{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:40:23Z","timestamp":1760179223032,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2020,9,23]],"date-time":"2020-09-23T00:00:00Z","timestamp":1600819200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100014881","name":"Guangzhou university","doi-asserted-by":"publisher","award":["20200115-9"],"award-info":[{"award-number":["20200115-9"]}],"id":[{"id":"10.13039\/501100014881","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Key R &amp; D Program of China","award":["2018YFB1005104"],"award-info":[{"award-number":["2018YFB1005104"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Fuzzy graph models enjoy the ubiquity of being in natural and human-made structures, namely dynamic process in physical, biological and social systems. As a result of inconsistent and indeterminate information inherent in real-life problems which are often uncertain, it is highly difficult for an expert to model those problems based on a fuzzy graph (FG). Vague graph structure (VGS) can deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problem, where fuzzy graphs may fail to reveal satisfactory results. Likewise, VGSs are very useful tools for the study of different domains of computer science such as networking, capturing the image, clustering, and also other issues like bioscience, medical science, and traffic plan. The limitations of past definitions in fuzzy graphs have led us to present new definitions in VGSs. Operations are conveniently used in many combinatorial applications. In various situations, they present a suitable construction means; therefore, in this research, three new operations on VGSs, namely, maximal product, rejection, residue product were presented, and some results concerning their degrees and total degrees were introduced. Irregularity definitions have been of high significance in the network heterogeneity study, which have implications in networks found across biology, ecology and economy; so special concepts of irregular VGSs with several key properties were explained. Today one of the most important applications of decision making is in medical science for diagnosing the patient\u2019s disease. Hence, we recommend an application of VGS in medical diagnosis.<\/jats:p>","DOI":"10.3390\/sym12101582","type":"journal-article","created":{"date-parts":[[2020,9,24]],"date-time":"2020-09-24T03:03:39Z","timestamp":1600916619000},"page":"1582","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":40,"title":["Vague Graph Structure with Application in Medical Diagnosis"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1427-5473","authenticated-orcid":false,"given":"Saeed","family":"Kosari","sequence":"first","affiliation":[{"name":"Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9615-3658","authenticated-orcid":false,"given":"Yongsheng","family":"Rao","sequence":"additional","affiliation":[{"name":"Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Huiqin","family":"Jiang","sequence":"additional","affiliation":[{"name":"Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xinyue","family":"Liu","sequence":"additional","affiliation":[{"name":"Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pu","family":"Wu","sequence":"additional","affiliation":[{"name":"Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0764-4135","authenticated-orcid":false,"given":"Zehui","family":"Shao","sequence":"additional","affiliation":[{"name":"Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,23]]},"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","doi-asserted-by":"crossref","first-page":"610","DOI":"10.1109\/21.229476","article-title":"Vague sets","volume":"23","author":"Gau","year":"1993","journal-title":"IEEE Trans. Syst. Man Cybern."},{"key":"ref_3","first-page":"607","article-title":"Introduction a la theories des sous-emsembles 503 ous","volume":"1","author":"Kauffman","year":"1973","journal-title":"Masson Cie"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/S0020-0255(71)80005-1","article-title":"Similarity relations and fuzzy ordering","volume":"3","author":"Zadeh","year":"1971","journal-title":"Inf. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2751","DOI":"10.1016\/j.ins.2008.02.012","article-title":"Is there a need for fuzzy logical","volume":"178","author":"Zadeh","year":"2008","journal-title":"Inf. Sci."},{"key":"ref_6","first-page":"365","article-title":"Residue product of fuzzy graph structures","volume":"34","author":"Akram","year":"2020","journal-title":"J. Multiple-Valued Log. Soft Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1007\/s12190-019-01237-2","article-title":"Certain fuzzy graph structures","volume":"61","author":"Akram","year":"2019","journal-title":"J. Appl. Math. Comput."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Akram, M., and Muzzamal, S. (2017). Certain concepts in Intuitionistic Neutrosophic Graph Structures. Information, 8.","DOI":"10.3390\/info8040154"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Akram, M., Naz, S., and Smarandache, F. (2019). Generalization of Maximizing Deviation and TOPSIS Method for MADM in Simplified Neutrosophic Hesitant Fuzzy Environment. Symmetry, 11.","DOI":"10.3390\/sym11081058"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Asif, M., Akram, M., and Ali, G. (2020). Pythagorean Fuzzy Matroids with Application. Symmetry, 12.","DOI":"10.3390\/sym12030423"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Sitara, M., Akram, M., and Yousaf Bhatti, M. (2019). Fuzzy graph structures with application. Mathematics, 7.","DOI":"10.3390\/math7010063"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3291","DOI":"10.3233\/IFS-152077","article-title":"Vague graphs with application","volume":"30","author":"Rashmanlou","year":"2016","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"571","DOI":"10.3233\/IFS-141333","article-title":"A study on bipolar fuzzy graphs","volume":"28","author":"Rashmanlou","year":"2015","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"371","DOI":"10.3233\/IFS-151762","article-title":"Product vague graphs and its applications","volume":"30","author":"Rashmanlou","year":"2016","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/03081079.2015.1072521","article-title":"Product of bipolar fuzzy graphs and their degree","volume":"45","author":"Rashmanlou","year":"2016","journal-title":"Int. J. Gen. Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"808","DOI":"10.1080\/18756891.2015.1063243","article-title":"Bipolar fuzzy graphs with Categorical properties","volume":"8","author":"Rashmanlou","year":"2015","journal-title":"Int. J. Comput. Intell. Syst."},{"key":"ref_17","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_18","first-page":"107","article-title":"The degree of a vertex in some fuzzy graphs","volume":"2","author":"Radha","year":"2009","journal-title":"Int. J. Algorithms Comput. Math."},{"key":"ref_19","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_20","unstructured":"Zadeh, L.A., Fu, K.S., and Shimura, M. (1975). Fuzzy Graphs, Academic Press. Fuzzy Sets and Their Applications."},{"key":"ref_21","first-page":"51","article-title":"Vague graphs","volume":"7","author":"Ramakrishna","year":"2009","journal-title":"Int. J. Comput. Cogn."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"16","DOI":"10.5373\/jarpm.1971.021614","article-title":"Ring sum in product intuitionistic fuzzy graphs","volume":"7","author":"Borzooei","year":"2015","journal-title":"J. Adv. Res. Pure Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1933","DOI":"10.3233\/IFS-151671","article-title":"Domination in vague graphs and its applications","volume":"29","author":"Borzooei","year":"2015","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"545","DOI":"10.14317\/jami.2015.545","article-title":"Degree of vertices in vague graphs","volume":"33","author":"Borzooei","year":"2015","journal-title":"J. Appl. Math. Inform."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"3681","DOI":"10.3233\/IFS-162114","article-title":"Regularity of vague graphs","volume":"30","author":"Borzooei","year":"2016","journal-title":"J. Intell. Fuzzy Syst."},{"key":"ref_26","first-page":"33","article-title":"On regular fuzzy graphs","volume":"12","author":"Radha","year":"2008","journal-title":"J. Phys. Sci."},{"key":"ref_27","first-page":"61","article-title":"Conjunction of two fuzzy graphs","volume":"3","author":"Radha","year":"2008","journal-title":"Int. Rev. Fuzzy Math."},{"key":"ref_28","first-page":"321","article-title":"Some sequences in fuzzy graphs","volume":"31","author":"Radha","year":"2008","journal-title":"Far East J. Appl. Math."},{"key":"ref_29","first-page":"517","article-title":"On irregular fuzzy graphs","volume":"6","author":"Radha","year":"2012","journal-title":"Appl. Math. Sci."},{"key":"ref_30","first-page":"91","article-title":"Irregular bipolar fuzzy graphs","volume":"2","author":"Samanta","year":"2012","journal-title":"Int. J. Appl. Fuzzy Sets"},{"key":"ref_31","first-page":"65","article-title":"Generalized graph structures","volume":"3","author":"Sampathkumar","year":"2006","journal-title":"Bull. Kerala Math. Assoc."},{"key":"ref_32","unstructured":"Dinesh, T. (2011). A Study on Graph Structures. Incidence Algebras and Their Fuzzy Analogues. [Ph.D. Thesis, Kannur University]."},{"key":"ref_33","first-page":"173","article-title":"On generalised fuzzy graph structures","volume":"5","author":"Ramakrishnan","year":"2011","journal-title":"Appl. Math. Sci."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"5","DOI":"10.37622\/AFM\/6.1.2011.5-12","article-title":"On generalised fuzzy graph structures II","volume":"6","author":"Ramakrishnan","year":"2011","journal-title":"Adv. Fuzzy Math."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Shao, Z., Kosari, S., Rashmanlou, H., and Shoaib, M. (2020). New concepts in intuitionistic fuzzy graph with application in water supplier systems. Mathematics, 8.","DOI":"10.3390\/math8081241"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/10\/1582\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:13:01Z","timestamp":1760177581000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/12\/10\/1582"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,23]]},"references-count":35,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2020,10]]}},"alternative-id":["sym12101582"],"URL":"https:\/\/doi.org\/10.3390\/sym12101582","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2020,9,23]]}}}