{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:40:51Z","timestamp":1760150451920,"version":"build-2065373602"},"reference-count":46,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2023,11,30]],"date-time":"2023-11-30T00:00:00Z","timestamp":1701302400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004242","name":"Princess Nourah bint Abdulrahman University","doi-asserted-by":"publisher","award":["PNURSP2023R87"],"award-info":[{"award-number":["PNURSP2023R87"]}],"id":[{"id":"10.13039\/501100004242","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The importance of symmetry in graph theory has always been significant, but in recent years, it has become much more so in a number of subfields, including but not limited to domination theory, topological indices, Gromov hyperbolic graphs, and the metric dimension of graphs. The purpose of this monograph is to initiate the idea of a multi polar q-rung orthopair fuzzy graphs (m-PqROPFG) as a fusion of multi polar fuzzy graphs and q-rung orthopair fuzzy graphs. Moreover, for a vertex of multi polar q-rung orthopair fuzzy graphs, the degree and total degree of the vertex are defined. Then, some product operations, inclusive of direct, Cartesian, semi strong, strong lexicographic products, and the union of multi polar q-rung orthopair fuzzy graphs (m-PqROPFGs), are obtained. Also, at first we define some degree based fuzzy topological indices of m-PqROPFG. Then, we compute Zareb indices of the first and second kind, Randic indices, and harmonic index of a m-PqROPFG.<\/jats:p>","DOI":"10.3390\/sym15122131","type":"journal-article","created":{"date-parts":[[2023,11,30]],"date-time":"2023-11-30T00:34:10Z","timestamp":1701304450000},"page":"2131","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Multi Polar q-Rung Orthopair Fuzzy Graphs with Some Topological Indices"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-4020-2731","authenticated-orcid":false,"given":"Andleeb","family":"Kausar","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, Riphah International University, I-14, Islamabad 44000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5529-0565","authenticated-orcid":false,"given":"Nabilah","family":"Abughazalah","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5362-9916","authenticated-orcid":false,"given":"Naveed","family":"Yaqoob","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Riphah International University, I-14, Islamabad 44000, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1016\/S0019-9958(65)90241-X","article-title":"Fuzzy sets","volume":"8","author":"Zadeh","year":"1965","journal-title":"Inf. Control"},{"key":"ref_2","unstructured":"Zhang, W.R. (1994, January 18\u201321). Bipolar fuzzy sets and relations: A computational framework forcognitive modeling and multiagent decision analysis. Proceedings of the 1st International Joint Conference of The North American Fuzzy Information Processing Society Biannual Conference, San Antonio, TX, USA."},{"key":"ref_3","unstructured":"Zhang, W.R. (1998, January 4\u20139). Bipolar fuzzy sets. Proceedings of the International Conference on Fuzzy Systems, Anchorage, AK, USA."},{"key":"ref_4","first-page":"416530","article-title":"m-Polar fuzzy sets: An extension of bipolar fuzzy sets","volume":"2014","author":"Chen","year":"2014","journal-title":"Sci. World J."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"8441","DOI":"10.3233\/JIFS-191087","article-title":"Pythagorean m-polar fuzzy sets and TOPSIS method for the selection of advertisement mode","volume":"37","author":"Naeem","year":"2019","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0165-0114(86)80034-3","article-title":"Intuitionistic fuzzy sets","volume":"20","author":"Atanassov","year":"1986","journal-title":"Fuzzy Sets Syst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"436","DOI":"10.1002\/int.21584","article-title":"Pythagorean membership grades, complex numbers, and decision making","volume":"28","author":"Yager","year":"2013","journal-title":"Int. J. Intell. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1133","DOI":"10.1002\/int.21738","article-title":"Some results for Pythagorean fuzzy sets","volume":"30","author":"Peng","year":"2015","journal-title":"Int. J. Intell. Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1977","DOI":"10.3233\/JIFS-190613","article-title":"A novel approach to censuses process by using Pythagorean m-polar fuzzy Dombi\u2019s aggregation operators","volume":"38","author":"Hashmi","year":"2020","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1007\/s40815-019-00763-2","article-title":"m-Polar neutrosophic topology with applications to multi-criteria decision-making in medical diagnosis and clustering analysis","volume":"22","author":"Hashmi","year":"2020","journal-title":"Int. Fuzzy Syst."},{"key":"ref_11","first-page":"1220","article-title":"Generalized orthopair fuzzy sets","volume":"25","author":"Yagar","year":"2017","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2139","DOI":"10.1002\/int.22007","article-title":"Another view on q-rung orthopair fuzzy sets","volume":"33","author":"Ali","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Riaz, M., Hamid, M.T., Afzal, D., Pamucar, D., and Chu, Y.M. (2021). Multi-criteria decision making in robotic agri-farming with q-rung orthopair m-polar fuzzy sets. PLoS ONE, 16.","DOI":"10.1371\/journal.pone.0246485"},{"key":"ref_14","unstructured":"Kaufmann, A. (1973). Introduction a la Theorie des Sous-Emsembles Flous, Masson et Cia Editeurs."},{"key":"ref_15","unstructured":"Rosenfeld, A. (1975). Fuzzy Graphs, Fuzzy Sets and Their Applications to Cognitive and Decision Processes, Academic Press."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1016\/0167-8655(87)90012-2","article-title":"Some remarks on fuzzy graphs","volume":"6","author":"Bhattacharya","year":"1987","journal-title":"Pattern Recognit. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/0020-0255(95)00238-3","article-title":"Cycles and cocycles of fuzzy graphs","volume":"90","author":"Mordeson","year":"1996","journal-title":"Inf. Sci."},{"key":"ref_18","first-page":"11","article-title":"Degree, order and size in intuitionistic fuzzy graphs","volume":"3","author":"Gani","year":"2010","journal-title":"Int. J. Algorithms Comput. Math."},{"key":"ref_19","first-page":"517","article-title":"On irregular fuzzy graphs","volume":"6","author":"Gani","year":"2012","journal-title":"Appl. Math. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"5548","DOI":"10.1016\/j.ins.2011.07.037","article-title":"Bipolar fuzzy graphs","volume":"181","author":"Akram","year":"2011","journal-title":"Inf. Sci."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.knosys.2012.08.022","article-title":"Bipolar fuzzy graphs with applications","volume":"39","author":"Akram","year":"2013","journal-title":"Knowl.-Based Syst."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1597","DOI":"10.1080\/09720529.2015.1117221","article-title":"m-Polar fuzzy graphs and m-polar fuzzy line graphs","volume":"20","author":"Akram","year":"2017","journal-title":"J. Discret. Math. Sci. Cryptogr."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/s11786-016-0277-x","article-title":"m-Polar fuzzy labeling graphs with application","volume":"10","author":"Akram","year":"2016","journal-title":"Math. Comput. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1142\/S1793005716500101","article-title":"Certain metrics in m-polar fuzzy graphs","volume":"12","author":"Akram","year":"2016","journal-title":"New Math. Nat. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1448","DOI":"10.1186\/s40064-016-3066-8","article-title":"On m-Polar Fuzzy Graph Structures","volume":"5","author":"Akram","year":"2016","journal-title":"SpringerPlus"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3145","DOI":"10.1007\/s00521-017-2894-y","article-title":"Novel applications of m-polar fuzzy competition graphs in decision support system","volume":"30","author":"Akram","year":"2018","journal-title":"Neural Comput. Appl."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Naz, S., Ashraf, S., and Akram, M. (2018). A novel approach to decision-making with Pythagorean fuzzy information. Mathematics, 6.","DOI":"10.3390\/math6060095"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2137","DOI":"10.3233\/JIFS-161944","article-title":"Operations on single valued neutrosophic graphs with application","volume":"32","author":"Naz","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1007\/s12190-014-0785-2","article-title":"m-Step fuzzy competition graphs","volume":"47","author":"Samanta","year":"2015","journal-title":"J. Appl. Math. Comput."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Habib, A., Akram, M., and Farooq, A. (2019). q-Rung orthopair fuzzy competition graphs with application in the soil ecosystem. Mathematics, 7.","DOI":"10.3390\/math7010091"},{"key":"ref_31","first-page":"83","article-title":"Cayley interval-valued fuzzy graphs","volume":"78","author":"Borzooei","year":"2016","journal-title":"UPB Sci. Bull. Ser. A Appl. Math. Phys."},{"key":"ref_32","first-page":"1756","article-title":"Some properties of highly irregular interval-valued fuzzy graphs","volume":"27","author":"Rashmanlou","year":"2013","journal-title":"World Appl. Sci. J."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1893","DOI":"10.3233\/IFS-151900","article-title":"New concepts of interval-valued intuitionistic (S, T)-fuzzy graphs","volume":"30","author":"Rashmanlou","year":"2016","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_34","first-page":"335","article-title":"Interval-valued Intuitionistic Fuzzy Competition Graph","volume":"34","author":"Talebi","year":"2020","journal-title":"J. Mult. Log. Soft Comput."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/0020-0255(94)90116-3","article-title":"Operations on fuzzy graphs","volume":"79","author":"Mordeson","year":"1994","journal-title":"Inf. Sci."},{"key":"ref_36","unstructured":"Mordeson, J.N., and Nair, P.S. (2012). Fuzzy Graphs and Fuzzy Hypergraphs, Physica Verlag."},{"key":"ref_37","first-page":"1","article-title":"Fuzzy graphs on composition, tensor and normal products","volume":"2","author":"Nirmala","year":"2012","journal-title":"Int. J. Sci. Res. Publ."},{"key":"ref_38","first-page":"203","article-title":"Some operations on fuzzy hypergraphs","volume":"132","author":"Gong","year":"2017","journal-title":"Ars Comb."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1059","DOI":"10.3233\/JIFS-16348","article-title":"Product of intuitionistic fuzzy graphs and degree","volume":"32","author":"Sahoo","year":"2017","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"6443","DOI":"10.3233\/JIFS-181488","article-title":"Product of interval-valued fuzzy graphs and degree","volume":"35","author":"Rashmanlou","year":"2018","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Yin, S., Li, H., and Yang, Y. (2019). Product operations on q-rung orthopair fuzzy graphs. Symmetry, 11.","DOI":"10.3390\/sym11040588"},{"key":"ref_42","unstructured":"Xu, J. (1997). Fuzzy Logic in Chemistry, Academic Press."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Mordeson, J.N., and Mathew, S. (2019). Advanced Topics in Fuzzy Graph Theory, Springer.","DOI":"10.1007\/978-3-030-04215-8"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"4156879","DOI":"10.1155\/2021\/4156879","article-title":"Connectivity indices of intuitionistic fuzzy graphs and their applications in internet routing and transport network flow","volume":"2021","author":"Naeem","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"6033","DOI":"10.3233\/JIFS-189077","article-title":"Some topological indices in fuzzy graphs","volume":"39","author":"Kalathian","year":"2020","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"101826","DOI":"10.1016\/j.asej.2022.101826","article-title":"Wiener index for an intuitionistic fuzzy graph and its application in water pipeline network","volume":"14","author":"Dinar","year":"2023","journal-title":"Ain Shams Eng. J."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/12\/2131\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:34:19Z","timestamp":1760132059000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/12\/2131"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,30]]},"references-count":46,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2023,12]]}},"alternative-id":["sym15122131"],"URL":"https:\/\/doi.org\/10.3390\/sym15122131","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,11,30]]}}}