The Relation Between some Polynomials and the Wiener Index of Fuzzy Graphs

Subject Areas : Fuzzy Optimization and Modeling Journal

Seyed Sadegh Salehi Amiri ^{1 *}

1 - Department of Mathematics, Babol Branch, Islamic Azad University, Babol, Iran.

Received: 2023-04-01 Accepted : 2023-06-07 Published : 2023-06-07

Keywords: Fuzzy graph, Fuzzy Wiener index, Fuzzy graph degree-based polynomial,

Abstract :

In this paper, the distance between two vertices in a fuzzy graph is defined in a new way. In addition, some new degree-based fuzzy graph polynomials are introduced. By using this definition, fuzzy graph polynomials, and a special lower triangular matrix, the Wiener index and the generalized Wiener index of a fuzzy graph are computed, which coincide with the Wiener index and the generalized Wiener index in the crisp graph. The result is used to compute the Wiener index of the sum, products, and composition of two fuzzy graphs.

References:

1. Bhutani, K. R. (1989). On automorphisms of fuzzy graphs. Pattern recognition letters, 9(3), 159-162.
2. Binu, M., Mathew, S., & Mordeson, J. N. (2019). Connectivity index of a fuzzy graph and its application to human trafficking. Fuzzy Sets and Systems, 360, 117-136.
3. Dobrynin, A. A., & Kochetova, A. A. (1994). Degree distance of a graph: A degree analog of the Wiener index. Journal of Chemical Information and Computer Sciences, 34(5), 1082-1086.
4. Fayazi, F., Mahmudi, F., & Gholami, A. (2020). Some properties of fuzzy Star graph and fuzzy Line graph. Jordan Journal of Mathematics and Statistics, 13(1), 139-151.
5. Kaufmann, A. (1973). Introduction a la theorie des Sour-ensembles flous, Masson et Cie: Paris, French.
6. Kalathian, S., Ramalingam, S., Raman, S., & Srinivasan, N. (2020). Some topological indices in fuzzy graphs. Journal of Intelligent & Fuzzy Systems, 39(5), 6033-6046.
7. Mathew, S., Mordeson, J. N., & Malik, D. S. (2018). Fuzzy graph theory (Vol. 363). Berlin, Germany: Springer .
8. Gani, A. N., & Ahamed, M. B. (2003). Order and size in fuzzy graphs. Bulletin of pure and applied sciences, 22(1), 145-148.
9. Rosenfeld, A., Fuzzy Graph, In: L.A. Zadeh, K.S. Fu and M. Shimura, Editors, Fuzzy setsand their applications to cognitive and decision Process, Academic press, New York, (1975), 77-95.
10. Sedghi, S., Shobe, N., Salahshour, M. A., & Gutman, I. (2013). Relations between the polynomials of a graph and wiener index. Kragujevac Journal of Science, (35), 77-88.
11. Wiener, H. (1947). Structural determination of paraffin boiling points. Journal of the American chemical society, 69(1), 17-20.
12. Yeh, Y. N., & Gutman, I. (1994). On the sum of all distances in composite graphs. Discrete Mathematics, 135(1-3), 359-365.
13. Zadeh, L. A. (1971). Similarity relations and fuzzy orderings. Information sciences, 3(2), 177-200.