On Laplacian energy of picture fuzzy graphs in site selection problem

1 Introduction

Various generalizations for the fuzzy sets (FSs) [1] and the intuitionistic fuzzy sets (IFSs) [8] have been provided by the researchers in the literature to model the uncertainties & the hesitancy inherent in many practical circumstances for having a better applicability and flexibility. Yager [22] stated that fuzzy set and IFS are not sufficient to address the manual perception in a practical sense and defined Pythagorean fuzzy set (PyFS) which has certainly expanded the coverage of information by modifying the existing constraint. Actually, the concept of belongingness (yes), non-belongingness (no) and indeterminacy (abstain) have been considered differently in the definitions of IFS and PyFS. “Smarandache [10] defined the neutrosophic set by introducing the degree of indeterminacy as an independent component. Some other generalizations may also be easily found in the literature.”

Further, Cuong [21] defined the concept of picture fuzzy set (PFS) in which all the four parameters, i.e., “degree of membership, degree of indeterminacy (neutral), degree of nonmembership and the degree of refusal have been taken into account”. In order to understand the importance of picture fuzzy information, we consider an example of a voting system - “Suppose the voters have been categorized into four different classes: one who votes for (yes), one who votes against (no), one who neither vote for nor against (abstain) and one who refused for voting (refusal). It may be noted that the concept of ‘refusal’ is found to be an additional component which was not being taken into account by any of the sets or by their generalizations (fuzzy set, intuitionistic fuzzy set, Pythagorean fuzzy set, neutrosophic set) stated above”. This introduced setup has the capability to handle the human’s imprecise perception in a more flexible fashion.

Kifayat et al. [38] studied the geometrical aspects and features of these generalizations - “fuzzy sets, intuitionistic fuzzy sets, Pythagorean fuzzy sets and picture fuzzy sets”. It may be noted that the phenomenon of the voting system stated above can not be represented closely and sufficiently by utilizing the Pythagorean fuzzy graphs/sets. For capturing the information content and utilizing the flexibility in a more broader sense, we extend the graph-theoretic literature of picture fuzzy set, picture fuzzy graph and its applications. Figure 1 clearly illustrates the wider scope of different extensions of fuzzy sets:

OPEN IN VIEWER

Fig. 1

Geometrical Representations of Generalizations of Fuzzy Set.

In many practical situations containing structural information viz. operation management, networking, systems analysis, economical interpretation, decision support system, the graph representation of the information has been found to be more effective and convenient for dealing with the information embedded among different objetcs/attributes/alternatives.

A brief literature survey on the Pythagorean fuzzy graph, picture fuzzy graph, other graph theoretic operations and applications has been given below:

“Based on the fuzzy relation [2], Kaufmann [3] proposed the concept of fuzzy graphs and Rosenfeld [4] subsequently developed the concept of fuzzy vertex and fuzzy edge. Some standard operations on the fuzzy graphs were studied by the Mordeson and Peng [9] along with their properties. Further, Parvathi et al. [11, 16] extended the notion of a fuzzy graph to intuitionistic fuzzy graph and analyzed various properties related to minmax intuitionistic fuzzy graph. Karunambigai et al. [17] proposed a category of constant and totally constant intuitionistic fuzzy graphs and subsequently Akram et al. [18] presented the concept of strong intuitionistic fuzzy graphs along with their properties. Also, Akram et al. [20] presented intuitionistic fuzzy hypergraphs with their applications and Alshehri et al. [24] defined the planarity, duality and multigraphs in context with intuitionistic fuzzy graphs. Sahoo and Pal [28] [33] proposed various types of product operations for intuitionistic fuzzy graphs, intuitionistic fuzzy tolerance graph with their applications.”

“Various researchers utilized the flexibility and its applicability to set forward some new ideas concerning the extended structures of intuitionistic fuzzy graphs and provided many interesting applications in clustering and decision-making problems and support systems [23, 35]. Naz et al. [39] proposed a generalization of the intuitionistic fuzzy graph, termed as the Pythagorean fuzzy graphs, and studied their applications in various decision making problems. Some graph-theoretic operations related with Pythagorean fuzzy graphs have been well studied by Verma et al. [36].”

Zuo et al. [40] introduced the new concept of picture fuzzy graph and its various types with different properties. Some important operations such as Cartesian product, composition, join, direct product, lexicographic and strong product on picture fuzzy graph have also been defined along with its utility and social networking application. Additionally, the significance of picture fuzzy graph has been explained with the help of social networking problem where the cluster of nodes and links forms a network which represents the agency/people/state and connection between the nodes respectively. The variation/vagueness in the relationship in the links can not be studied by the classical graphs and hence Samanta and Pal [27] utilized type 1 fuzzy graph in place of classical fuzzy graph.

Further, Das et al. [41, 43] comprehensively proposed a lot of research in the field of picture fuzzy graph, planar graphs with its applications using the introduced novel notions of m-step picture fuzzy competition graphs, picture fuzzy economic competition graphs and picture fuzzy competition hypergraphs [42]. The applications have been dealt in the field of academics, environment and companies etc. Pal et al. [44] explained the modern trends in study of fuzzy theory in the field of medicine. Some useful applications in various fields, e.g., decision making, medical diagnosis, pattern recognition, service quality analysis etc are available in literature.

In a different context, it may be noted that Meenakshi et al. [26] explained the energy of graph connecting the graph more closely to the chemical quantity known as π- electron energy of conjugated hydro carbon molecule. Nowadays, the analogous energies are being calculated using the eigenvalues of the graph matrices which technically makes the study more powerful in both the cases of chemistry as well as mathematics. This proves the importance of calculation of energy of a graph. The concept of energy of a graph [6, 13] has been utilized in chemical engineering applications-the molecular orbital theory of conjugated molecules [5, 15]. The concept of energy to fuzzy graphs has been extended by Narayanan and Mathew [19] with some bounds on the energy of fuzzy graphs. Further, Praba et al. [25] extended the energy concept for intuitionistic fuzzy graph with important results. Sridhara and Kanna [31] studied the bounds on the energy and Laplacian energy of graphs with various important observations and results. Recently, the application of energy of Pythagorean fuzzy graphs has been studied in the decision making example of a satellite communication system and in the evaluation of the schemes of reservoir operation [37].

In the present communication, we introduce the novel concept of adjacency matrix, energy and Laplacian energy for picture fuzzy graphs with applications. A new methodology for solving a selection problem based on the proposed notions of picture fuzzy graph has also been provided with an example. In whole, the purpose of the proposed work is to further expand the fuzzy graph related concepts under picture fuzzy environment. Such extensions and enrichment will certainly help in widening the span and coverage of the information significantly.

The rest of the manuscript has been presented as follows. In Section 1, some basic definitions and fundamental notions in connection with the picture fuzzy set, Pythagorean fuzzy graph, picture fuzzy graph, energy etc. Considering the fact that picture fuzzy graph has the sufficient strength to formulate the impreciseness, vagueness and incompleteness embedded in the information of an application, we have proposed the definition of adjacency matrix of such graph, its spectrum and energy/Laplacian energy with upper/lower bounds in Section 2. In reference with picture fuzzy directed graph, similar studies and results have been presented in Section 3. Further, we present a new algorithm to solve hydropower plant site selection problem by utilizing the notion of energy/Laplacian energy of picture fuzzy graph in Section 4. In Section 5, some comparative findings and advantages of the proposed approach have been provided. The paper has been finally concluded in Section 6.

Here, we present some fundamental and basic definitions in relation with the Pythagorean fuzzy graph, picture fuzzy graph and energy etc. The generalization process from intuitionistic fuzzy sets to picture fuzzy sets may be well understood through the following definitions:

Let U be the universe of discourse with μ _M  : U → [0, 1] and ν _M  : U → [0, 1] being membership and non-membership grade respectively.

The set M ={ < α, μ _M  (α) , ν _M  (α) > ∣ α ∈ U } is called

Intuitionistic fuzzy set [8] in U if it satisfies the condition 0 ≤ μ _M  (α) + ν _M  (α) ≤1 with the degree of indeterminacy given by π M ( α ) = 1 - μ M ( α ) - ν M ( α ) .

Definition 1. [21] “A picture fuzzy set A in U (universe of discourse) is given by A = { < α , μ A ( α ) , η A ( α ) , ν A ( α ) > ∣ α ∈ U } ; where μ _A  : U → [0, 1], η _A  : U → [0, 1] and ν _A  : U → [0, 1] denote the degree of membership, degree of neutral membership (abstain) and degree of non-membership respectively and for every α ∈ U satisfy the condition μ A ( α ) + η A ( α ) + ν I ( α ) ≤ 1 . The degree of refusal for any picture fuzzy set A and α ∈ U is given by r A ( α ) = 1 - ( μ A ( α ) + η A ( α ) + ν A ( α ) ) .

In this paper, throughout PFS (U) denotes the set of all picture fuzzy sets.”

Definition 2. [21] “An intuitionistic fuzzy graph on U, denoted by G ˜ = ( P , Q ) , where P is an intuitionistic fuzzy set on U and Q is an intuitionistic fuzzy relation in U × U such that μ Q ( α , β ) ≤ min { μ P ( α ) , μ P ( β ) } , ν Q ( α , β ) ≥ max { ν P ( α ) , ν P ( β ) } ; satisfying the constraint condition 0 ≤ μ Q ( α , β ) + ν Q ( α , β ) ≤ 1 , ∀ α , β ∈ U . The set P is called the intuitionistic fuzzy vertex set of the graph G ˜ and Q is called the intuitionistic fuzzy edge set of the graph G ˜ .”

Definition 3. [39] “A Pythagorean fuzzy graph on U, denoted by G ˆ = ( M , N ) , where M is a Pythagorean fuzzy set on U and N is a Pythagorean fuzzy relation in U × U such that μ N ( α , β ) ≤ min { μ M ( α ) , μ M ( β ) } , ν N ( α , β ) ≥ max { ν M ( α ) , ν M ( β ) } ; satisfying the constraint condition 0 ≤ μ N 2 ( α , β ) + ν N 2 ( α , β ) ≤ 1 , ∀ α , β ∈ U . The set M is called the Pythagorean fuzzy vertex set of the graph G ˆ and N is called the Pythagorean fuzzy edge set of the graph G ˆ .”

Definition 4. [40] “A picture fuzzy graph on (S, R), denoted by G = (M, N), where M is a picture fuzzy set on S and N is a picture fuzzy relation in R = S × S such that μ N ( α , β ) ≤ min { μ M ( α ) , μ M ( β ) } , η N ( α , β ) ≥ max { η M ( α ) , η M ( β ) } , ν N ( α , β ) ≥ min { ν M ( α ) , ν M ( β ) } ; satisfying the constraint condition 0 ≤ μ _N  (α, β) + η _N  (α, β) + ν _N  (α, β) ≤1, ∀ α, β ∈ S. The set M is called the picture fuzzy vertex set of the graph G and N is called the picture fuzzy edge set of the graph G.”

Definition 5. A picture fuzzy graph G = (M, N) is called a

Regular picture fuzzy graph [40] if it satisfies the condition ∑ α , α ≠ β μ N ( α , β ) = constant , ∑ α , α ≠ β η N ( α , β ) = constant , ∑ α , α ≠ β ν N ( α , β ) = constant ; ∀ α , β ∈ S .

Definition 7.[25] Let G ˜ = ( P , Q ) be a intuitionistic fuzzy graph and A ( G ˜ ) be its adjacency matrix. Consider λ _i be the eigenvalues of A μ ( G ˜ ) and γ _i be the eigenvalues of A ν ( G ˜ ) . Then the energy of the intuitionistic fuzzy graph is given by E ( G ˜ ) ) = ( ∑ i | λ i | , ∑ i | γ i | ) .

3 Notion of energy of picture fuzzy graph

In this section, we propose some novel concepts of adjacency matrix, spectrum, energy and Laplacian energy of picture fuzzy graph as follows:

Let G = (S, R) be a picture fuzzy graph, where S is the picture fuzzy vertex set and R is the picture fuzzy edge set.

Definition 8. The adjacency matrix A (G) of the graph G is a square matrix defined as A ( G ) = [ a ij ] , where a ij = ( μ R ( α i , α j ) , η R ( α i , α j ) , ν R ( α i , α j ) ) . Here, μ _R  (α _i , α _j ) , η _R  (α _i , α _j )  &  ν _R  (α _i , α _j ) are the degree of membership, degree of neutral membership (abstain) and degree non-membership respectively.

Definition 9. The spectrum of the adjacency matrix A (G) of the picture fuzzy graph G = (S, R) is given by (Θ, Φ, Ψ), where Θ, Φ and Ψ are the set of the eigenvalues of matrices A ( μ R ( α i , α j ) ) = [ μ R ( α i , α j ) ] ; A ( η R ( α i , α j ) ) = [ η R ( α i , α j ) ] ; and A ( ν R ( α i , α j ) ) ) = [ ν R ( α i , α j ) ] ; respectively.

Definition 10. The energy E (G) of the picture fuzzy graph G is defined as E ( G ) = ( E ( μ R ( α i , α j ) ) , E ( η R ( α i , α j ) ) , E ( ν R ( α i , α j ) ) ) = ( ∑ i = 1 , θ i ∈ Θ m | θ i | , ∑ i = 1 , φ i ∈ Φ m | φ i | , ∑ i = 1 , ψ i ∈ Ψ m | ψ i | ) .

For illustrating the proposed definitions, we consider the following example of a picture fuzzy graph:

Example 1. Suppose G = (S, R) be a picture fuzzy graph given in Figure 2.

OPEN IN VIEWER

Fig. 2

Graph G = (S, R).

In view of the definitions proposed above, the adjacency matrix A (G) is given by A ( G ) = ( ( 0.0 , 0.0 , 0.0 ) ( 0.4 , 0.3 , 0.2 ) ( 0.0 , 0.0 , 0.0 ) ( 0.4 , 0.2 , 0.3 ) ( 0.4 , 0.3 , 0.2 ) ( 0.0 , 0.0 , 0.0 ) ( 0.5 , 0.2 , 0.2 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.5 , 0.2 , 0.2 ) ( 0.0 , 0.0 , 0.0 ) ( 0.3 , 0.5 , 0.2 ) ( 0.4 , 0.2 , 0.3 ) ( 0.0 , 0.0 , 0.0 ) ( 0.3 , 0.5 , 0.2 ) ( 0.0 , 0.0 , 0.0 ) ) . Using the adjacency matrix A (G), the spectrum of the picture fuzzy graph G can be evaluated as Spec ( μ R ( α i , α j ) ) = { - 0.8063 , 0.8063 , - 0.0992 , 0.0992 } ; Spec ( η R ( α i , α j ) ) = { - 0.5114 , 0.5114 , - 0.0917 , 0.0917 } ; Spec ( ν R ( α i , α j ) ) = { - 0.4561 , 0.4561 , - 0.0438 , 0.0438 } . Hence, the spectrum of G may be presented as Spec ( G ) = { ( - 0.8063 , - 0.5114 , - 0.4561 ) , ( 0.8063 , 0.5114 , 0.4561 ) , ( - 0.0992 , - 0.0917 , - 0.0438 ) , ( 0.0992 , 0.0917 , 0.0438 ) } . Now, the calculation of energy of the picture fuzzy graph can be done as E ( μ R ( α i , α j ) ) = 1.811 ; E ( η R ( α i , α j ) ) = 1.2062 ; E ( ν R ( α i , α j ) ) = 0.998 . Hence, the energy of G is E ( G ) = ( 1.811 , 1.2062 , 0.998 ) .

Next, we present the studies and various important results related to eigenvalues of adjacency matrix, upper bound and lower bound of energy of picture fuzzy graph.

Theorem 1. Let G = (S, R) be a picture fuzzy graph and A (G) be its adjacency matrix. If θ₁≥ θ₂ ≥ … ≥ θ _m , φ₁ ≥ φ₂ ≥ … ≥ φ _m and ψ₁ ≥ ψ₂ ≥ … ≥ ψ _m are the absolute eigenvalues of A (μ _R  (α _i , α _j )), A (η _R  (α _i , α _j )) and A (ν _R  (α _i , α _j )), respectively, then:

∑ i = 1 , θ i ∈ Θ m θ i = 0 , ∑ i = 1 , φ i ∈ Φ m φ i = 0 and ∑ i = 1 , ψ i ∈ Ψ m ψ i = 0 ;

Proof:

Since the diagonal entries of adjacency matrix A (G) are zero, therefore the trace of the matrix is zero. As the trace of a matrix is equal to sum of its eigenvalues, the proof is obvious.

Further, throughout the manuscript we denote M μ = : ∑ 1 ≤ i < j ≤ n ( μ R ( α i , α j ) ) 2 ; M η = : ∑ 1 ≤ i < j ≤ n ( η R ( α i , α j ) ) 2 ; and M ν = : ∑ 1 ≤ i < j ≤ n ( ν R ( α i , α j ) ) 2 . Also, we denote | A μ | = : det ( A ( μ R ( α i , α j ) ) ) ; | A η | = : det ( A ( η R ( α i , α j ) ) ) ; and | A ν | = : det ( A ( ν R ( α i , α j ) ) ) .

Theorem 2. Let G = (S, R) be the picture fuzzy graph with m vertices and A (G) be its adjacency matrix. Then

2 M μ + m ( m - 1 ) | A μ | 2 m ≤ E ( μ R ( α i , α j ) ) ≤ 2 mM μ ;

Proof: Using Cauchy-Schwarz inequality for the vectors (1, 1, …, 1) and (|θ₁|, |θ₂|, …, |θ _m |) with m entries, we get ∑ i = 1 m | θ i | ≤ m ∑ i = 1 m | θ i | 2 .(3.1)

Also, ( ∑ i = 1 m | θ i | ) 2 = ∑ i = 1 m | θ i | 2 + 2 ∑ 1 ≤ i < j ≤ m m | θ i θ j | = 0 .(3.2) The characteristic polynomial of A (G) is given by

∏ i = 1 m ( λ - θ i ) = | A ( G ) - λ I | . Now, comparing the coefficients of λ^n-2 in above polynomial, we get ∑ 1 ≤ i < j ≤ m m | θ i θ j | = - M μ .(3.3) Using equation (3.3) in equation (3.2), we have ∑ i = 1 m | θ i | 2 = 2 M μ .(3.4) Substituting equation (3.4) in equation (3.1), we have ∑ 1 ≤ i < j ≤ m m | θ i | ≤ m 2 M μ = 2 m M μ . Therefore, E ( μ R ( α i , α j ) ) ≤ 2 m M μ .(3.5) Next, ( E ( μ R ( α i , α j ) ) ) 2 = ( ∑ i = 1 m | θ i | ) 2 = ∑ i = 1 m | θ i | 2 + 2 ∑ 1 ≤ i < j ≤ m m | θ i θ j | = 2 M μ + 2 m ( m - 1 ) 2 AM { | θ i θ j | } . Since the arithmetic mean is greater than or equal to the geometric mean, i.e., AM {|θ _i θ _j |} ≥ GM {|θ _i θ _j |}, therefore, E ( μ R ( α i , α j ) ) ≥ 2 M μ + m ( m - 1 ) GM { | θ i θ j | } .(3.6) Also,

GM { | θ i θ j | } = ( ∏ 1 ≤ i < j ≤ m | θ i θ j | ) 2 m ( m - 1 ) = ( ∏ 1 ≤ i < j ≤ m | θ i | m - 1 ) 2 m ( m - 1 ) = ( ∏ i = 1 m | θ i | ) 2 m = | A μ | 2 m .(3.7) Substituting equation (3.7) in equation (3.6), we get E ( μ R ( α i , α j ) ) ≥ 2 M μ + m ( m - 1 ) | A μ | 2 m .(3.8) Thus, from equations (3.5) and (3.8), we have 2 M μ + m ( m - 1 ) | A μ | 2 m ≤ E ( μ R ( α i , α j ) ) ≤ 2 mM μ . On similar lines, we can show that 2 M η + m ( m - 1 ) | A η | 2 m ≤ E ( η R ( α i , α j ) ) ≤ 2 mM η and 2 M ν + m ( m - 1 ) | A ν | 2 m ≤ E ( ν R ( α i , α j ) ) ≤ 2 mM ν .

Theorem 3. Let G = (S, R) be a picture fuzzy graph on m vertices and A (G) be its adjacency matrix. If m ≤ 2M _μ , m ≤ 2M _η and m ≤ 2M _ν , then:

E ( μ R ( α i , α j ) ) ≤ 2 M μ m + ( m - 1 ) { 2 M μ - ( 2 M μ m ) 2 } ;

Proof: Since A (G) of the picture fuzzy graph G is symmetric with trace zero, therefore, θ 1 ≥ 2 ∑ 1 ≤ i < j ≤ m μ R ( α i , α j ) m , where θ₁, θ₂, …, θ _m are the eigenvalues of A (G). In view of the results obtained in Theorem 1, we write ∑ i = 2 m θ i 2 = 2 M μ - θ 1 2 .(3.9) Using Cauchy- Schwarz inequality for the vectors (1, 1, …, 1) & (|θ₁|, |θ₂|, …, |θ _m |) with (m - 1) entries, we get E ( μ R ( α i , α j ) ) - θ 1 = ∑ i = 2 m θ i 2 ≤ ( m - 1 ) ∑ i = 2 m θ i 2 .(3.10) Substituting equation (3.9) in equation (3.10) and after rearranging, we have E ( μ R ( α i , α j ) ) ≤ θ 1 + ( m - 1 ) ( 2 M μ - θ 1 2 ) .(3.11) Since the function F ( α ) = α + ( m - 1 ) ( 2 M μ - α 2 ) decreases on the interval ( 2 M μ m , 2 M μ ) , as 1 ≤ 2 M μ m , therefore, 2 M μ m ≤ 2 M μ m ≤ 2 ( μ R ( α i , α j ) ) m ≤ θ 1 ≤ 2 M μ . Thus the equation (3.11) implies E ( μ R ( α i , α j ) ) ≤ 2 M μ m + ( m - 1 ) { 2 M μ - ( 2 M μ m ) 2 } . On similar lines, we can also show that E ( η R ( α i , α j ) ) ≤ 2 M η m + ( m - 1 ) { 2 M η - ( 2 M η m ) 2 } ; and

E ( ν R ( α i , α j ) ) ≤ 2 M ν m + ( m - 1 ) { 2 M ν - ( 2 M ν m ) 2 } .

It may be noted that the results obtained in the above theorem provide the upper bound for the energy of the picture fuzzy graph, with the conditions m ≤ 2M _μ , m ≤ 2M _η and m ≤ 2M _ν .

Theorem 4. Let G = (S, R) be a picture fuzzy graph on m vertices. Then E ( G ) ≤ m 2 ( 1 + m ) .

Proof: If n ≤ 2Mμ, then by calculus it is easy to show that f ( M μ ) = 2 M μ m + ( m - 1 ) { 2 M μ - ( 2 M μ m ) 2 }

obtains a maximum value when M μ = m 2 + m m 4 . Substituting this value of M _μ in the above Theorem 3, we get E ( μ R ( α i , α j ) ) ≤ m ( 1 + m ) 2 . Similarly, the results for other energy components can be obtained. Hence, the theorem is proved.

Next, we study another important notion of energy of graph, known as Laplacian Energy of Picture Fuzzy Graph and discuss its various graph-theoretic aspects.

Definition 11. Let G = (S, R) be a picture fuzzy graph on m vertices. The degree matrix D (G) = [d _ij ], of G is a m × m diagonal matrix defined as: d ij = { d G ( α i ) if i = j ; 0 otherwise .

Example: We consider an example of picture fuzzy graph on 4 vertices. The degree matrix D (G) would be a 4 × 4 diagonal matrix presented by D ( G ) = ( ( 0.4 , 0.2 , 0.1 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.5 , 0.7 , 0.6 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.2 , 0.3 , 0.4 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.1 , 0.2 , 0.5 ) ) .

Definition 12. Let G = (S, R) be a picture fuzzy graph on m vertices. The Laplacian matrix of a picture fuzzy graph G is defined as L (G) = D (G) - A (G) ; where D (G)  &  A (G) are degree and adjacency matrix of the picture fuzzy graph G, respectively.

Example: We consider the picture fuzzy graph on 4 vertices whose Laplacian matrix may be represented by L ( G ) = D ( G ) - A ( G ) = ( ( 0.4 , 0.2 , 0.1 ) ( - 0.4 , - 0.2 , - 0.3 ) ( - 0.4 , - 0.3 , - 0.2 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.5 , 0.7 , 0.6 ) ( 0.0 , 0.0 , 0.0 ) ( - 0.3 , - 0.5 , - 0.2 ) ( 0.0 , 0.0 , 0.0 ) ( - 0.2 , - 0.1 , - 0.6 ) ( 0.2 , 0.3 , 0.4 ) ( 0.0 , 0.0 , 0.0 ) ( - 0.5 , - 0.2 , - 0.3 ) ( 0.0 , 0.0 , 0.0 ) ( - 0.5 , - 0.2 , - 0.2 ) ( 0.1 , 0.2 , 0.5 ) ) .

Definition 13. The spectrum of the Laplacian matrix L (G) of the picture fuzzy graph G = (S, R) is given by {(Δ, ϒ, Ω)}, where Δ, ϒ and Ω are the set of the eigenvalues of L (μ _R  (α _i , α _j )), L (η _R  (α _i , α _j )) and L (ν _R  (α _i , α _j ))), respectively.

Example: Using the example given in the definition of the Laplacian matrix (Definition 1), we calculate the spectrum ofL (G) as given below: spec ( μ R ( α i , α j ) ) = { - 0.20962 , 0.4000 , 0.50481 + 0.3561 i , 0.50481 - 0.3561 i } ; spec ( η R ( α i , α j ) ) = { - 0.023607 , 0.42361 , 0.5 - 0.1 i , 0.5 + 0.1 i } ; spec ( ν R ( α i , α j ) ) = { - 0.027074 , 0.4000 , 0.61354 - 0.25845 i , 0.61354 + 0.25845 i } .

Theorem 5. Let G = (S, R) be a picture fuzzy graph on m vertices and L (G) be its Laplacian matrix. If δ₁ ≥ δ₂ ≥ … ≥ δ _m , υ₁ ≥ υ₂ ≥ … ≥ υ _m and ω₁ ≥ ω₂ ≥ … ≥ ω _m are the absolute eigenvalues of L (μ _R  (α _i , α _j )), L (η _R  (α _i , α _j )) and L (ν _R  (α _i , α _j ))) respectively, then

∑ i = 1 , δ i ∈ Δ m δ i = 2 ∑ 1 ≤ i < j ≤ m μ R ( α i , α j ) ; ∑ i = 1 , υ i ∈ ϒ m υ i = 2 ∑ 1 ≤ i < j ≤ m η R ( α i , α j ) ; and ∑ i = 1 , ω i ∈ Ω m ω i = 2 ∑ 1 ≤ i < j ≤ m ν R ( α i , α j ) .

Proof: The theorem can be proved on the similar lines as the proof of Theorem 1.

Definition 14. The Laplacian energy of the picture fuzzy graph G = (S, R), denoted by LE (G), is defined as LE ( G ) = ( LE ( μ R ( α i , α j ) ) , LE ( η R ( α i , α j ) ) , LE ( η R ( α i , α j ) ) ) = ( ∑ i = 1 m | ρ i | , ∑ i = 1 m | ξ i | , ∑ i = 1 m | ς i | ) ;

where ρ i = δ i - 2 ∑ 1 ≤ i < j ≤ m μ R ( α i , α j ) m ; ξ i = υ i - 2 ∑ 1 ≤ i < j ≤ m η R ( α i , α j ) m ; ς i = ω i - 2 ∑ 1 ≤ i < j ≤ m ν R ( α i , α j ) m .

Example: Here, we use the example used in the definition of Laplacian matrix (Definition 1) to present the evaluated Laplacian energy of the picture fuzzy graph as: LE ( μ R ( α i , α j ) ) = 1.845103 , LE ( η R ( α i , α j ) ) = 1.467021 , LE ( ν R ( α i , α j ) ) = 1.758581 .

Theorem 6. Let G = (S, R) be a picture fuzzy graph on m vertices and L (G) be its Laplacian matrix. If δ₁ ≥ δ₂ ≥ … ≥ δ _m , υ₁ ≥ υ₂ ≥ … ≥ υ _m and ω₁ ≥ ω₂ ≥ … ≥ ω _m are the absolute eigenvalues of L (μ _R  (α _i , α _j )), L (η _R  (α _i , α _j )) and L (ν _R  (α _i , α _j ))), respectively, then

∑ i = 1 m ρ i = 0 , ∑ i = 1 m ξ i = 0 and ∑ i = 1 m ς i = 0 ;

Proof: The proof of the theorem is obvious.

Theorem 7. Let G = (S, R) be a picture fuzzy graph on m vertices and L (G) be its Laplacian matrix. Then

LE ( μ R ( α i , α j ) ) ≤ 2 mM μ + m ∑ i = 1 m ( d R ( α i , α j ) ( α i ) - 2 ∑ 1 ≤ i < j ≤ m μ R ( α i , α j ) m ) 2 ;

Proof: The proof can be given on the similar lines as the proof of the Theorem 2.

Theorem 8. Let G = (S, R) be a picture fuzzy graph on m vertices and L (G) be its Laplacian matrix. Then

LE ( μ R ( α i , α j ) ) ≥ 2 M μ + 1 2 ∑ i = 1 m ( d R ( α i , α j ) ( α i ) - 2 ∑ 1 ≤ i < j ≤ m μ R ( α i , α j ) m ) 2 ;

Proof: The proof can be given on the similar lines as the proof of the Theorem 2.

The results obtained in the above theorems provide us the upper bound and lower bound of the Laplacian energy of the picture fuzzy graph G.

Theorem 9. Let G = (S, R) be a picture fuzzy graph on m vertices and L (G) be its Laplacian matrix. Then

LE ( μ R ( α i , α j ) ) ≤ | ρ 1 | + ( m - 1 ) ( 2 N μ - ρ 1 2 ) ;

Proof: The proof can be given on the similar lines as the proof of the Theorem 3.

In case of the directed graph, the adjacency matrix A (G) of a picture fuzzy directed graph is not necessarily symmetric. Therefore, the eigenvalues of the adjacency matrix may be complex numbers. This section generalizes the concept of energy and Laplacian energy for picture fuzzy directed graphs.

Definition 15. The spectrum of the adjacency matrix A (G) of the picture fuzzy directed graph G = ( S , R → ) is given by {(Θ, Φ, Ψ)}, where Θ, Φ and Ψ are the set of the eigenvalues of A ( μ R → ( α i , α j ) ) , A ( η R → ( α i , α j ) ) and A ( ν R → ( α i , α j ) ) ) , respectively.

Definition 16. The energy of the picture fuzzy directed graph G is given as: E ( G ) = ( E ( μ R → ( α i , α j ) ) , E ( η R → ( α i , α j ) ) , E ( ν R → ( α i , α j ) ) ) = ( ∑ i = 1 , θ i ∈ Θ m | Re ( θ i ) | , ∑ i = 1 , φ i ∈ Φ m | Re ( φ i ) | , ∑ i = 1 , ψ i ∈ Ψ m | Re ( ψ i ) | ) ; where Re (θ _i ), Re (φ _i ) and Re (ψ _i ) represents the real part of the eigenvalues θ _i , φ _i and ψ _i , respectively.

Theorem 10. Let G = ( S , R → ) be a picture fuzzy directed graph and A (G) be its adjacency matrix. If θ₁ ≥ θ₂ ≥ … ≥ θ _m , φ₁ ≥ φ₂ ≥ … ≥ φ _m and ψ₁ ≥ ψ₂ ≥ … ≥ ψ _m are the absolute eigenvalues of A (μ _R  (α _i , α _j ), A (η _R  (α _i , α _j ) and A (ν _R  (α _i , α _j ), respectively, then ∑ i = 1 , θ i ∈ Θ m Re ( θ i ) = 0 , ∑ i = 1 , φ i ∈ Φ m Re ( φ i ) = 0 and ∑ i = 1 , ψ i ∈ Ψ m Re ( ψ i ) = 0 .

Definition 17. Let G = ( S , R → ) be a picture fuzzy graph on m vertices. The out-degree matrix D ^out  (G) = [d _ij ], of G is a m × m diagonal matrix defined as: d ij = { d G out ( α i ) if i = j ; 0 otherwise .

Definition 18. Let G = ( S , R → ) be a picture fuzzy directed graph on m vertices. The Laplacian matrix of a picture fuzzy directed G, denoted by L (G) is defined as L ( G ) = D out ( G ) - A ( G ) ; where D ^out  (G)  &  A (G) are the out degree matrix and adjacency matrix of the picture fuzzy directed graph G, respectively.

Definition 19. The spectrum of the Laplacian matrix L (G) of the picture fuzzy directed graph G = ( S , R → ) is given by {(Δ, ϒ, Ω)}, where Δ, ϒ and Ω are the set of the eigenvalues of L ( μ R → ( α i , α j ) ) , L ( η R → ( α i , α j ) ) and L ( ν R → ( α i , α j ) ) , respectively.

Theorem 11. Let G = ( S , R → ) be a picture fuzzy graph on m vertices and L (G) be its Laplacian matrix. If δ₁ ≥ δ₂ ≥ … ≥ δ _m , υ₁ ≥ υ₂ ≥ … ≥ υ _m and ω₁ ≥ ω₂ ≥ … ≥ ω _m are the absolute eigenvalues of L ( μ R → ( α i , α j ) ) , L ( η R → ( α i , α j ) ) and L ( ν R → ( α i , α j ) ) respectively, then ∑ i = 1 , δ i ∈ Δ m Re ( δ i ) = tr ( L ( μ R → ( α i , α j ) ) ) ; ∑ i = 1 , υ i ∈ ϒ m Re ( υ i ) = tr ( L ( η R → ( α i , α j ) ) ) ; and ∑ i = 1 , ω i ∈ Ω m Re ( ω i ) = tr ( L ( ν R → ( α i , α j ) ) ) .

Proof: The proof can be given on the similar lines as the proof of the Theorem 1.

Definition 20. The Laplacian energy of the picture fuzzy directed graph G = ( S , R → ) denoted by LE (G) is defined as LE ( G ) = ( LE ( μ R → ( α i , α j ) ) , LE ( η R → ( α i , α j ) ) , LE ( η R → ( α i , α j ) ) ) = ( ∑ i = 1 m | ρ i | , ∑ i = 1 m | ξ i | , ∑ i = 1 m | ς i | ) ; where ρ i = Re ( δ i ) - ∑ i = 1 , δ i ∈ Δ m Re ( δ i ) m ; ξ i = Re ( υ i ) - ∑ i = 1 , υ i ∈ ϒ m Re ( υ i ) m ; ς i = Re ( ω i ) - ∑ i = 1 , ω i ∈ Ω m Re ( ω i ) m .

Theorem 12. Let G = ( S , R → ) be a picture fuzzy directed graph on m vertices and L (G) be its Laplacian matrix. If δ₁ ≥ δ₂ ≥ … ≥ δ _m , υ₁ ≥ υ₂ ≥ … ≥ υ _m and ω₁ ≥ ω₂ ≥ … ≥ ω _m are the absolute eigenvalues of L ( μ R → ( α i , α j ) ) , L ( η R → ( α i , α j ) ) and L ( ν R → ( α i , α j ) ) ) , respectively, then ∑ i = 1 m ρ i = 0 , ∑ i = 1 m ξ i = 0 and ∑ i = 1 m ς i = 0 .

Proof: The proof of the theorem is obvious.

For illustrating the proposed definitions, we consider the following example of a picture directed fuzzy graph:

Example 2. Suppose G = (S, R) be a picture fuzzy graph as given in Figure 3.

OPEN IN VIEWER

Fig. 3

Graph G = (S, R).

In view of the above definitions and Figure 3, the adjacency matrix of picture fuzzy directed graph can be given by A ( G ) = ( ( 0.0 , 0.0 , 0.0 ) ( 0.4 , 0.2 , 0.3 ) ( 0.4 , 0.3 , 0.2 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.3 , 0.5 , 0.2 ) ( 0.0 , 0.0 , 0.0 ) ( 0.2 , 0.1 , 0.6 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.5 , 0.2 , 0.3 ) ( 0.0 , 0.0 , 0.0 ) ( 0.5 , 0.2 , 0.2 ) ( 0.0 , 0.0 , 0.0 ) ) . The spectrum of the picture fuzzy directed graph G can be computed as Spec ( μ R → ( α i , α j ) ) = { 0.48586 , - 0.17786 + 0.3976 i , - 0.17786 - 0.3976 i , - 0.13014 } ; Spec ( η R → ( α i , α j ) ) = { 0.3387 , - 0.1208 + 0.27689 i , - 0.1208 - 0.27689 i , - 0.0970 } ; Spec ( ν R → ( α i , α j ) ) = { 0.39225 , - 0.11764 + 0.32108 i , - 0.11764 - 0.32108 i , - 0.156972 } . Hence, the spectrum of picture fuzzy directed graph G may be presented as Spec ( G ) = { ( 0.48556 , 0.3387 , 0.39225 ) , ( - 0.17786 + 0.3976 i , - 0.1208 + 0.27689 i , - 0.11764 + 0.32108 i ) , ( - 0.17786 - 0.3976 i , - 0.1208 - 0.27689 i , - 0.11764 - 0.32108 i ) , ( - 0.13014 , - 0.0970 , - 0.156972 ) } . The calculation of the components for energy of picture fuzzy directed graph G has been listed below: E ( μ R → ( α i , α j ) ) = - 0.13014 ; E ( η R → ( α i , α j ) ) = - 0.0970 ; E ( ν R → ( α i , α j ) ) = - 0.156972 . Hence, the energy of picture fuzzy directed graph G is E ( G ) = ( - 0.13014 , - 0.0970 , - 03156972 ) . Next, the out-degree matrix D (G) and the Laplacian matrix L (G) of the picture fuzzy directed graph G are given by D out ( G ) = ( ( 0.8 , 0.5 , 0.5 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.2 , 0.1 , 0.6 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.5 , 0.2 , 0.3 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 0.0 , 0.0 , 0.0 ) ( 1.0 , 0.4 , 0.5 ) ) and