International Journal of Analysis and Applications Volume 19, Number 2 (2021), 252-263 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-19-2021-252 FUZZY ZAGREB INDICES AND SOME BOUNDS FOR FUZZY ZAGREB ENERGY MAHESH KALE∗, S. MINIRANI Department of Basic Sciences and Humanities, MPSTME, SVKM’s NMIMS Deemed to be University, Mumbai, India ∗Corresponding author: mnk.maths@gmail.com Abstract. Topological indices M1/M2 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. 1. 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 [7–12] have contributed significantly in the developments of fuzzy graphs. Received January 12th, 2021; accepted February 15th, 2021; published March 2nd, 2021. 2010 Mathematics Subject Classification. 05C07, 05C072, 15B15. Key words and phrases. fuzzy graphs; fuzzy Zagreb indices; fuzzy Zagreb matrices; fuzzy Zagreb energies. ©2021 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 252 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-19-2021-252 Int. J. Anal. Appl. 19 (2) (2021) 253 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. 2. 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]. Definition 2.1. Let G=(V,E) be a simple graph. Degree of vertex u is denoted by du and e = uv ∈ E is an edge in graph G. First Zagreb index M1(G) is defined as M1(G) = ∑ uv∈E [du + dv] = ∑ u∈V d2u Second Zagreb index M2(G) is defined as M2(G) = ∑ uv∈E du.dv and Hyper Zagreb index HM(G) is defined as HM(G) = ∑ uv∈E [du + dv] 2 Definition 2.2. A fuzzy graph G=(V,σ,µ); which also can simply be denoted by G=(σ,µ), is a graph with vertex-membership function σ : V → [0, 1] and edge-strength function µ : V ×V → [0, 1] such that it satisfies the relation µ(x,y) ≤ min{σ(x),σ(y)}, ∀x,y ∈ V . Corresponding crisp graph is denoted by G=(σ∗,µ∗). Also, we denote strength of vertex u by µ(u), it represents the minimum of strengths of edges incident to the vertex u µ(u) = ∧ uvi∈µ∗ µ(u,vi) 3. 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. Int. J. Anal. Appl. 19 (2) (2021) 254 Definition 3.1. The fuzzy Zagreb first index of G=(σ,µ) is defined as FM1 (G) = ∑ uv∈µ∗ [σ(u)µ(u) + σ(v)µ(v)] Equivalently the index can also be defined as FM1 (G) = ∑ u∈σ∗ [σ(u)µ(u)du] Definition 3.2. The fuzzy Zagreb second index of G=(σ,µ) is defined as FM2 (G) = ∑ uv∈µ∗ [σ(u)µ(u).σ(v)µ(v)] Definition 3.3. The fuzzy hyper Zagreb index of G=(σ,µ) is defined as FHM (G) = ∑ uv∈µ∗ [σ(u)µ(u) + σ(v)µ(v)] 2 Definition 3.4. If G=(σ,µ) is a fuzzy graph and σ∗ = {u1,u2, . . . ,un} then first fuzzy Zagreb matrix is defined as F Z (1) = (fz(1))i,j, where (fz(1))i,j =   σ(ui)µ(ui) + σ(uj)µ(uj) , if i 6= j and ui.uj ∈ µ∗ 0 , if ui.uj /∈ µ∗ 0 , if i = j and second fuzzy Zagreb matrix is defined as F Z (2) = (fz(2))i,j, where (fz(2))i,j =   σ(ui)µ(ui).σ(uj)µ(uj) , if i 6= j and ui.uj ∈ µ∗ 0 , if ui.uj /∈ µ∗ 0 , if i = j Definition 3.5. If G=(σ,µ) is a fuzzy graph and σ∗ = {u1,u2, . . . ,un}, if F Z (1) is the first fuzzy Zagreb matrix with its eigen values ξ (1) 1 , ξ (1) 2 , ... , ξ (1) n then the first fuzzy Zagreb energy is defined as FZ E (1) = n∑ i=1 |ξ(1)i | Definition 3.6. If G=(σ,µ) is a fuzzy graph and σ∗ = {u1,u2, . . . ,un}, if F Z (2) is the second fuzzy Zagreb matrix with ξ (2) 1 , ξ (2) 2 , ... , ξ (2) n as its eigen values then the second fuzzy Zagreb energy is defined as FZ E (2) = n∑ i=1 |ξ(2)i | Example 3.1. Consider the fuzzy graphs G=(σ,µ) as shown in fig.1 Here σ∗ = {u1,u2,u3,u4,u5,u6} with membership values σ(u1) = 0.4, σ(u2) = 0.2, σ(u3) = 0.6, σ(u4) = 0.5, σ(u5) = 0.7, σ(u6) = 0.8 and strengths of edges µ(u1u2) = 0.1, µ(u2u3) = 0.2, µ(u2u4) = 0.1, µ(u3u4) = 0.4, µ(u4u5) = 0.4, µ(u5u6) = 0.5, µ(u5u1) = 0.3, µ(u1u6) = 0.3 then we get FM1 = 1.68, FM2 = 0.1107 and FHM = 0.0468. The first and second fuzzy Zagreb matrices are given by Int. J. Anal. Appl. 19 (2) (2021) 255 Figure 1. F Z (1) = u1 u2 u3 u4 u5 u6    u1 0 0.06 0 0 0.25 0.28 u2 0.06 0 0.14 0.07 0 0 u3 0 0.14 0 0.17 0 0 u4 0 0.07 0.17 0 0.26 0 u5 0.25 0 0 0.26 0 0.45 u6 0.28 0 0 0 0.45 0 and F Z (2) = u1 u2 u3 u4 u5 u6    u1 0 0.0008 0 0 0.0084 0.0096 u2 0.0008 0 0.024 0.001 0 0 u3 0 0.024 0 0.006 0 0 u4 0 0.001 0.006 0 0.0105 0 u5 0.0084 0 0 0.0105 0 0.0504 u6 0.0096 0 0 0 0.0504 0 Eigen values of FZ (1) are given by ξ (1) 1 = 0.7067, ξ (1) 2 = −0.5276, ξ (1) 3 = −0.2595, ξ (1) 4 = 0.2487, ξ (1) 5 = −0.1704, ξ (1) 6 = 0.0021 hence the first fuzzy Zagreb energy is FZE (1) = 1.915. Eigen values of FZ (2) are given by ξ (2) 1 = 0.0544, ξ (2) 2 = −0.0515, ξ (2) 3 = 0.0249, ξ (2) 4 = −0.0245, ξ (2) 5 = −0.00404, ξ(2)6 = 0.00069 hence the second fuzzy Zagreb energy is FZE (2) = 0.16. 4. 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. Int. J. Anal. Appl. 19 (2) (2021) 256 For simplicity of notations, fuzzy Zagreb first matrix will be denoted by FZ, its (i,j) th element by (fz)i,j and the corresponding fuzzy Zagreb energy by FZE = n∑ i=1 |ξi| , where ξ1 ≥ ξ2 ≥ . . . ≥ ξn are eigen values of FZ. Here FZ is a real symmetric matrix and hence all its eigen values are real. For non-negative integer k, the kth spectral moment of FZ is given by Nk = n∑ i=1 (ξi) k = tr(FZk) Although elementary, the following result deserves tobe stated as: Theorem 4.1. If G=(V,σ,µ) and H=(W,σ′,µ′) are fuzzy isomorphic graphs then FM1 (G) = FM1 (H) and FM2 (G) = FM2 (H) Proof. If G=(V,σ,µ) and H=(W,σ′,µ′) are fuzzy isomorphic graphs then there exists a bijective function f : V → W such that for every u ∈ V and uv ∈ µ∗, σ(u) = σ′(f(u)) and µ(uv) = µ′(uv). Hence we can easily conclude that FM1 (G) = ∑ uv∈µ∗ [σ(u)µ(u) + σ(v)µ(v)] = ∑ f(u)f(v)∈µ′∗ [σ′(f(u))µ′(f(u)) + σ′(f(v))µ′(f(v))] = FM1 (H) and also FM2 (G) = ∑ uv∈µ∗ [σ(u)µ(u).σ(v)µ(v)] = ∑ f(u)f(v)∈µ′∗ [σ′(f(u))µ′(f(u)).σ′(f(v))µ′(f(v))] = FM2 (H) � Theorem 4.2. If G=(V,σ,µ) with V = {u1,u2, . . . ,un} is a fuzzy graph and FZ is its fuzzy Zagreb matrix then (1) N1 = tr(FZ ) = 0 (2) N2 = tr(FZ 2) = 2.FHM Int. J. Anal. Appl. 19 (2) (2021) 257 Proof. (1) By definition of FZ, all its diagonal elements are zero, hence trace of FZ is zero. (2) The diagonal elements of FZ2 are given by (fz2)i,i = n∑ j=1 (fz)i,j.(fz)j,i = n∑ j=1 (fz)2i,j = ∑ ui.uj∈µ∗ i,j∈{1,2,...,n} [σ(ui)µ(ui) + σ(uj)µ(uj)] 2 hence N2 = tr(FZ 2) = n∑ i=1 ∑ ui.uj∈µ∗ i,j∈{1,2,...,n} [σ(ui)µ(ui) + σ(uj)µ(uj)] 2 = 2. ∑ ui.uj∈µ∗ i,j∈{1,2,...,n} [σ(ui)µ(ui) + σ(uj)µ(uj)] 2 = 2.FHM � Theorem 4.3. Let G=(V,σ,µ) be a fuzzy graph with |V | = n vertices. Let FZ be the corresponding fuzzy Zagreb matrix with eigen values ξ1, ξ2, . . . , ξn and FZE is its fuzzy Zagreb energy then√ 2(FHM ) + n(n− 1)|det(FZ )| 2 n ≤ FZE ≤ √ 2n.(FHM ) Proof. Upper Bound: Applying Cauchy-Schwartz inequality to the n numbers (1, 1, . . . , 1) and (ξ1,ξ2, . . . ,ξn) we get, n∑ i=1 (1. |ξi|) ≤ ( n∑ i=1 1 )1/2 . ( n∑ i=1 |ξi| 2 )1/2 (4.1) n∑ i=1 |ξi| ≤ √ n. √√√√ n∑ i=1 |ξi| 2 Also we have, ( n∑ i=1 |ξi| )2 = n∑ i=1 |ξi| 2 + 2. ∑ 1≤i 0 and FZE is its fuzzy Zagreb energy then FZE ≥ 2 n FM1 + (n− 1) + ln [ n |det (FZ )| 2FM1 ] Proof. Consider G=(V,σ,µ) is a fuzzy graph and ξ1 ≥ ξ2 ≥ . . . ≥ ξn > 0 are eigen values of corresponding Fuzzy Zagreb matrix FZ . For simplicity, consider a function f(x) = 1 − x − ln(x), x > 0. Elementary calculations shows thatf(x)is decreasing function in (0, 1] and it is increasing function for x ≥ 1. Hence f(x) ≥ f(1) = 0 for x > 0 gives, x ≥ 1 + ln(x) for x > 0. FZE = n∑ i=1 |ξi| = |ξ1| + n∑ i=2 |ξi| ≥ |ξ1| + n∑ i=2 [1 + ln |ξi|] = |ξ1| + (n− 1) + ln [ n∏ i=2 |ξi| ] = |ξ1| + (n− 1) + ln [ n∏ i=1 |ξi| ] − ln |ξ1| = |ξ1| + (n− 1) + ln |det (FZ )|− ln |ξ1| As the function g(x) = x + (n − 1) + ln |det(FZ )| − ln x is increasing function in 1 ≤ x ≤ n, hence for |ξ1| ≥ 2n FM1 , FZE ≥ 2 n FM1 + (n− 1) + ln |det(FZ )|− ln [ 2 n FM1 ] (4.14) FZE ≥ 2 n FM1 + (n− 1) + ln [ n |det (FZ )| 2FM1 ] � Theorem 4.6. If G=(V,σ,µ) is a fuzzy graph and σ∗ = {u1,u2, . . . ,un}, FZ is the corresponding fuzzy Zagreb matrix with eigen values ξ1 ≥ ξ2 ≥ . . . ≥ ξn > 0 and FZE is its fuzzy Zagreb energy then FZE ≤ 2(FHM ) + 2 n FM1 − ( 2 n FM1 )2 − ln [ n |det (FZ )| 2FM1 ] Int. J. Anal. Appl. 19 (2) (2021) 261 Proof. Consider G=(V,σ,µ) is a fuzzy graph and ξ1 ≥ ξ2 ≥ . . . ≥ ξn > 0 are the eigen values of FZ . Recall that, 2FM1 = 2 ∑ uiuj∈µ∗ [σ(ui)µ(ui) + σ(uj)µ(uj)] ≥ n, hence 2n FM1 ≥ 1. For simplicity, consider a function f(x) = x2 −x− ln(x), x > 0. f(x) is decreasing function in 0 < x ≤ 1 and it is increasing function in x ≥ 1. Hence f(x) ≥ f(1) = 0 for x > 0, gives x ≤ x2 − ln(x) for x > 0. FZE = n∑ i=1 |ξi| = |ξ1| + n∑ i=2 |ξi| ≤ |ξ1| + n∑ i=2 [ |ξi| 2 − ln |ξi| ] = |ξ1| + n∑ i=1 |ξi| 2 −|ξ1| 2 − ln n∏ i=1 |ξi| + ln |ξ1| = 2(FHM ) + |ξ1|− |ξ1| 2 − ln [ |det(FZ )| |ξ1| ] As the function g(x) = 2(FHM ) + x−x2 − ln [|det(FZ )|] + ln(x) is increasing function in 0 < x ≤ 1 and it is decreasing function in x ≥ 1, also x ≥ 2 n FM1 ≥ 1 we get, (4.15) FZE ≤ 2(FHM ) + 2 n FM1 − ( 2 n FM1 )2 − ln [ n |det(FZ )| 2FM1 ] � Illustration: Consider the fuzzy graph G=(σ,µ) as shown in fig.1. Here FZE = 1.915, FHM = 0.468 and FM1 = 1.68 FZE ≥ √ 2(FHM ) + n(n− 1)|det(FZ)| 2 n = √ 2(0.468) + 6(5)|0.00000861| 2 6 = 1.2453 and FZE ≤ √ 2n.(FHM ) = √ 2(6).(0.468) = 2.3698 hence it verifies the theorem 4.3. FZE ≤ 2(FHM ) n + √√√√(n− 1) { 2(FHM ) − ( 2 n (FHM ) )2} = 2(0.468) 6 + √√√√(5) { 2(0.468) − ( 2 6 (0.468) )2} = 2.291 Int. J. Anal. Appl. 19 (2) (2021) 262 hence it verifies the theorem 4.4. FZE ≥ 2 n FM1 + (n− 1) + ln [ n |det(FZ )| 2FM1 ] = 2 6 (1.68) + 5 + ln [ 6(0.00000861) 2(1.68) ] = −5.5228 hence the theorem 4.5 is verified. FZE ≤ 2(FHM ) + 2 n FM1 − ( 2 n FM1 )2 − ln [ n |det(FZ )| 2FM1 ] = 2(0.468) + 2 6 (1.68) − ( 2 6 (1.68) )2 − ln [ 6(0.00000861) 2(1.68) ] = 12.2652 hence the theorem 4.6 is verified. 5. 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. 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