# Bounds for fuzzy Zagreb Estrada index

Document Type : Original paper

Authors

1 NMIMS Deemed tobe University, Mumbai.

2 NMIMS Deemed to University

Abstract

Let $G(V,\sigma ,\mu )$ be a fuzzy graph of order $n$, where $\sigma(u)$ is the vertex membership, $\mu(u,v)$ is membership value of an edge and $\mu (u)$ is the strength of vertex. The first fuzzy Zagreb index is the sum $\sigma ({{u}_{i}})\mu ({{u}_{i}})+\sigma ({{u}_{j}})\mu ({{u}_{j}})$ where ${{{u}_{i}}{{u}_{j}}\in {{\mu }}}$ and the corresponding fuzzy Zagreb matrix is the square matrix of order $n$ whose $(i,j)^{th}$ entry whenever $i\neq j$, is $\sigma ({{u}_{i}})\mu ({{u}_{i}})+\sigma ({{u}_{j}})\mu ({{u}_{j}})$ and zero otherwise. In this paper, we introduce the Zagreb Estrada index of fuzzy graphs and establish some bounds for it.

Keywords

Main Subjects

#### References

[1] N. Anjali and S. Mathew, Energy of a fuzzy graph, Annals Fuzzy Math. Inform. 6 (2013), no. 3, 455–465.
[2] M. Binu, S. Mathew, and J.N. Mordeson, Wiener index of a fuzzy graph and application to illegal immigration networks, Fuzzy Sets and Systems 384 (2020), 132–147.
[3] B. Borovicanin, K.C. Das, B. Furtula, and I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem. 78 (2017), no. 1, 17–100.
[4] A.N. Gani and K. Radha, The degree of a vertex in some fuzzy graphs, Int. J. Algorithms Comput. Math. 2 (2009), 107–116.
[5] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. total ϕ-electron energy of alternant hydrocarbons, Chemical Physics Letters 17 (1972), no. 4, 535–538.
[6] S.R. Islam, S. Maity, and M. Pal, Comment on “Wiener index of a fuzzy graph and application to illegal immigration networks”, Fuzzy Sets and Systems 384 (2020), 148–151.
[7] S.R. Islam and M. Pal, First Zagreb index on a fuzzy graph and its application, Journal of Intelligent & Fuzzy Systems 40 (2021), no. 6, 10575–10587.
[8] , Hyper-Wiener index for fuzzy graph and its application in share market, Journal of Intelligent & Fuzzy Systems 41 (2021), no. 1, 2073–2083.
[9] M. Kale and S. Minirani, On Zagreb indices of graphs with a deleted edge, Annals Pure Appl. Math. 21 (2020), no. 1, 1–14.
[10] , Fuzzy Zagreb indices and some bounds for fuzzy Zagreb energy, Int. J. Analysis Appl. 19 (2021), no. 2, 252–263.
[11] A. Lotfi Zadeh, Fuzzy sets, Information and control 8 (1965), no. 3, 338–353.
[12] S. Mathew and M.S. Sunitha, Types of arcs in a fuzzy graph, Inform. Sci. 179 (2009), no. 11, 1760–1768.
[13] J.N. Mordeson and S. Mathew, Advanced Topics in Fuzzy Graph Theory, Springer, 2019.
[14] J.N. Mordeson and P.S. Nair, Fuzzy Graphs and Fuzzy Hypergraphs, vol. 46, Physica, 2012.
[15] A. Nagoorgani and V.T. Chandrasekaran, A First Look at Fuzzy Graph Theory, Allied Publication Pvt. Ltd, 2010.
[16] A. Nagoorgani and K. Ponnalagu, A new approach on solving intuitionistic fuzzy linear programming problem, Appl. Math. Sci. 6 (2012), no. 70, 3467–3474.
[17] M. Pal, S. Samanta, and G. Ghorai, Modern Trends in fuzzy graph theory, Springer, 2020.
[18] S.S. Rahimi and F. Fayazi, Laplacian energy of a fuzzy graph, 5 (2014), no. 1, 1–10.
[19] A. Rosenfeld, Fuzzy graphs, Fuzzy sets and their applications to cognitive and decision processes, Elsevier, 1975, pp. 77–95.
[20] M.S. Sunitha and A. Vijayakumar, Complement of a fuzzy graph, Indian J. Pure Appl. Math. 33 (2002), no. 9, 1451–1464.
[21] M.G. Thomason, Convergence of powers of a fuzzy matrix, J. Math. Analysis Appl. 57 (1977), no. 2, 476–480.
[22] H.-J. Zimmermann, Fuzzy set theory and mathematical programming, Fuzzy Sets Theory and Applications, Springer, 1986, pp. 99–114.