{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,23]],"date-time":"2026-06-23T05:18:01Z","timestamp":1782191881570,"version":"3.54.5"},"reference-count":36,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,4,23]],"date-time":"2019-04-23T00:00:00Z","timestamp":1555977600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11626079"],"award-info":[{"award-number":["11626079"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003787","name":"Natural Science Foundation of Hebei Province","doi-asserted-by":"publisher","award":["F2015402033"],"award-info":[{"award-number":["F2015402033"]}],"id":[{"id":"10.13039\/501100003787","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The q-rung orthopair fuzzy graph is an extension of intuitionistic fuzzy graph and Pythagorean fuzzy graph. In this paper, the degree and total degree of a vertex in q-rung orthopair fuzzy graphs are firstly defined. Then, some product operations on q-rung orthopair fuzzy graphs, including direct product, Cartesian product, semi-strong product, strong product, and lexicographic product, are defined. Furthermore, some theorems about the degree and total degree under these product operations are put forward and elaborated with several examples. In particular, these theorems improve the similar results in single-valued neutrosophic graphs and Pythagorean fuzzy graphs.<\/jats:p>","DOI":"10.3390\/sym11040588","type":"journal-article","created":{"date-parts":[[2019,4,24]],"date-time":"2019-04-24T03:14:28Z","timestamp":1556075668000},"page":"588","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["Product Operations on q-Rung Orthopair Fuzzy Graphs"],"prefix":"10.3390","volume":"11","author":[{"given":"Songyi","family":"Yin","sequence":"first","affiliation":[{"name":"School of Management Engineering and Business, Hebei University of Engineering, Handan 056038, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hongxu","family":"Li","sequence":"additional","affiliation":[{"name":"School of Management Engineering and Business, Hebei University of Engineering, Handan 056038, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yang","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Management Engineering and Business, Hebei University of Engineering, Handan 056038, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2019,4,23]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1222","DOI":"10.1109\/TFUZZ.2016.2604005","article-title":"Generalized orthopair fuzzy sets","volume":"25","author":"Yager","year":"2017","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_2","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_3","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. Intell. Syst."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"958","DOI":"10.1109\/TFUZZ.2013.2278989","article-title":"Pythagorean membership grades in multicriteria decision making","volume":"22","author":"Yager","year":"2014","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Bai, K.Y., Zhu, X.M., Wang, J., and Zhang, R.T. (2018). Some partitioned maclaurin symmetric mean based on q-rung orthopair fuzzy information for dealing with multi-attribute group decision making. Symmetry, 10.","DOI":"10.3390\/sym10090383"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Xu, Y., Shang, X.P., Wang, J., Wu, W., and Huang, H.Q. (2018). Some q-rung dual hesitant fuzzy Heronian mean operators with their application to multiple attribute group decision-making. Symmetry, 10.","DOI":"10.3390\/sym10100472"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2255","DOI":"10.1002\/int.22028","article-title":"Exponential operation and aggregation operator for q-rung orthopair fuzzy set and their decision-making method with a new score function","volume":"33","author":"Peng","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1002\/int.21927","article-title":"Some q-rung orthopair fuzzy aggregation operators and their applications to multiple-attribute decision making","volume":"33","author":"Liu","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Li, L., Zhang, R.T., Wang, J., Shang, X.P., and Bai, K.Y. (2018). A novel approach to multi-attribute group decision-making with q-rung picture linguistic information. Symmetry, 10.","DOI":"10.3390\/sym10050172"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1002\/int.21933","article-title":"Some q-rung orthopai fuzzy Bonferroni mean operators and their application to multi-attribute group decision making","volume":"33","author":"Liu","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"802","DOI":"10.1002\/int.21968","article-title":"Minkowski-type distance measures for generalized orthopair fuzzy sets","volume":"33","author":"Du","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1426","DOI":"10.1002\/int.21985","article-title":"Some q-rung orthopair fuzzy Heronian mean operators in multiple attribute decision making","volume":"33","author":"Wei","year":"2018","journal-title":"Int. J. Intell. Syst."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Wang, R., and Li, Y.L. (2018). A Novel approach for green supplier selection under a q-rung orthopair fuzzy environment. Symmetry, 10.","DOI":"10.3390\/sym10120687"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"564","DOI":"10.1002\/int.22065","article-title":"Correlation and correlation coefficient of generalized orthopair fuzzy sets","volume":"34","author":"Du","year":"2019","journal-title":"Int. J. Intell. Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"652","DOI":"10.1002\/int.22071","article-title":"Multiple-attribute group decision-making based on power Bonferroni operators of linguistic q-rung orthopair fuzzy numbers","volume":"34","author":"Liu","year":"2019","journal-title":"Int. J. Intell. Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1002\/int.22060","article-title":"New q-rung orthopair fuzzy partitioned Bonferroni mean operators and their application in multiple attribute decision making","volume":"34","author":"Yang","year":"2019","journal-title":"Int. J. Intell. Syst."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Wang, J., Gao, H., Wei, G.W., and Wei, Y. (2019). Methods for multiple-attribute group decision making with q-rung interval-valued orthopair fuzzy information and their applications to the selection of green suppliers. Symmetry, 11.","DOI":"10.3390\/sym11010056"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1057\/jors.1977.45","article-title":"Graph theory with applications","volume":"28","author":"Bondy","year":"1977","journal-title":"J. Oper. Res. Soc."},{"key":"ref_19","unstructured":"Rosenfeld, A. (1975). Fuzzy graphs. Fuzzy Sets and Their Applications to Cognitive and Decision Processes, Academic Press."},{"key":"ref_20","first-page":"48","article-title":"Intuitionistic fuzzy graphs","volume":"18","author":"Parvathi","year":"2006","journal-title":"Sci. World J."},{"key":"ref_21","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_22","doi-asserted-by":"crossref","unstructured":"Shahzadi, S., and Akram, M. (2018). Graphs in an intuitionistic fuzzy soft environment. Axioms, 7.","DOI":"10.3390\/axioms7020020"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1000","DOI":"10.1007\/s40815-017-0368-0","article-title":"A novel decision-making method based on rough fuzzy information","volume":"20","author":"Zafar","year":"2018","journal-title":"Int. J. Fuzzy Syst."},{"key":"ref_24","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_25","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_26","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/S0898-1221(99)00140-6","article-title":"Fuzzy logic and the calculi of fuzzy rules, fuzzy graphs, and fuzzy probabilities","volume":"37","author":"Zadeh","year":"1999","journal-title":"Comput. Math. Appl."},{"key":"ref_27","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_28","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_29","first-page":"145","article-title":"Order and size in fuzzy graph","volume":"22E","author":"Gani","year":"2003","journal-title":"Bull. Pure Appl. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Mordeson, J.N., and Nair, P.S. (2001). Fuzzy Graphs and Fuzzy Hypergraphs, Physica Verlag.","DOI":"10.1007\/978-3-7908-1854-3"},{"key":"ref_31","first-page":"107","article-title":"The degree of a vertex in some fuzzy graphs","volume":"2","author":"Gani","year":"2009","journal-title":"Int. J. Algorithms Comput. Math."},{"key":"ref_32","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_33","first-page":"203","article-title":"Some operations on fuzzy hypergraphs","volume":"132","author":"Gong","year":"2017","journal-title":"ARS Combin."},{"key":"ref_34","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_35","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_36","doi-asserted-by":"crossref","unstructured":"Klement, E., and Mesiar, R. (2018). L-Fuzzy Sets and Isomorphic Lattices: Are All the \u201cNew\u201d Results Really New?. Mathematics, 6.","DOI":"10.3390\/math6090146"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/4\/588\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:46:34Z","timestamp":1760186794000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/4\/588"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,4,23]]},"references-count":36,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2019,4]]}},"alternative-id":["sym11040588"],"URL":"https:\/\/doi.org\/10.3390\/sym11040588","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,4,23]]}}}