Skip to main content
Springer Nature Link
Log in

Results on soft graphs and soft multigraphs with application in controlling urban traffic flows

  • Fuzzy systems and their mathematics
  • Published:
Soft Computing Aims and scope Submit manuscript

Abstract

In this paper, first we introduce the notions of size, order, and degree of vertices and edges in a soft graph, and we give some examples for a better understanding of these concepts. Then, we define the fundamental concepts of the adjacency matrix of soft graphs, complement of soft graphs, and regularity based on the degree of vertices of soft graphs, and we investigate some related results. Moreover, we introduce the concept of soft multiset and soft multigraph and define the notions of union, intersection, complement, and sum, and then we investigate the relations among them. Finally, we give an application of soft graphs in controlling urban traffic flows.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

Data availability

Enquiries about data availability should be directed to the authors.

References

  • Afsharmanesh S, Borzooei RA, Deldar M (2022) Domination in fuzzy incidence graphs based on valid edges. J Appl Math Comput 68:101–124

    Article  MathSciNet  MATH  Google Scholar 

  • Akram M, Nawaz S (2015) Operations on soft graphs. Fuzzy Inform Eng 7(4):423–449

    Article  MathSciNet  MATH  Google Scholar 

  • Akram M, Nawaz S (2015) On fuzzy soft graphs. Italian J Pure Appl Math 34:497–514

    MathSciNet  MATH  Google Scholar 

  • Akram M, Nawaz S (2016) Fuzzy soft graphs with applications. J Intell Fuzzy Syst 30(6):3619–3632

    Article  MATH  Google Scholar 

  • Aktas H, Cagman N (2007) Soft sets and soft graphs. Inf Sci 177:2727–2735

    Article  MATH  Google Scholar 

  • Ali MI, Feng F, Liu X, Min WK, Shabir M (2009) On some new operations in soft set theory. Comput Math Appl 57:1547–1553

    Article  MathSciNet  MATH  Google Scholar 

  • Babitha KV, Sunil JJ (2013) On soft multi sets. Ann Fuzzy Math Inform 5(1):35–44

    MathSciNet  MATH  Google Scholar 

  • Baghernejad M, Borzooei RA (2019) Vague multigraphs. Soft Comput 23:12607–12619

    Article  MATH  Google Scholar 

  • Baowen L, Peizhuang W, Xihui L, Yong S (1988) Fuzzy bags and relations with set-valued statistics. Comput Math Appl 15(10):811–818

    Article  MathSciNet  MATH  Google Scholar 

  • Blizard WD (1988) Multiset theory. Notre Dame J Formal Logic 30(1):36–66

    Article  MathSciNet  MATH  Google Scholar 

  • Bondy JA, Murty USR (1976) Graph theory with applications. Nurth-Holand add publisher

    Book  MATH  Google Scholar 

  • Borzooei RA, Almallah R (2022) Inverse fuzzy multigraphs and planarity with application in decision-making. Soft Comput 26:1531–1539

    Article  Google Scholar 

  • Diestel R (2017) Graph theory, 5th edn. Springer

    Book  MATH  Google Scholar 

  • Fathalian M, Borzooei RA, Hamidi M (2019) Fuzzy magic labeling of simple graphs. J Appl Math Comput 60:369–385

    Article  MathSciNet  MATH  Google Scholar 

  • Herawan T, Mustafa MD (2009) On multi-soft sets construction in information systems. In: ICIC 2009 LNAI, vol 5755, pp 101–110

  • Jean SP, Gosh SK, Tripathy BK (2001) On the theory of bags and lists. Inf Sci 132:241–254

    Article  MathSciNet  Google Scholar 

  • Khalili M, Borzooei RA, Deldar M (2021) Matching numbers in fuzzy graphs. J Appl Math Comput 67(1–2):1–22

    Article  MathSciNet  MATH  Google Scholar 

  • Maji PK, Roy AR, Biswas R (2001) Fuzzy soft-sets. J Fuzzy Math 9(3):589–602

    MathSciNet  MATH  Google Scholar 

  • Maji PK, Roy AR, Biswas R (2002) An application of soft set in a decision making problem. Comput Math Appl 44(8–9):1077–1083

    Article  MathSciNet  MATH  Google Scholar 

  • Maji PK, Roy AR, Biswas R (2003) Soft set theory. Comput Math Appl 45(4–5):555–562

    Article  MathSciNet  MATH  Google Scholar 

  • Molodtsov DA (1999) Soft set theory-first results. Comput Math Appl 37:19–31

    Article  MathSciNet  MATH  Google Scholar 

  • Rosenfeld A (1975) Fuzzy sets and their applications to cognitive and decision processes. In: Proceedings of The U.S.-Japan seminar on fuzzy setsand their application, Held at the University of California, Berkeley, California, July 1–4, 1974, Academic Press, NewYork, NY, USA, pp 77–95

  • Roy AR, Maji PK (2007) A fuzzy soft set theoretic approach to decision making problems. J Comput Appl Math 203(2):412–418

    Article  MATH  Google Scholar 

  • Samanta S, Pal M (2013) Concept of fuzzy planar graphs. Proc Sci Inform Conf 5–7:557–563

  • Thumbakara RK, George B (2014) Soft graphs. General Math Notes 21(2):75–86

    Google Scholar 

  • Tripathy BK, Arun KR (2015) A new approach to soft sets, soft multisets and their properties. Int J Reason Based Intell Syst 7(3/4):244–253

    MathSciNet  Google Scholar 

  • Wang Q, Zhan J, Borzooei RA (2017) A study on soft rough semigroups and corresponding decision making applications. Open Math 15:1400–1413

    Article  MathSciNet  MATH  Google Scholar 

  • Yager RR (1986) On the theory of bags. General Syst 13:23–37

    Article  MathSciNet  Google Scholar 

  • Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353

    Article  MATH  Google Scholar 

  • Zadeh LA (1975) The Concept of a linguistic and application to approximate reasoning I. Inf Sci 8:199–249

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran

    M. Baghernejad

  2. Department of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

    R. A. Borzooei

Authors
  1. M. Baghernejad
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. R. A. Borzooei
    View author publications

    You can also search for this author inPubMed Google Scholar

Contributions

All authors have equal contributions.

Corresponding author

Correspondence to R. A. Borzooei.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interest.

Human and animal rights

This article does not contain any studies with human participants or animals performed by any of the authors.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baghernejad, M., Borzooei, R.A. Results on soft graphs and soft multigraphs with application in controlling urban traffic flows. Soft Comput 27, 11155–11175 (2023). https://doi.org/10.1007/s00500-023-08650-7

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00500-023-08650-7

Keywords