FUZZY ZAGREB INDICES AND SOME BOUNDS FOR FUZZY ZAGREB ENERGY

. Topological indices M 1 / M 2 known as ﬁrst/second Zagreb indices are deﬁned as the sum of the sum/product of degrees of pairs of adjacent vertices of a simple graph. These indices and their properties have been studied in detail under chemical graph theory. In this paper we introduce the concepts of ﬁrst, second and hyper Zagreb indices of fuzzy graphs. We also study the Zagreb matrices and the associated Zagreb energies of fuzzy graphs. Some bounds for these energies are also obtained.


Introduction
Topological indices are numerical quantities of structural molecular graphs. They are studied and applied in various fields by engineers, pharmacist, graph theorist and mathematicians. I. Gutman [1] in 1972, introduced the first Zagreb index and Randec in [2] introduced Randec index, which are oldest among the topological indices. I Gutman, Eliasi, Kulli, KC Das and many other experts have contributed in the developments of different Zagreb indices, Randic indices of simple graphs.
In case of classical graphs, both the vertices and edges have membership value one, but in case of fuzzy graphs both vertices and edges are equally important along with their fuzzy membership values. If the description of objects or their relationships or both are vague in nature, then we design a Fuzzy Graph model. In 1965, Zadeh [3] introduced the concept of fuzzy sets and fuzzy relations. Further Rosenfeld [4], Zimmerman [5], Thomson [6] and many experts in [7][8][9][10][11][12] have contributed significantly in the developments of fuzzy graphs.
In [13], Anjali and Mathew introduced energy of fuzzy graphs and in [14] authors introduced Laplacian energy of fuzzy graphs. In [15,16], authors discussed Weiner index of fuzzy graph and found relationships between connectivity index and Wiener index of a fuzzy graph. Recently, authors in [17,18], discussed transitive blocks, Hamiltonian fuzzy graphs and their applications in fuzzy interconnection networks, human trafficking.
The paper is structured as follows: In section 2, we discuss the preliminary definitions required for the development of the content. In Section 3, we introduce the important definitions of fuzzy Zagreb indices, fuzzy Zagreb matrices and corresponding energies. Section 4 provides some bounds for the fuzzy Zagreb energies.

Preliminaries
In this section, we recall some definitions of Zagreb indices and some notions of fuzzy graphs which will play an important role in the subsequent sections of the paper. Basics of Zagreb indices can be referred in [19,20]. Basics of graphs and fuzzy graphs can be referred in [4,21].
Also, we denote strength of vertex u by µ(u), it represents the minimum of strengths of edges incident to the vertex u

Fuzzy Zagreb Indices
In this section, we introduce the definitions of fuzzy Zagreb first index, fuzzy Zagreb second index and fuzzy Zagreb hyper index along with associated fuzzy Zagreb matrices and the fuzzy Zagreb energies. These definitions are required to discuss the main results.
Equivalently the index can also be defined as Definition 3.2. The fuzzy Zagreb second index of G=(σ, µ) is defined as Definition 3.3. The fuzzy hyper Zagreb index of G=(σ, µ) is defined as Definition 3.4. If G=(σ, µ) is a fuzzy graph and σ * = {u 1 , u 2 , . . . , u n } then first fuzzy Zagreb matrix is and second fuzzy Zagreb matrix is defined as F Z (2) = (f z (2) ) i,j , where n then the first fuzzy Zagreb energy is defined as 2 , ... , ξ n as its eigen values then the second fuzzy Zagreb energy is defined as Consider the fuzzy graphs G=(σ, µ) as shown in fig.1 Here The first and second fuzzy Zagreb matrices are given by Figure 1.    (2 ) are given by ξ 6 = 0.00069 hence the second fuzzy Zagreb energy is FZE (2 ) = 0.16.

Main Results
In this paper we will discuss fuzzy Zagreb first index and the corresponding first fuzzy Zagreb matrix and first fuzzy Zagreb energy. The analogous study of second fuzzy Zagreb quantities along with first and second fuzzy Eztrada Zagreb energies will be communicated in forthcoming paper.
is increasing function in 0 < x ≤ 1 and it is decreasing function in x ≥ 1, also x ≥ 2 n FM 1 ≥ 1 we get,  hence the theorem 4.6 is verified.

Applications
In case of human trafficking, objects can be considered as vertices which are reasons for human trafficking while each link between these reasons can be considered as an edge. So each edge has strength of the routes between vertices. Concepts of indices can be applied to measure of susceptibility of certain routs which need to be eliminated with respect to human trafficking. Similarly, in case of internet routing, fuzzy Zagreb indices can be used to identify the nature of particular vertex or strength of the whole system so that it reduces the time consumption in the particular area.

Conclusion
Fuzzy Zagreb first, second and hyper indices as well as first, second fuzzy Zagreb matrices and their energies are studied. Some bounds for first fuzzy Zagreb energy are studied along with illustration. Further study on these fuzzy Zagreb indices may reveal more analogous results of these kind and will be discussed in the forthcoming papers.
Acknowledgements: The authors are highly grateful to the anonymous reviewers for their helpful comments and suggestions for improving the paper.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.