Note for Soft MultiExpert Graph and MultiSoft MultiExpert Graph

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Published: Mar 18, 2026
DOI: https://doi.org/10.26554/integrajimcs.20263146
Keywords:
Soft Graph, Soft Expert Graph, Soft Multi Expert Graph, Multi Soft Multi Expert Graph, Multi Soft Graft

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Takaaki Fujita
Independent Researcher, Tokyo, Japan

Abstract

This paper studies multilevel extensions of soft-set-based graph models for uncertainty aware decision support. We first recall soft, multisoft, soft expert, and multisoft multi expert sets, which encode parameterized and expert-dependent approximations of a universe. Building on these notions, we introduce the MultiSoft Graph, a family of induced subgraphs of a given graph indexed by multilevel parameter blocks, and show that it strictly generalizes classical soft graphs while inducing a canonical multisoft set on the vertex set. We then define Soft MultiExpert Graphs and MultiSoft MultiExpert Graphs, providing a unified framework that jointly handles graph topology, multiple parameters, and expert opinions.

Article Details

Issue
Vol. 3 No. 1 (2026): March
Section
Articles

References

[1] Pradip Kumar Maji, Ranjit Biswas, and A Ranjan Roy. Soft set theory. Computers & Mathematics with Applications, 45(4-5):555–562, 2003.

[2] Muhammad Saqlain and Xiao Long Xin. Interval valued, m-polar and m-polar interval valued neutrosophic hypersoft sets. Neutrosophic Sets and Systems, 34(1):52–76, 2020.

[3] Florentin Smarandache. Extension of soft set to hypersoft set, and then to plithogenic hypersoft set. Neutrosophic Sets and Systems, 22(1):168–170, 2018.

[4] Fujita Takaaki and Arif Mehmood. Iterative multifuzzy set, iterative multineutrosophic set, iterative multisoft set, and multiplithogenic sets. Neutrosophic Computing and Machine Learning, 41(1):1–30, 2025.

[5] Shawkat Alkhazaleh, Abdul Razak Salleh, Nasruddin Hassan, and Abd Ghafur Ahmad. Multisoft sets. In Proceedings of the 2nd International Conference on Mathematical Sciences, pages 910–917, 2010.

[6] Murat Kirici. Decision making method related to pythagorean fuzzy soft sets with infectious diseases application. Journal of King Saud University - Computer and Information Sciences, 34(8):5968–5978, 2021.

[7] Ashraf Al-Quran, Nasruddin Hassan, and Shawkat Alkhazaleh. Fuzzy parameterized complex neutrosophic soft expert set for decision under uncertainty. Symmetry, 11(3):382, 2019.

[8] Ashraf Al-Quran, Nasruddin Hassan, and Emad A Marei. A novel approach to neutrosophic soft rough set under uncertainty. Symmetry, 11(3):384, 2019.

[9] Siyang Yang. Extending superhypersoft framework: Weighted soft sets for priority-based decision-making in engineering ethics risk analysis based on big data technology. Neutrosophic Sets and Systems, 86(1):119–125, 2025.

[10] Florentin Smarandache. Practical applications of indetermsoft set and indetermhypersoft set and introduction to treesoft set as an extension of the multisoft set. Infinite Study, 2022.

[11] Takaaki Fujita and Florentin Smarandache. Harnessing quantum superposition in soft set theory: Introducing quantum hypersoft and superhypersoft sets. European Journal of Pure and Applied Mathematics, 18(3):6607–6607, 2025.

[12] Florentin Smarandache. Foundation of the superhypersoft set and the fuzzy extension superhypersoft set: A new vision. Neutrosophic Systems with Applications, 11(1):48–51, 2023.

[13] Takaaki Fujita, Raed Hatamleh, and Ahmed Salem Heilat. Contrasoft set and contrarough set with using upside-down logic. Statistics, Optimization & Information Computing, 13(1), 2025.

[14] Muhammad Ihsan, Atiqe Ur Rahman, and Muhammad Haris Saeed. Single valued neutrosophic hypersoft expert set with application in decision making. Neutrosophic Sets and Systems, 44:118–129, 2021.

[15] Muhammad Akram, Ghous Ali, Xindong Peng, and Muhammad Zain Ul Abidin. Hybrid group decision-making technique under spherical fuzzy n-soft expert sets. Artificial Intelligence Review, 55:4117–4163, 2021.

[16] Vakkas Ulucay, Zeynep Baser, and Seyyed Ahmad Edalatpanah. Euclidean distance on treesoft sets and application for electric vehicle selection. Journal of Fuzzy Extension and Applications, 6(1):e232762, 2025.

[17] Florentin Smarandache. Treesoft set vs. hypersoft set and fuzzy-extensions of treesoft sets. Biblio Publishing, 2024.

[18] Hairong Luo. Forestsoft set approach for estimating innovation and entrepreneurship education in universities through a hierarchical and uncertainty-aware analytical framework. Neutrosophic Sets and Systems, 86:332–342, 2025.

[19] Li Song, Jianyong Liu, Han Ding, and Wenhui Zhang. Forestsoft set for mechanical automation production control systems analysis based on an intelligent manufacturing environment. Neutrosophic Sets and Systems, 85:229–254, 2025.

[20] Jinta Jose, Bobin George, and Rajesh K Thumbakara. Soft graphs: A comprehensive survey. New Mathematics and Natural Computation, 20(01):1–52, 2024.

[21] Muhammad Akram and Saira Nawaz. Operations on soft graphs. Fuzzy Information and Engineering, 7(4):423–449, 2015.

[22] Hao Jin. Valuation of building facade design effects using forestsoft hypergraphs and virtual simulation technology. Neutrosophic Sets and Systems, 86(1):11, 2025.

[23] Yifan Feng, Haoxuan You, Zizhao Zhang, Rongrong Ji, and Yue Gao. Hypergraph neural networks. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pages 3558–3565, 2019.

[24] Claude Berge. Hypergraphs: Combinatorics of Finite Sets, volume 45. Elsevier, 1984.

[25] Florentin Smarandache. Extension of hypergraph to n-superhypergraph and to plithogenic n-superhypergraph, and extension of hyperalgebra to n-ary (classical-/neutro-/anti-) hyperalgebra. Infinite Study, 2020.

[26] Masoud Ghods, Zahra Rostami, and Florentin Smarandache. Introduction to neutrosophic restricted superhypergraphs and neutrosophic restricted superhypertrees and several of their properties. Neutrosophic Sets and Systems, 50:480–487, 2022.

[27] Dmitriy Molodtsov. Soft set theory-first results. Computers & Mathematics with Applications, 37(4-5):19–31, 1999.

[28] Piyu Li, Zhi Kong, Wen-Li Liu, and Chang-Tao Xue. On multi-soft rough sets. In 29th Chinese Control And Decision Conference (CCDC), pages 3051–3055. IEEE, 2017.

[29] Ashraf Al-Quran and Nasruddin Hassan. The complex neutrosophic soft expert set and its application in decision making. Journal of Intelligent & Fuzzy Systems, 34(1):569–582, 2018.

[30] Muhammad Ihsan, Muhammad Saeed, and Atiqe Ur Rahman. Multi-attribute decision-making application based on pythagorean fuzzy soft expert set. International Journal of Information and Decision Sciences, 16(4):383–408, 2024.

[31] Abid Khan, Muhammad Zainul Abidin, and Muhammad Amad Sarwar. Another view on soft expert set and its application in multi-criteria decision-making. Mathematics, 13(2):252, 2025.

[32] Mehmet Şahin, İrfan Deli, and Vakkas Uluçay. Bipolar neutrosophic soft expert sets. Infinite Study, 2015.

[33] Takaaki Fujita. Note for soft multiexpert set and multisoft multiexpert set. Preprint, 2025.

[34] S Satham Hussain, R Jahir Hussain, Ghulam Muhiuddin, and P Anitha. Operations on neutrosophic vague soft graphs. Neutrosophic Sets and Systems, 51:254–266, 2022.

[35] Vakkas Uluçay. Q-neutrosophic soft graphs in operations management and communication network. Soft Computing, 25:8441–8459, 2021.

[36] Takaaki Fujita. A comprehensive discussion on fuzzy hypersoft expert, superhypersoft, and indetermsoft graphs. Neutrosophic Sets and Systems, 77:241–263, 2025.

[37] Azriel Rosenfeld. Fuzzy graphs. In Fuzzy Sets and Their Applications to Cognitive and Decision Processes, pages 77–95. Elsevier, 1975.

[38] John N Mordeson and Premchand S Nair. Fuzzy graphs and fuzzy hypergraphs, volume 46. Physica, 2012.

[39] Said Broumi, Mohamed Talea, Assia Bakali, and Florentin Smarandache. Single valued neutrosophic graphs. Journal of New Theory, (10):86–101, 2016.

[40] Liangsong Huang, Yu Hu, Yuxia Li, PK Kishore Kumar, Dipak Koley, and Arindam Dey. A study of regular and irregular neutrosophic graphs with real life applications. Mathematics, 7(6):551, 2019.

[41] Said Broumi, Mohamed Talea, Assia Bakali, and Florentin Smarandache. Interval valued neutrosophic graphs. Critical Review, 12:5–33, 2016.

[42] Xiaolong Shi, Saeed Kosari, Hossein Rashmanlou, Said Broumi, and S Satham Hussain. Properties of interval-valued quadripartitioned neutrosophic graphs with real-life application. Journal of Intelligent & Fuzzy Systems, 44(5):7683–7697, 2023.

[43] Hong-Yu Yao, Chun-Yang Zhang, Qian-Xi Tang, and CL Philip Chen. Investigating all uncertainties in hypergraph representation learning and inference. IEEE Transactions on Fuzzy Systems, 33(8):2867–2881, 2025.

[44] Mehbooba P Shareef, Babita Roslind Jose, Jimson Mathew, and PB Ramkumar. Indeterminacy-driven trade-off in reinforcement learning on neutrosophic fuzzy hypergraphs for explainable item recommendation with path-compliant rewards. IEEE Transactions on Emerging Topics in Computational Intelligence, 2025.

[45] Mehbooba Shareef, Babita Roslind Jose, Jimson Mathew, and Dayananda Pruthviraja. Advancing explainable mooc recommendation systems: a morphological operations-based framework on partially ordered neutrosophic fuzzy hypergraphs. Artificial Intelligence Review, 58(2):46, 2024.

[46] Muhammad Akram, Hafsa M Malik, Sundas Shahzadi, and Florentin Smarandache. Neutrosophic soft rough graphs with application. Axioms, 7(1):14, 2018.

[47] Fazeelat Sultana, Muhammad Gulistan, Mumtaz Ali, Naveed Yaqoob, Muhammad Khan, Tabasam Rashid, and Tauseef Ahmed. A study of plithogenic graphs: applications in spreading coronavirus disease (covid-19) globally. Journal of Ambient Intelligence and Humanized Computing, 14(10):13139–13159, 2023.

[48] Takaaki Fujita and Florentin Smarandache. A review of the hierarchy of plithogenic, neutrosophic, and fuzzy graphs: Survey and applications. In Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond (Second Volume). Biblio Publishing, 2024.