×

A new type-2 soft set: type-2 soft graphs and their applications. (English) Zbl 1397.05149

Summary: The correspondence between a vertex and its neighbors has an essential role in the structure of a graph. Type-2 soft sets are also based on the correspondence of primary parameters and underlying parameters. In this study, we present an application of type-2 soft sets in graph theory. We introduce vertex-neighbors based type-2 soft sets over \(X\) (set of all vertices of a graph) and \(\mathcal{E}\) (set of all edges of a graph). Moreover, we introduce some type-2 soft operations in graphs by presenting several examples to demonstrate these new concepts. Finally, we describe an application of type-2 soft graphs in communication networks and present procedure as an algorithm.

MSC:

05C72 Fractional graph theory, fuzzy graph theory
05C76 Graph operations (line graphs, products, etc.)
05C90 Applications of graph theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Rosenfeld, A.; Zadeh, L. A.; Fu, K. S.; Shimura, M., Fuzzy graphs, Fuzzy Sets and Their Applications, 77-95, (1975), New York, NY, USA: Academic Press, New York, NY, USA
[2] Kauffman, A., Introduction a la Theorie des Sous-emsembles Flous, Masson et Cie, 1, (1973) · Zbl 0302.02023
[3] Zadeh, L. A., Fuzzy sets, Information and Computation, 8, 338-353, (1965) · Zbl 0139.24606
[4] Bhattacharya, P., Some remarks on fuzzy graphs, Pattern Recognition Letters, 6, 5, 297-302, (1987) · Zbl 0629.05060 · doi:10.1016/0167-8655(87)90012-2
[5] Mordeson, J. N.; Nair, P. S., Fuzzy Graphs and Fuzzy Hypergraphs. Fuzzy Graphs and Fuzzy Hypergraphs, Studies in Fuzziness and Soft Computing, 46, (2000), Heidelberg, Germany: Physica-Verlag HD, Heidelberg, Germany · Zbl 0947.05082 · doi:10.1007/978-3-7908-1854-3_4
[6] Akram, M., Bipolar fuzzy graphs with applications, Knowledge-Based Systems, 39, 1-8, (2013) · doi:10.1016/j.knosys.2012.08.022
[7] Akram, M.; Dudek, W. A., Intuitionistic fuzzy hypergraphs with applications, Information Sciences, 218, 182-193, (2013) · Zbl 1293.05242 · doi:10.1016/j.ins.2012.06.024
[8] Akram, M.; Akmal, R., Intuitionistic fuzzy graph structures, Kragujevac Journal of Mathematics, 41, 219-237, (2017) · Zbl 1468.05244
[9] Akram, M.; Dudek, W. A., Interval-valued fuzzy graphs, Computers & Mathematics with Applications, 61, 2, 289-299, (2011) · Zbl 1211.05133 · doi:10.1016/j.camwa.2010.11.004
[10] Samanta, S.; Pal, M.; Rashmanlou, H.; Borzooei, R. A., Vague graphs and strengths, Journal of Intelligent and Fuzzy Systems, 30, 6, 3675-3680, (2016) · Zbl 1361.05108 · doi:10.3233/IFS-162113
[11] Samanta, S.; Pal, M., Fuzzy Planar Graphs, IEEE Transactions on Fuzzy Systems, 23, 6, 1936-1942, (2015) · doi:10.1109/TFUZZ.2014.2387875
[12] Samanta, S.; Pal, M., Irregular bipolar fuzzy graphs, International Journal of Fuzzy Systems, 2, 91-102, (2012)
[13] Molodtsov, D., Soft set theory—first results, Computers & Mathematics with Applications, 37, 4-5, 19-31, (1999) · Zbl 0936.03049 · doi:10.1016/S0898-1221(99)00056-5
[14] Maji, P. K.; Biswas, R.; Roy, A. R., Soft set theory, Computers & Mathematics with Applications, 45, 4-5, 555-562, (2003) · Zbl 1032.03525 · doi:10.1016/S0898-1221(03)00016-6
[15] Aktaş, H.; Çağman, N., Soft sets and soft groups, Information Sciences, 177, 13, 2726-2735, (2007) · Zbl 1119.03050 · doi:10.1016/j.ins.2006.12.008
[16] Ali, M. I.; Feng, F.; Liu, X.; Min, W. K.; Shabir, M., On some new operations in soft set theory, Computers & Mathematics with Applications, 57, 9, 1547-1553, (2009) · Zbl 1186.03068 · doi:10.1016/j.camwa.2008.11.009
[17] Sezgin, A.; Atagün, A. O., On operations of soft sets, Computers & Mathematics with Applications, 61, 5, 1457-1467, (2011) · Zbl 1217.03040 · doi:10.1016/j.camwa.2011.01.018
[18] Maji, P. K.; Biswas, R.; Roy, A. R., Fuzzy soft sets, Journal of Fuzzy Mathematics, 9, 589-602, (2001) · Zbl 0995.03040
[19] Ali, M. I.; Shabir, M.; Feng, F., Representation of graphs based on neighborhoods and soft sets, International Journal of Machine Learning and Cybernetics, 8, 5, 1525-1535, (2017) · doi:10.1007/s13042-016-0525-z
[20] Akram, M.; Nawaz, S., Operation on soft graph, Fuzzy Information and Engineering, 7, 423-449, (2015)
[21] Akram, M.; Nawaz, S., Certian types of soft graphs, Scientific Bulletin-University Politehnica of Bucharest, Series A, 78, 67-82, (2016) · Zbl 1513.05312
[22] Akram, M.; Nawaz, S., On fuzzy soft graphs, Italian Journal of Pure and Applied Mathematics, 34, 497-514, (2015) · Zbl 1329.05248
[23] Bondy, J. A.; Murty, U. S. R., Graph Theory with Applications, (1976), New York, NY, USA: Macmillan Press, New York, NY, USA · Zbl 1226.05083 · doi:10.1007/978-1-349-03521-2
[24] Brouwer, A. E.; Cohen, A. M.; Neumaier, A., Distance-Regular Graphs, 18, (1989), New York, NY, USA: Springer, New York, NY, USA · Zbl 0747.05073 · doi:10.1007/978-3-642-74341-2
[25] Tutte, W. T., Graph Theory as I Have Known It, (1998), Oxford, England: Oxford University Press, Oxford, England · Zbl 0915.05041 · doi:10.2307/3621554
[26] Chatterjeea, R.; Majumdar, P.; Samanta, S. K., Type-2 soft sets, Journal of Intelligent & Fuzzy Systems, 29, 885-898, (2015) · Zbl 1352.68240
[27] Chatterjeea, R.; Majumdar, P.; Samanta, S. K., Distance, entropy and similarity measures of Type-2 soft sets · doi:10.1063/1.4980963
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.