X. Integral Cayley graphs over dihedral groups. Journal of Algebraic Combinatorics. – 2018. – Vol. 47, no. 4. – pp. 585-601.</mixed-citation></citation-alternatives></ref><ref id="cit35"><label>35</label><citation-alternatives><mixed-citation xml:lang="ru">Huang J., Li. Sh. Integral and distance integral Cayley graphs over generalized dihedral groups. Journal of Algebraic Combinatorics. – 2021. – V. 53, № 4. – P. 921-943.</mixed-citation><mixed-citation xml:lang="en">Huang J., Li. Sh. Integral and distance integral Cayley graphs over generalized dihedral groups. Journal of Algebraic Combinatorics. – 2021. – Vol. 53, no. 4. – pp. 921-943.</mixed-citation></citation-alternatives></ref></ref-list><fn-group><fn fn-type="conflict"><p>The authors declare that there are no conflicts of interest present.</p></fn></fn-group></back></article>
