q -Rung orthopair fuzzy graphs under Hamacher operators

Abstract

A q-rung orthopair fuzzy set (q-ROFS) is more practical and powerful than intuitionistic fuzzy set (IFS) and Pythagorean fuzzy set (PFS) to model uncertainty in various decision-making problems. In this research article, we introduce the notion of q-rung orthopair fuzzy Hamacher graphs (q-ROFHGs). We utilize the Hamacher operators because they are flexible and parameterized in decision making. We determine the energy of q-ROFHGs as well as the energy of splitting and shadow q-ROFHGs. In addition, we propose the Randić energy of q-ROFHG and its some substantial results. Further, we present the idea of q-rung orthopair fuzzy Hamacher digraphs (q-ROFHDGs). We solve a decision-making numerical example related to the selection of best housing society for investment by calculating the energy and Randić energy of q-ROFHDGs and an algorithm to exhibit the applicability of the presented concepts in decision making. Finally, we present the conclusion.

Keywords

Energy Splitting graph Shadow graph Randić energy

1 Introduction

Despite of the boosting complexity in many disciplines, decision makers have to tackle the problems in various difficult situations. The existence of ambiguity and vagueness in decision-making problems has made it difficult to represent attribute values properly. Researchers invented many tools to exhibit ambiguity and vagueness. In 1965, Zadeh [45] inaugurated the innovative concept of fuzzy set (FS). The drawback of FS was that it contained only membership degree (MD). IFS was primarily proposed by Attanasov [6], characterized by an MD and a non-membership degree (NMD) having the constraint, 0 ≤ μ + ν ≤ 1, where μ and ν represent the MD and NMD, respectively. However, IFS has some limitations. Motivated by this, Yager [39 , 42] presented the remarkable notion of PFS, which relax the constraint of IFS with the sum of squares of MD and NMD should be no greater than one. PFSs are more general and incorporate more amount of vagueness than IFS. For example, if the MD of an element to a set is 0.8 and the NMD is 0.4, then 0.8 + 0.4 > 1, therefore IFS fails to manage this information. On the other hand, (0.8) ² + (0.4) ² < 1 . Thus the PFS is capable to address this issue. Afterthat, Zhang and Xu [51] presented the idea of Pythagorean fuzzy number (PFN). After the formation of PFS, Yager [40] the idea of q-ROFS of prodigious reputation, with the condition, 0 ≤ μ^q + ν^q ≤ 1 . Consider the case with the MD 0.7 and NMD 0.8. For q = 3, it can be represented by (0.7) ^q + (0.8) ^q ≤ 1, and (0.7, 0.8) _q is a q-rung orthopair fuzzy number (q-ROFN). q-ROFSs enlarge the space of orthopairs and the results can be represented more accurately by increasing the value of q .

Researchers presented various operators, including min-max, Einstein, Frank, Product, Lukasiewicz, Hamacher, Azcel-Alsina and Dombi operators. Menger [29] proposed t-norms (triangular norms) and t-conorms (triangular conorms). Hamacher [16] introduced Hamacher operations, including Hamacher sum and Hamacher product. Liu and Wang [27] presented q-rung orthopair fuzzy aggregation operators. Liu et al. presented certain aggregation operators [26, 28] under q-rung orthopair fuzzy environment. Wu et al. [38] proposed Pythagorean fuzzy Hamacher aggregation operators and provide their application in decision-making. Further, Darko and Liang [9] presented q-rung orthopair fuzzy Hamacher aggregation operators with application. Euler laid the foundation of graph theory in 1735 by solving Koingsberg bridges^, problem. A graph is a convenient too to model and represent relationships digrammatically between various objects called entities. Kaufmann [18] initiated fuzzy graphs (FGs), based on [46]. Rosenfeld [33] discussed the notion of FG and its structure. Parvathi and Karunambigai [31] presented the concept of intuitionistic fuzzy graph (IFG). Naz et al. [30] initiated the idea of Pythagorean fuzzy graph (PFG). Akram et al. [2] introduced different graphs and elaborate their significant characteristics under Pythagorean fuzzy circumstances. Habib et al. [15] initiated the concept of q-rung orthopair fuzzy graphs (q-ROFGs) and provide its application in the soil system. In addition, Lin et al. [25] studied and discussed product operations on q-ROFGs by applying min - max operator. For further study on decision making techniques, the readers are referred to [4 , 47–50].

Graph energy is basically related to its spectrum. Gutman [12] initiated the idea of graph energy in chemistry, due to its relevance to the total π electron energy of some molecules and found lower and upper limits of the graph energy [13]. A graph without edges $K_{n}^{c}$ has energy zero, while the complete graph has energy 2 (n - 1) , where n represents the number of vertices. Vaidya and Popat [35] investigated the energy of splitting and shadow graphs. Gutman and Zhou [14] studied the Laplacian energy of a graph. The Randić matrix R (G) = (x_ij) of a graph G whose vertex v_i has degree d_i is defined by $x_{ij} = \frac{1}{\sqrt{d_{i} d_{j}}},$ if the vertices v_i and v_j are adjacent and x_ij = 0 otherwise. The Randić energy is the sum of absolute values of the eigenvalues of R (G) . The energy of a FG was first discussed by Anjali and Mathew [7]. Sharbaf and Fayazi [34] presented the Laplacian energy of a FG. Praba et al. [32] introduced the energy of IFGs. The Laplacian energy of an IFG was studied by Basha and Kartheek [8]. Later, Akram et al. [1] presented the energy and Laplacian energy of PFGs. Different techniques and decision making methods were developed by the scholars to cope the new challenges in different fields [3 , 44]. q-ROFSs are more powerful and general in contrast with IFSs and PFSs as they cover greater amount of information for orthopairs. Moreover, the Hamacher t-norm and t-conorm as a generalization of the algebraic and Einstein t-norm and t-conorm, are more flexible and parameterized. Inspired by the existing studies, we present the novel concept of q-rung orthopair fuzzy Hamacher graphs (q-ROFHGs) as they are highly effective and flexible to express the opinions of the decision makers in comparison with various existing theories. As the energy of q-ROFGs have not yet been discussed and studied, therefore motivated from the previous work, in this paper, we present the energy and Randić energy of q-ROFGs and q-ROFDGs. We employ graph terminology on q-ROFSs using Hamacher operators. In this paper, we introduce the energy and Randić energy of q-ROFHGs and q-ROFHDGs. The following points motivate us to write this article:

The spectrum of graph theory has a wide range of applications in computer science, chemistry, physics and other branches of mathematics. Graph spectrum plays a vital role to pattern recognition and appears in combinatorial optimization problems in mathematics.

The concept of energy of a graph is adopted by the energy in chemistry. Many mathematical properties of energy of graphs are being investigated. Inspired by the idea of energy of fuzzy graphs, we have extended the concept of energy and Randić energy to q-ROFGs.

The q-ROFS as an extension of IFS and PFS is an effective tool to model the situations, as it permits more space of admissible orthopairs than IFS and PFS.

In this article, we utilize the Hamacher aggregation operators as they are generalized and parameterized. Therefore, we apply the Hamacher aggregation operators to determine the energy and Randić energy of fuzzy graphs based on q-ROF environment.

Following are the contributions of this article:

The concept of q-ROFHG based on Hamacher t-norm and t-conorm is introduced.

Energy and Randić energy of q-ROFHGs are discussed by providing interesting results. Besides this, the energy of a splitting q-ROFHG and the energy of a shadow q-ROFHG is developed.

An algorithm to deal the numerical problems is presented with q-ROF data.

The presented algorithm is further applied to a problem to get a fair decision.

This paper is organized as follows: In section 2, some basic notions are discussed. Section 3 proposes the idea of q-ROFHGs and q-ROFHDGs and determines their energy. We also discuss the energy of a splitting q-ROFHG and the energy of shadow q-ROFHG. In Section 4, Randić energy of q-ROFHGs and q-ROFHDGs are discussed. Section 5 provides numerical examples by using q-rung orthopair fuzzy Hamacher averaging (q-ROFHA) operator, q-rung orthopair fuzzy Hamacher weighted averaging (q-ROFHWA) operator, q-rung orthopair fuzzy Hamacher geometric (q-ROFHG) operator and q-rung orthopair fuzzy Hamacher weighted geometric (q-ROFHWG) operator. Finally, in Section 6, we present the conclusion. For the sake of convenience, we have elaborated the mathematical symbols as follows:

Symbols	Mathematical description
κ	Membership function of a q-ROFS
λ	Nonmembership function of a q-ROFS
π	Hesitancy degree of a q-ROFS
⇔	If and only if
$T$	t-norm
	s-norm
$G = (U, V)$	q-ROFHG, where $U$ is the vertex set and $V$ is the edge set
G^* = (Y, E)	Crisp graph, where Y is the vertex set and E is the edge set
$A (G)$	Adjacency matrix of q-ROFHG $G$
$E (G)$	Energy of q-ROFHG $G$
η	Eigenvalues of the adjacency matrix corresponding to the membership function κ
ρ	Eigenvalues of the adjacency matrix corresponding to the nonmembership function λ
$S (G)$	Splitting q-ROFHG
$S h (G)$	Shadow q-ROFHG
c ∼ m	Vertex $g_{c}$ and $g_{m}$ are adjacent
c ≁ m	Vertex $g_{c}$ and $g_{m}$ are not adjacent
$RE (G)$	Randić energy of q-ROFHG
$d_{G} (g_{c})$	Degree of a vertex $g_{c}$ in q-ROFHG

2 Preliminaries

In this section, some basic notions are recalled for the better understanding of presented work.

IFS and PFS have a lot of applications in decision making. However, in some substantial decision making problems, these sets are unable to model the data. Due to this limitation of IFS and PFS, Yager [40] generalized the concept of IFS and PFS by introducing q-ROFS with the constraint κ^q + λ^q ≤ 1 (q ≥ 1) . q-ROFS allows managing of information more generally than IFS and PFS, because this model still applies in cases of IFS and PFS.

Definition 2.1. [40] A q-ROFS $U$ on a fixed set Y is represented as: $U = {(g, κ_{U} (g), λ_{U} (g)) | g \in Y},$ characterized by a membership function $κ_{U}$ and a non-membership function $λ_{U},$ where: $κ_{U} : Y \to [0, 1]$ and $λ_{U} : Y \to [0, 1],$ such that $0 \leq κ_{U}^{q} (g) + λ_{U}^{q} (g) \leq 1, q \geq 1$ for all $g \in Y .$ $π_{U} (g) = \sqrt[q]{1 - κ_{U}^{q} (g) - λ_{U}^{q} (g)}$ is called a q-rung orthopair hesitancy degree of $g$ in Y. Moreover, α = (κ_α, λ_α) is called a q-ROFN with constraints, κ_α, λ_α ∈ [0, 1] , $κ_{α}^{q} + λ_{α}^{q} \leq 1$ and $π_{α} = \sqrt[q]{1 - κ_{α}^{q} - λ_{α}^{q}} .$

Definition 2.2. [40] Let Y be a fixed set. Consider three q-ROFSs $U = {(g, κ_{U} (g), λ_{U} (g)) : g \in Y},$ $U_{1} = {(g, κ_{U_{1}} (g), λ_{U_{1}} (g)) : g \in Y}$ and $U_{2} = {(g, κ_{U_{2}} (g), λ_{U_{2}} (g)) : g \in Y}$ in Y, then

(i) $U_{1} \subseteq U_{2}$ ⇔ $κ_{U_{1}} (g) \leq κ_{U_{2}} (g)$ and $λ_{U_{1}} (g) \geq λ_{U_{2}} (g),$ for all $g \in Y;$

(ii) $U_{1} = U_{2}$ ⇔ $κ_{U_{1}} (g) = κ_{U_{2}} (g)$ and $λ_{U_{1}} (g) = λ_{U_{2}} (g),$ for all $g \in Y;$

(iii) $\bar{U} = {(g, λ_{U} (g), κ_{U} (g)) : g \in Y} .$

Example 2.1. Consider a fixed set Y having only one element $g,$ $κ_{U} (g) = 0.9,$ $λ_{U} (g) = 0.7 .$ Then $U = {(g, 0.9, 0.7) : g \in Y}$ is a 4-ROFS and $U = (0.9, 0.7)$ is a q-ROFN.

Definition 2.3. [15] A q-rung orthopair fuzzy relation (q-ROFR) $V$ on Y × Y is represented as: $V = {(gh, κ_{V} (gh), λ_{V} (gh)) | gh \in Y \times Y},$ characterized by a membership function $κ_{V}$ and a non-membership function $λ_{V},$ where: $κ_{V} : Y \times Y \to [0, 1]$ and $λ_{V} : Y \times Y \to [0, 1],$ such that $0 \leq κ_{V}^{q} (gh) + λ_{V}^{q} (gh) \leq 1, q \geq 1$ for all $gh \in Y \times Y .$

The q-rung orthopair fuzzy preference relation (q-ROFPR) can be stated as follows:

Definition 2.4. [30] Let $V = {g_{1}, g_{2}, \dots, g_{n}}$ be a set of alternatives. A q-ROFPR on V is represented by a matrix Q = (b_aj) _n×n, where $b_{aj} = (g_{a} g_{j}, κ (g_{a} g_{j}), λ (g_{a} g_{j}))$ for all a, j = 1, 2, ⋯ , n . Here κ_aj represents the degree to which the object $g_{a}$ is preferred to the object $g_{j}$ and λ_aj represents the degree to which the object $g_{a}$ is not preferred to the object $g_{j},$ and $π_{aj} = \sqrt[q]{1 - κ_{aj}^{q} - λ_{aj}^{q}}$ is the hesitancy degree with the following conditions κ_aj, λ_aj ∈ [0, 1] , $κ_{aj}^{q} + λ_{aj}^{q} \leq 1,$ κ_aj = λ_ja, κ_aa = λ_aa = 0.5, for all $a, j = 1, 2, \dots, n .$

For comparison of between two q-ROFNs, a score and accuracy function is provided as:

Definition 2.5. [40] Suppose $B = (κ_{B}, λ_{B})$ is a q-ROFN. The score function $S (B)$ (SF) and accuracy function (AF) $A (B)$ of $B$ can be defined as:

$S (B) = \frac{1}{2} (1 + κ_{B}^{q} - λ_{B}^{q}), where S (B) \in [0, 1],$ $A (B) = κ_{B}^{q} + λ_{B}^{q}, where A (B) \in [0, 1] .$

A ranking method for q-ROFNs, based on score and accuracy function is described as follows:

Definition 2.6. [51] Suppose Y is the fixed set. $B_{1} = (κ_{B_{1}}, λ_{B_{1}})$ and $B_{2} = (κ_{B_{2}}, λ_{B_{2}})$ are two q-ROFNs on Y. $S (B_{1}),$ $S (B_{2})$ and $A (B_{1}),$ $A (B_{2})$ are the SF and AF of $B_{1}$ and $B_{2},$ respectively. Then

If $S (B_{1}) < S (B_{2}),$ then $B_{1} < B_{2};$ ( $B_{1}$ is less than $B_{2}$ )

If $S (B_{1}) > S (B_{2}),$ then $B_{1} > B_{2};$ ( $B_{1}$ is greater than $B_{2}$ )

If $S (B_{1}) = S (B_{2}),$ then the AF is defined as:

If $A (B_{1}) < A (B_{2}),$ then $B_{1} < B_{2};$

If $A (B_{1}) > A (B_{2}),$ then $B_{1} > B_{2};$

If $A (B_{1}) = A (B_{2}),$ then $B_{1} = B_{2} .$

Some special cases of t-norm and t-conorm particularly Hamacher sum and product is described as follows:

Definition 2.7. [19] A binary function $T$ : [0, 1] × [0, 1] → [0, 1] is said to be t-norm if for all $g, h, l \in [0, 1],$ it fulfills the following:

$T$ $(g, 1) = g;$

$T$ $(g, h) =$ $T$ $(h, g);$

$T$ $(g,$ $T$ $(h, l)) =$ $T$ $($ $T$ $(g, h), l);$

$T$ $(g, h) \leq$ $T$ $(l, a)$ if $g \leq l$ and $h \leq a .$

Definition 2.8. [19] A binary function : [0, 1] × [0, 1] → [0, 1] is said to be t-conorm if there is a t-norm $T$ such that $(g, h) = 1 - T (1 - g, 1 - h)$ for all $(g, h) \in [0, 1]^{2} .$

Hamacher[16] established Hamacher product and Hamacher sum which are the generalizations of existing t-norm and t-conorms operations, defined by $\begin{matrix} T_{ϱ}^{H} (g, h) \\ = {\begin{matrix} \frac{g h}{ϱ + (1 - ϱ) (g + h - g h)}, if & ϱ > 0; \\ \frac{g h}{g + h - g h}, if & ϱ = 0, \end{matrix} \end{matrix}$ $\begin{matrix} {T^{*}}_{ϱ}^{H} (g, h) \\ = {\begin{matrix} \frac{g + h - g h - (1 - ϱ) g h}{1 - (1 - ϱ) (g h)}, if & ϱ > 0; \\ \frac{g + h - 2 g h}{1 - g h}, if & ϱ = 0 . \end{matrix} \end{matrix}$ For ϱ = 1, the Hamacher t-norm and t-conorm convert into the algebraic t-norm and t-conorm as follows: $\begin{matrix} P & = & gh, \\ P^{*} (g, h) & = & g + h - gh . \end{matrix}$ For ϱ = 2, the Hamacher t-norm and t-conorm convert into the Einstein t-norm and t-conorm [36] as follows: $\begin{matrix} I (g, h) & = & \frac{gh}{1 + (1 - g) (1 - h)}, \\ I^{*} (g, h) & = & \frac{g + h}{1 + gh} . \end{matrix}$

3 q-Rung orthopair fuzzy hamacher graphs

In this section, we introduce the concept of q-ROFG using Hamacher operator. We also discuss the energy of a q-ROFG and its crucial properties. In particular, we develop the energy of a splitting q-ROFG and the energy of a shadow q-ROFG.

Definition 3.1. A q-ROFHG on a fixed set Y is an ordered pair $G = (U, V),$ where $U = (κ_{U}, λ_{U}) : Y \to [0, 1]$ is a q-ROFS in Y and $V = (κ_{V}, λ_{V}) : Y \times Y \to [0, 1]$ is a q-ROFR on Y such that:

$\begin{matrix} κ_{V} (gj) & \leq \frac{κ_{U} (g) κ_{U} (j)}{ϱ + (1 - ϱ) (κ_{U} (g) + κ_{U} (j) - κ_{U} (g) κ_{U} (j))}, \\ λ_{V} (gj) & \leq \frac{λ_{U} (g) + λ_{U} (j) - λ_{U} (g) λ_{U} (j) - (1 - ϱ) λ_{U} (g) λ_{U} (j)}{1 - (1 - ϱ) λ_{U} (g) λ_{U} (j)}, \end{matrix}$

for all $g, j \in Y,$ where $0 \leq κ_{V}^{q} (gj) + λ_{V}^{q} (gj) \leq 1 .$ We call $U$ and $V$ the q-rung orthopair fuzzy vertex set and q-rung orthopair fuzzy edge set of $G,$ respectively. Here, $V$ is a symmetric q-ROFR on $U .$ If $V$ is not symmetric on $U,$ then $G = (U, \vec{V})$ is called q-ROFHDG.

Example 3.1. Let $G = (U, V)$ be a 3-ROFHG on G^* = (Y, E) , as shown in Fig. 1, where $Y = {g_{1}, g_{2}, g_{3}, g_{4}, g_{5}}$ and $E = {g_{1} g_{2}, g_{1} g_{3}, g_{1} g_{4}, g_{2} g_{4}, g_{2} g_{5}, g_{3} g_{5}, g_{4} g_{5}} .$ The set of vertices $U$ and set of edges $V$ of $G$ are defined on Y and E, respectively.

$\begin{matrix} U = & (\frac{g_{1}}{(0.8, 0.7)}, \frac{g_{2}}{(0.9, 0.6)}, \frac{g_{3}}{(0.5, 0.9)}, \\ \frac{g_{4}}{(0.7, 0.8)}, \frac{g_{5}}{(0.6, 0.7)}), \\ V = & (\frac{g_{1} g_{2}}{(0.72, 0.78)}, \frac{g_{1} g_{3}}{(0.44, 0.90)}, \frac{g_{1} g_{4}}{(0.59, 0.86)}, \\ \frac{g_{2} g_{4}}{(0.63, 0.82)}, \frac{g_{2} g_{5}}{(0.55, 0.78)}, \frac{g_{3} g_{5}}{(0.36, 0.90)}, \frac{g_{4} g_{5}}{(0.47, 0.86)}) . \end{matrix}$

Fig. 1

3-rung orthopair fuzzy Hamacher graph

Now we define the energy of a q-ROFHG and discuss its properties.

The adjacency matrix (AM) $A (G) = (A (κ_{V} (g_{c} g_{m})), A (λ_{V} (g_{c} g_{m})))$ of a q-ROFHG $G = (U, V)$ is defined to be a square matrix $A (G) = [c_{cm}],$ $c_{cm} = (κ_{V} (g_{c} g_{m}), λ_{V} (g_{c} g_{m})),$ where $κ_{V} (g_{c} g_{m})$ and $λ_{V} (g_{c} g_{m})$ indicate the MD and NMD between $g_{c}$ and $g_{m},$ respectively.

Definition 3.2. Suppose a q-ROFHG $G = (U, V) .$ The spectrum of AM of a q-ROFHG $A (G)$ is defined as (C, D) , where C and D are the sets of eigenvalues of $A (κ_{V} (g_{c} g_{m}))$ and $A (λ_{V} (g_{c} g_{m})),$ respectively.

Definition 3.3. The energy of a q-ROFHG $G = (U, V)$ is defined as: $\begin{matrix} E (G) & = & (E (κ_{V} (g_{c} g_{m})), E (λ_{V} (g_{c} g_{m}))) \\ = & (\sum_{c = 1}^{n} | η_{c} |, \sum_{c = 1}^{n} | ρ_{c} |), \end{matrix}$ where η_c ∈ C and ρ_c ∈ D .

Theorem 3.1. Suppose $G = (U, V)$ is a q-ROFHG, having its AM $A (G)$ . Suppose η₁, η₂, …, η_n are the eigenvalues of $A (κ_{V} (g_{c} g_{m}))$ with the condition η_c ≥ η_m (c < m) and ρ₁, ρ₂, …, ρ_n are the eigenvalues of $A (λ_{V} (g_{c} g_{m}))$ with the condition ρ_c ≥ ρ_m (c < m), then:

$\begin{matrix} (i) & \sum_{c = 1}^{n} η_{c} = 0 and \sum_{c = 1}^{n} ρ_{c} = 0, \\ (ii) & \sum_{c = 1}^{n} η_{c}^{2} = 2 \sum_{1 \leq c < m \leq n} κ_{V}^{2} (g_{c} g_{m}) \\ and \sum_{c = 1}^{n} ρ_{c}^{2} = 2 \sum_{1 \leq c < m \leq n} λ_{V}^{2} (g_{c} g_{m}), \end{matrix}$ where η_c ∈ C and ρ_c ∈ D .

Proof. (i) Obvious.

(ii) Using the trace properties of a matrix, we have: $tr (A^{2} (κ_{V} (g_{c} g_{m})) (G)) = \sum_{c = 1}^{n} η_{c}^{2}, (η_{c} \in C)$ where: $\begin{matrix} tr (A^{2} (κ_{V} (g_{c} g_{m})) (G)) \\ = (0 + κ_{V}^{2} (g_{1} g_{2}) + \dots + κ_{V}^{2} (g_{1} g_{n})) \\ + (κ_{V}^{2} (g_{2} g_{1}) + 0 + \dots + κ_{V}^{2} (g_{2} g_{n})) \\ ⋮ \\ + (κ_{V}^{2} (g_{n} g_{1}) + κ_{V}^{2} (g_{n} g_{2}) + \dots + 0) \\ = 2 \sum_{1 \leq c < m \leq n} κ_{V}^{2} (g_{c} g_{m}) . \end{matrix}$ Hence: $\sum_{c = 1}^{n} η_{c}^{2} = 2 \sum_{1 \leq c < m \leq n} κ_{V}^{2} (g_{c} g_{m}), (η_{c} \in C) .$

Similarly, it is easy to show that $\sum_{c = 1}^{n} ρ_{c}^{2} = 2 \sum_{1 \leq c < m \leq n} λ_{V}^{2} (g_{c} g_{m}), (ρ_{c} \in D) .$ □

Example 3.2. Let $G = (U, V)$ be a 4-ROFHG on G^* = (Y, E) , as shown in Fig. 2, where $Y = {g_{1}, g_{2}, g_{3}, g_{4}, g_{5}, g_{6}, g_{7}}$ and $E = {g_{1} g_{2}, g_{1} g_{4}, g_{2} g_{4}, g_{3} g_{5}, g_{4} g_{5}, g_{5} g_{7}, g_{6} g_{7}} .$ The set of vertices $U$ and set of edges $V$ of $G$ are defined on Y and E, respectively.

$\begin{matrix} U = & (\frac{g_{1}}{(0.6, 0.9)}, \frac{g_{2}}{(0.8, 0.8)}, \frac{g_{3}}{(0.7, 0.9)}, \frac{g_{4}}{(0.9, 0.6)}, \\ \frac{g_{5}}{(0.7, 0.8)}, \frac{g_{6}}{(0.3, 0.9)}, \frac{g_{7}}{(0.5, 0.8)}), \\ V = & (\frac{g_{1} g_{2}}{(0.50, 0.90)}, \frac{g_{1} g_{4}}{(0.55, 0.90)}, \frac{g_{2} g_{4}}{(0.73, 0.82)}, \\ \frac{g_{3} g_{5}}{(0.51, 0.92)}, \frac{g_{4} g_{5}}{(0.61, 0.84)}, \frac{g_{5} g_{7}}{(0.40, 0.87)}, \frac{g_{6} g_{7}}{(0.23, 0.91)}) . \end{matrix}$

The AM of a 4-ROFHG is given below: $A (G) = [\begin{matrix} (0, 0) & (0.50, 0.90) & (0, 0) & (0.55, 0.90) & (0, 0) & (0, 0) & (0, 0) \\ (0.50, 0.90) & (0, 0) & (0, 0) & (0.73, 0.83) & (0, 0) & (0, 0) & (0, 0) \\ (0, 0) & (0, 0) & (0, 0) & (0, 0) & (0.51, 0.92) & (0, 0) & (0, 0) \\ (0.55, 0.90) & (0.73, 0.82) & (0, 0) & (0, 0) & (0.61, 0.84) & (0, 0) & (0, 0) \\ (0, 0) & (0, 0) & (0.51, 0.92) & (0.61, 0.84) & (0, 0) & (0, 0) & (0.40, 0.87) \\ (0, 0) & (0, 0) & (0, 0) & (0, 0) & (0, 0) & (0, 0) & (0.23, 0.91) \\ (0, 0) & (0, 0) & (0, 0) & (0, 0) & (0.40, 0.87) & (0.23, 0.91) & (0, 0) \end{matrix}] .$

Fig. 2

4-rung orthopair fuzzy Hamacher graph

The spectrum of a 4-ROFHG $G$ is given as follows:

$\begin{matrix} Spec (κ_{V} (g_{i} g_{j})) = & {- 1.0842, - 0.5214, - 0.3854, \\ - 0.1656, 0.1799, 0.6337, 1.3430}, \\ Spec (λ_{V} (g_{i} g_{j})) = & {- 1.7184, - 0.9234, - 0.8596, \\ - 0.4288, 0.5959, 1.3383, 1.9961} . \end{matrix}$

Thus,

$Spec (G)$ = {(-1.0842, - 1.7184), (-0.5214, - 0.9234) , (-0.3854, - 0.8596) , (-0.1656, - 0.4288) , (0.1799, 0.5959) , (0.6337, 1.3383) , (1.3430, 1.9961)} .

Now, energy of a 4-ROFHG $G$ is given as follows: $E (κ_{V} (g_{c} g_{m})) = 4.3132$ and $E (λ_{V} (g_{c} g_{m})) = 7.8605 .$ Therefore, $E (G) = (4.3132, 7.8605) .$

Theorem 3.2. Suppose a q-ROFHG $G = (U, V),$ having n vertices and $A (G) = (A (κ_{V} (g_{c} g_{m})), A (λ_{V} (g_{c} g_{m})))$ is the AM of $G .$ Then:

$\begin{matrix} (i) \sqrt{2 (\sum_{1 \leq c \leq m \leq n} (κ_{V}^{2} (g_{c} g_{m}))) + n (n - 1) | \det (A (κ_{V} (g_{c} g_{m}))) |^{\frac{2}{n}}} \\ \leq E (κ_{V} (g_{c} g_{m})) \leq \sqrt{2 n (\sum_{1 \leq c \leq m \leq n} (κ_{V}^{2} (g_{c} g_{m})))}, \end{matrix}$ $\begin{matrix} (ii) \sqrt{2 (\sum_{1 \leq c \leq m \leq n} (λ_{V}^{2} (g_{c} g_{m}))) + n (n - 1) | \det (A (λ_{V} (g_{c} g_{m}))) |^{\frac{2}{n}}} \\ \leq E (λ_{V} (g_{c} g_{m})) \leq \sqrt{2 n (\sum_{1 \leq c \leq m \leq n} (λ_{V}^{2} (g_{c} g_{m})))} . \end{matrix}$

Now we define the energy of a splitting q-ROFHG and the energy of a shadow q-ROFHG.

Definition 3.4. The splitting q-ROFHG $S (G)$ of a q-ROFHG $G$ is a graph in which a new vertex $g^{'}$ is added to each vertex $g$ , such that $g^{'}$ is adjacent to every vertex that is adjacent to $g$ in $G$ with the same MD and NMD.

Theorem 3.3. Suppose a splitting q-ROFHG $S (G)$ of a q-ROFHG $G .$ Then $E (S (G)) = \sqrt{5} E (G) .$

Proof. Suppose a q-ROFHG having the vertex set ${g_{1}, g_{2}, \dots, g_{n}} .$ The AM of $G$ is $A (G) = (A (κ_{V} (g_{c} g_{m})) (G), A (λ_{V} (g_{c} g_{m})) (G)),$ where:

$\begin{matrix} A (κ_{V} (g_{c} g_{m})) (G) \\ = [\begin{matrix} 0 & κ_{V} (g_{1} g_{2}) & \dots & κ_{V} (g_{1} g_{n}) \\ κ_{V} (g_{2} g_{1}) & 0 & \dots & κ_{V} (g_{2} g_{n}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ κ_{V} (g_{n} g_{1}) & κ_{V} (g_{n} g_{2}) & \dots & 0 \end{matrix}] . \end{matrix}$ Consider new vertices $g_{1}^{'}, g_{2}^{'}, \dots, g_{n}^{'}$ corresponding to the vertices $g_{1}, g_{2}, \dots, g_{n}$ which are added in $G$ to obtain $S (κ_{V} (g_{c} g_{m})) (G),$ such that $N (g_{c}) = N (g_{i}^{'}),$ c = 1, 2, …, n . Now $A (κ_{V} (g_{c} g_{m})) (S (G))$ is a block matrix given as follows: $A (κ_{V} (g_{c} g_{m})) (S (G)) = [\begin{matrix} 0 & κ_{V} (g_{1} g_{2}) & \dots & κ_{V} (g_{1} g_{n}) & 0 & κ_{V} (g_{1} g_{2}^{'}) & \dots & κ_{V} (g_{1} g_{n}^{'}) \\ κ_{V} (g_{2} g_{1}) & 0 & \dots & κ_{V} (g_{2} g_{n}) & κ_{V} (g_{2} g_{1}^{'}) & 0 & \dots & κ_{V} (g_{2} g_{n}^{'}) \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ κ_{V} (g_{n} g_{1}) & κ_{V} (g_{n} g_{2}) & \dots & 0 & κ_{V} (g_{n} g_{1}^{'}) & κ_{V} (g_{n} g_{2}^{'}) & \dots & 0 \\ 0 & κ_{V} (g_{1}^{'} g_{2}) & \dots & κ_{V} (g_{1}^{'} g_{n}) & 0 & 0 & \dots & 0 \\ κ_{V} (g_{2}^{'} g_{1}) & 0 & \dots & κ_{V} (g_{2}^{'} g_{n}) & 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ κ_{V} (g_{n}^{'} g_{1}) & κ_{V} (g_{n}^{'} g_{2}) & \dots & 0 & 0 & 0 & \dots & 0 \end{matrix}],$ that is, $\begin{matrix} A (κ_{V} (g_{c} g_{m})) (S (G)) & = & [\begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix}] \otimes A (κ_{V} (g_{c} g_{m})) (G) . \end{matrix}$ As the eigenvalues of $[\begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix}]$ and $A (κ_{V} (g_{c} g_{m})) (G)$ are $\frac{1 \pm \sqrt{5}}{2}$ and η_c, c = 1, 2, …, n, respectively. Therefore, $\begin{matrix} E (κ_{V} (g_{c} g_{m})) (S (G)) = \sum_{c = 1}^{n} | (\frac{1 \pm \sqrt{5}}{2}) (η_{c}) | \\ = \sum_{c = 1}^{n} (\frac{1 + \sqrt{5}}{2} + \frac{1 - \sqrt{5}}{2}) (| η_{c} |) = \sqrt{5} \sum_{c = 1}^{n} (| η_{c} |) . \end{matrix}$ In a similar way, we can show that $E (λ_{V} (g_{c} g_{m})) (S (G)) = \sqrt{5} \sum_{c = 1}^{n} (| ρ_{c} |) .$ Hence, $E (S (G)) = \sqrt{5} E (G) .$ □

Definition 3.5. The shadow q-ROFHG $S h (G)$ of a connected q-ROFHG $G$ is formed by taking two copies of $G,$ say $G^{'}$ and $G^{″} .$ Join each vertex $g^{'}$ in $G^{'}$ to the neighbors of the corresponding vertex $g^{″}$ in $G^{″}$ with the same MD and NMD.

Theorem 3.4. Suppose a shadow q-ROFHG $S h (G)$ of a q-ROFHG $G .$ Then $E (S h (G)) = 2 E (G) .$

Proof. Suppose a q-ROFHG having the vertex set ${g_{1}, g_{2}, \dots, g_{n}} .$ The AM of $G$ is $A (G) = (A (κ_{V} (g_{c} g_{m})) (G), A (λ_{V} (g_{c} g_{m})) (G)),$ where: $\begin{matrix} A (κ_{V} (g_{c} g_{m})) (G) \\ = [\begin{matrix} 0 & κ_{V} (g_{1} g_{2}) & \dots & κ_{V} (g_{1} g_{n}) \\ κ_{V} (g_{2} g_{1}) & 0 & \dots & κ_{V} (g_{2} g_{n}) \\ ⋮ & ⋮ & ⋱ & ⋮ \\ κ_{V} (g_{n} g_{1}) & κ_{V} (g_{n} g_{2}) & \dots & 0 \end{matrix}] . \end{matrix}$ Consider new vertices $g_{1}^{'}, g_{2}^{'}, \dots, g_{n}^{'}$ corresponding to the vertices $g_{1}, g_{2}, \dots, g_{n}$ which are added in $G$ to obtain $S h (κ_{V} (g_{c} g_{m})) (G),$ such that $N (g_{c}) = N (g_{c}^{'}),$ c = 1, 2, …, n . Now $A (κ_{V} (g_{c} g_{m})) (S h (G))$ is a block matrix given as follows: $A (κ_{V} (g_{c} g_{m})) (S h (G)) = [\begin{matrix} 0 & κ_{V} (g_{1} g_{2}) & \dots & κ_{V} (g_{1} g_{n}) & 0 & κ_{V} (g_{1} g_{2}^{'}) & \dots & κ_{V} (g_{1} g_{n}^{'}) \\ κ_{V} (g_{2} g_{1}) & 0 & \dots & κ_{V} (g_{2} g_{n}) & κ_{V} (g_{2} g_{1}^{'}) & 0 & \dots & κ_{V} (g_{2} g_{n}^{'}) \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ κ_{V} (g_{n} g_{1}) & κ_{V} (g_{n} g_{2}) & \dots & 0 & κ_{V} (g_{n} g_{1}^{'}) & κ_{V} (g_{n} g_{2}^{'}) & \dots & 0 \\ 0 & κ_{V} (g_{1}^{'} g_{2}) & \dots & κ_{V} (g_{1}^{'} g_{n}) & 0 & κ_{V} (g_{1}^{'} g_{2}) & \dots & κ_{V} (g_{1}^{'} g_{n}^{'}) \\ κ_{V} (g_{2}^{'} g_{1}) & 0 & \dots & κ_{V} (g_{2}^{'} g_{n}) & κ_{V} (g_{2}^{'} g_{1}^{'}) & 0 & \dots & κ_{V} (g_{2}^{'} g_{n}^{'}) \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ κ_{V} (g_{n}^{'} g_{1}) & κ_{V} (g_{n}^{'} g_{2}) & \dots & 0 & κ_{V} (g_{n}^{'} g_{1}^{'}) & κ_{V} (g_{n}^{'} g_{2}^{'}) & \dots & 0 \end{matrix}],$ that is, $\begin{matrix} A (κ_{V} (g_{c} g_{m})) (S h (G)) \\ = [\begin{matrix} A (κ_{V} (g_{c} g_{m})) (G) & A (κ_{V} (g_{c} g_{m})) (G) \\ A (κ_{V} (g_{c} g_{m})) (G) & A (κ_{V} (g_{c} g_{m})) (G) \end{matrix}] \\ = [\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}] \otimes A (κ_{V} (g_{c} g_{m})) (G) . \end{matrix}$ As the eigenvalues of $[\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix}]$ and $A (κ_{V} (g_{c} g_{m})) (G)$ are 0,2 and η_c, c = 1, 2, …, n, respectively. Therefore, $\begin{matrix} E (κ_{V} (g_{c} g_{m})) (S h (G)) & = & \sum_{c = 1}^{n} | 2 (η_{c}) | = 2 \sum_{c = 1}^{n} (| η_{c} |) . \end{matrix}$ In a similar way, we can show that $E (λ_{V} (g_{c} g_{m})) (S h (G)) = 2 \sum_{c = 1}^{n} (| ρ_{c} |) .$ Hence, $E (S (G)) = 2 E (G) .$ □

Example 3.3. Let $G = (U, V)$ be a 2-ROFHG on G^* = (Y, E) , as shown in Fig. 3, where $Y = {g_{1}, g_{2}, g_{3}, g_{4}}$ and $E = {g_{1} g_{2}, g_{1} g_{3}, g_{1} g_{4}} .$ The set of vertices $U$ and set of edges $V$ of $G$ are defined on Y and E, respectively. $\begin{matrix} U = & (\frac{g_{1}}{(0.8, 0.5)}, \frac{g_{2}}{(0.7, 0.7)}, \frac{g_{3}}{(0.9, 0.2)}, \frac{g_{4}}{(0.3, 0.8)}), \end{matrix}$ $\begin{matrix} V = & (\frac{g_{1} g_{2}}{(0.58, 0.75)}, \frac{g_{1} g_{3}}{(0.72, 0.45)}, \frac{g_{1} g_{4}}{(0.27, 0.81)}) . \end{matrix}$ The AM and energy of $G$ are given below:

$\begin{matrix} A (G) \\ = [\begin{matrix} (0, 0) & (0.58, 0.75) & (0.73, 0.55) & (0.27, 0.83) \\ (0.58, 0.75) & (0, 0) & (0, 0) & (0, 0) \\ (0.73, 0.55) & (0, 0) & (0, 0) & (0, 0) \\ (0.27, 0.83) & (0, 0) & (0, 0) & (0, 0) \end{matrix}], \end{matrix}$

$E (G) = (1.9414, 2.4932) .$

Fig. 3

2-rung orthopair fuzzy Hamacher graph

Fig. 4

Splitting 2-rung orthopair fuzzy Hamacher graph

The AM and the energy of a splitting q-ROFHG of the q-ROFHG presented in Fig. 3 are given below: $A (S (G)) = [\begin{matrix} (0, 0) & (0.58, 0.75) & (0.73, 0.55) & (0.27, 0.83) & (0, 0) & (0.58, 0.75) & (0.73, 0.55) & (0.27, 0.83) \\ (0.58, 0.75) & (0, 0) & (0, 0) & (0, 0) & (0.58, 0.75) & (0, 0) & (0, 0) & (0, 0) \\ (0.73, 0.55) & (0, 0) & (0, 0) & (0, 0) & (0.73, 0.55) & (0, 0) & (0, 0) & (0, 0) \\ (0.27, 0.83) & (0, 0) & (0, 0) & (0, 0) & (0.27, 0.83) & (0, 0) & (0, 0) & (0, 0) \\ (0, 0) & (0.58, 0.75) & (0.73, 0.55) & (0.27, 0.83) & (0, 0) & (0, 0) & (0, 0) & (0, 0) \\ (0.58, 0.75) & (0, 0) & (0, 0) & (0, 0) & (0, 0) & (0, 0) & (0, 0) & (0, 0) \\ (0.73, 0.55) & (0, 0) & (0, 0) & (0, 0) & (0, 0) & (0, 0) & (0, 0) & (0, 0) \\ (0.27, 0.83) & (0, 0) & (0, 0) & (0, 0) & (0, 0) & (0, 0) & (0, 0) & (0, 0) \end{matrix}]$

$\begin{matrix} E (S (G)) & = & (4.3410, 5.5748) \\ E (S (G)) & = & \sqrt{5} (1.9413, 2.4931) \\ E (S (G)) & = & \sqrt{5} E (G) . \end{matrix}$ The splitting q-ROFHG of the q-ROFHG in Fig. 3 is shown in Fig. 4.

Fig. 5

Shadow 2-rung orthopair fuzzy Hamacher graph

The AM and the energy of a shadow q-ROFHG, shown in Fig. 5 are given below:

A (S h (G)) = [\begin{matrix} (0, 0) & (0.58, 0.75) & (0.73, 0.55) & (0.27, 0.83) & (0, 0) & (0.58, 0.75) & (0.73, 0.55) & (0.27, 0.83) \\ (0.58, 0.75) & (0, 0) & (0, 0) & (0, 0) & (0.58, 0.75) & (0, 0) & (0, 0) & (0, 0) \\ (0.73, 0.55) & (0, 0) & (0, 0) & (0, 0) & (0.73, 0.55) & (0, 0) & (0, 0) & (0, 0) \\ (0.27, 0.83) & (0, 0) & (0, 0) & (0, 0) & (0.27, 0.83) & (0, 0) & (0, 0) & (0, 0) \\ (0, 0) & (0.58, 0.75) & (0.73, 0.55) & (0.27, 0.83) & (0, 0) & (0.58, 0.75) & (0.73, 0.55) & (0.27, 0.83) \\ (0.58, 0.75) & (0, 0) & (0, 0) & (0, 0) & (0.58, 0.75) & (0, 0) & (0, 0) & (0, 0) \\ (0.73, 0.55) & (0, 0) & (0, 0) & (0, 0) & (0.73, 0.55) & (0, 0) & (0, 0) & (0, 0) \\ (0.27, 0.83) & (0, 0) & (0, 0) & (0, 0) & (0.27, 0.83) & (0, 0) & (0, 0) & (0, 0) \end{matrix}]

$\begin{matrix} E (S h (G)) & = & (3.8826, 4.9862) \\ E (S h (G)) & = & 2 (1.9413, 2.4931) \\ E (S h (G)) & = & 2 E (G) . \end{matrix}$

Definition 3.6. Let $\vec{G} = (U, \vec{V})$ be a q-ROFHDG. The spectrum of the AM of a q-ROFHDG $A (\vec{G})$ is defined as $(C, D),$ where $C$ and $D$ are the sets of eigenvalues of $A (κ_{\vec{G}} (g_{c} g_{m}))$ and $A (λ_{\vec{V}} (g_{c} g_{m})),$ respectively.

Definition 3.7. Let $\vec{G} = (U, \vec{V})$ be a q-ROFHDG on n vertices. The energy of $\vec{G}$ is defined as: $\begin{matrix} E (\vec{G}) & = & (E (κ_{\vec{V}} (g_{c} g_{m})), E (λ_{\vec{V}} (g_{c} g_{m}))) \\ = & (\sum_{c = 1}^{n} | Re (x_{c}) |, \sum_{c = 1}^{n} | Re (y_{c}) |), \end{matrix}$ where $x_{c} \in C$ and $y_{c} \in D .$ Note that Re (x_c) and Re (y_c) represent the real parts of the eigenvalues x_c and y_c respectively.

4 Randić energy of q-rung orthopair fuzzy hamacher graphs

In this section, we discuss the Randić energy $(RE)$ of a q-ROFHG and its related properties.

Definition 4.1. Suppose a q-ROFHG, $G = (U, V)$ on n vertices. An n × n matrix $R (G) = (R (κ_{G} (g_{c} g_{m}), λ_{G} (g_{c} g_{m}))) = [g_{cm}]$ of $G$ is defined to be Randić matrix with the following conditions:

$g_{cm} = {\begin{matrix} 0, & if c = m; \\ \frac{1}{\sqrt{d_{G} (g_{c}) d_{G} (g_{m})}}, & if c \sim m; \\ 0, & if c ≁ m . \end{matrix}$

Definition 4.2. The spectrum of Randić matrix of a q-ROFHG $R (G)$ is defined as $(C_{R}, D_{R}),$ where $C_{R}$ and $D_{R}$ are the sets of eigenvalues of $R (κ_{V} (g_{c} g_{m}))$ and $R (λ_{V} (g_{c} g_{m})),$ respectively.

Definition 4.3. The Randić energy of a q-ROFHG $G = (U, V)$ is defined as: $\begin{matrix} R E (G) & = & (R E (κ_{V} (g_{c} g_{m})), R E (λ_{V} (g_{c} g_{m}))) \\ = & (\sum_{c = 1}^{n} | η_{c} |, \sum_{c = 1}^{n} | ρ_{c} |), \end{matrix}$ where $η_{c} \in C_{R}$ and $ρ_{c} \in D_{R} .$

Lemma 4.1. Suppose a q-ROFHG $G = (U, V)$ on n vertices and $R (G) = (R (κ (g_{c} g_{m})), λ (g_{c} g_{m})))$ be its Randić matrix. Then

$tr (R (G)) = 0;$

$tr (R^{2} (G)) = 2 \sum_{c \sim m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})};$

$tr (R^{3} (G)) = 2 \sum_{c \sim m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})}$ $(\sum_{r \sim c, r \sim m} \frac{1}{d_{G} (g_{r})});$

$tr (R^{4} (G)) = \sum_{c = 1}^{n} (\sum_{c \sim m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})})^{2} + \sum_{c \neq m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})} (\sum_{r \sim c, r \sim m} \frac{1}{d_{G} (g_{m})})^{2} .$

Proof.

Since the sum of diagonal elements of $R (G)$ is zero. Therefore $tr (R (G)) = 0 .$

Now, we find the trace of $R^{2} (G) .$ For c = m $\begin{matrix} R^{2} (κ (g_{c} g_{c})) (G) \\ = \sum_{m = 1}^{n} R (κ (g_{c} g_{m})) (G) R (κ (g_{m} g_{c})) (G) \\ = \sum_{m = 1}^{n} R (κ (g_{c} g_{m}))^{2} (G) \\ = \sum_{c \sim m} R (κ (g_{c} g_{m}))^{2} (G) \\ = \sum_{c \sim m} \frac{1}{d_{(κ)} (g_{c}) d_{(κ)} (g_{m})} . \end{matrix}$ For c ≠ m $\begin{matrix} \begin{matrix} R^{2} (κ (g_{c} g_{m})) (G) \\ = \sum_{r = 1}^{n} R (κ (g_{c} g_{r})) (G) R (κ (g_{r} g_{m})) (G) \\ = R (κ (g_{c} g_{c})) (G) R (κ (g_{c} g_{m})) (G) \\ + R (κ (g_{c} g_{m})) (G) R (κ (g_{m} g_{m})) (G) + \\ \sum_{r \sim c, r \sim m} R (κ (g_{c} g_{r})) (G) R (κ (g_{r} g_{m})) (G) \\ = \frac{1}{\sqrt{d_{(κ)} (g_{c}) d_{(κ)} (g_{m})}} \sum_{r \sim c, r \sim m} \frac{1}{(d_{κ}) (g_{r})} . \\ tr (R^{2} (κ (g_{c} g_{m}))) (G) \\ = \sum_{c = 1}^{n} \sum_{c \sim m} \frac{1}{\sqrt{d_{(κ)} (g_{c}) d_{(κ)} (g_{m})}} \\ = 2 \sum_{c \sim m} \frac{1}{\sqrt{d_{(κ)} (g_{c}) d_{(κ)} (g_{m})}} . \end{matrix} \end{matrix}$ Similarly, $tr (R^{2} (λ (g_{c} g_{m}))) (G) = 2 \sum_{c \sim m} \frac{1}{d_{(λ)} (g_{c}) d_{(λ)} (g_{m})} .$ Hence, $tr (R^{2} (G)) = 2 \sum_{c \sim m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})} .$

$\begin{matrix} R^{3} (κ (g_{c} g_{c})) (G) \\ = \sum_{m = 1}^{n} R (κ (g_{c} g_{m})) (G) R^{2} (κ (g_{m} g_{r})) (G) \\ = \sum_{c \sim m} \frac{1}{\sqrt{d_{(κ)} (g_{c}) d_{(κ)} (g_{m})}} R^{2} (κ (g_{c} g_{m})) (G) \\ = \sum_{c \sim m} \frac{1}{d_{(κ)} (g_{c}) d_{(κ)} (g_{m})} (\sum_{r \sim c, r \sim m} \frac{1}{d_{(κ)} (g_{r})}) . \end{matrix}$ Therefore,

$\begin{matrix} tr (R^{3} (κ (g_{c} g_{c}))) (G) \\ = \sum_{c = 1}^{n} \sum_{c \sim m} \frac{1}{d_{(κ)} (g_{c}) d_{(κ)} (g_{m})} (\sum_{r \sim c, r \sim m} \frac{1}{d_{(κ)} (g_{r})}) \\ = 2 \sum_{c \sim m} \frac{1}{d_{(κ)} (g_{c}) d_{(κ)} (g_{m})} (\sum_{r \sim c, r \sim m} \frac{1}{d_{(κ)} (g_{r})}) . \end{matrix}$

It is easy to show that $tr (R^{3} (κ (g_{c} g_{c}))) (G) = 2 \sum_{c \sim m} \frac{1}{d_{(λ)} (g_{c}) d_{(λ)} (g_{m})} (\sum_{r \sim c, r \sim m} \frac{1}{d_{(λ)} (g_{r})}) .$ Hence, $tr (R^{3} (G)) = 2 \sum_{c \sim m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})} (\sum_{r \sim c, r \sim m} \frac{1}{d_{G} (g_{r})}) .$

Now, we determine $tr (R^{4} (κ (g_{c} g_{m}))) (G) .$ $\begin{matrix} tr (R^{4} (κ (g_{c} g_{m}))) (G) \\ = \sum_{c, m = 1}^{n} | R^{2} (κ (g_{c} g_{m})) (G) |^{2} \\ = \sum_{c = m} | R^{2} (κ (g_{c} g_{m})) (G) |^{2} \\ + \sum_{c \neq m} | R^{2} (κ (g_{c} g_{m})) (G) |^{2} \\ = \sum_{c = 1}^{n} (\sum_{c \sim m} \frac{1}{d_{(κ)} (g_{c}) d_{(κ)} (g_{m})})^{2} \\ + \sum_{c \neq m} \frac{1}{d_{(κ)} (g_{c}) d_{(κ)} (g_{m})} (\sum_{r \sim c, r \sim m} \frac{1}{d_{κ} (g_{r})})^{2} . \end{matrix}$ Similarly, it is easy to show that $\begin{matrix} tr (R^{4} (λ (g_{c} g_{m}))) (G) \\ = \sum_{c = 1}^{n} (\sum_{c \sim m} \frac{1}{d_{(λ)} (g_{c}) d_{(λ)} (g_{m})})^{2} \\ + \sum_{c \neq m} \frac{1}{d_{(λ)} (g_{c}) d_{(λ)} (g_{m})} (\sum_{r \sim c, r \sim m} \frac{1}{d_{λ} (g_{r})})^{2} . \end{matrix}$ Hence, $\begin{matrix} tr (R^{4} (G)) = \sum_{c = 1}^{n} (\sum_{c \sim m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})})^{2} \\ + \sum_{c \neq m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})} (\sum_{r \sim c, r \sim m} \frac{1}{d_{G} (g_{r})})^{2} . \end{matrix}$

□ Theorem 4.1. Suppose a q-ROFHG

G = (U, V)

on n vertices. Then

R E (G) \leq \sqrt{2 n \sum_{c \sim m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})}} .

Moreover,

R E (G) = \sqrt{2 n \sum_{c \sim m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})}}

if and only if

G = (U, V)

is a q-ROFHG with only isolated vertices, or end vertices.

Proof. Let $G = (U, V)$ be a q-ROFHG on n vertices. The variance of the numbers |η_c|, c = 1, 2, …, n is given as follows: $| η_{c} | = \frac{1}{n} (\sum_{c = 1}^{n} | η_{c} |^{2}) - (\frac{1}{n} (\sum_{c = 1}^{n} | η_{c} |)^{2}) \geq 0 .$ Now, $(\sum_{c = 1}^{n} | η_{c} |^{2}) = (\sum_{c = 1}^{n} η_{c}^{2}) = tr (R^{2} (κ (g_{c} g_{m}))) (G) .$ Further, $\frac{1}{n} tr (R^{2} (κ (g_{c} g_{m}))) (G) - (\frac{1}{n} R E (κ (g_{c} g_{m})) (G))^{2} \geq 0$

$\begin{matrix} \Leftrightarrow \frac{1}{n} tr (R^{2} (κ (g_{c} g_{m}))) (G) & \geq (\frac{1}{n} R E (κ (g_{c} g_{m})) (G))^{2} \\ \Leftrightarrow R E (κ (g_{c} g_{m})) (G) & \leq \sqrt{ntr (R^{2} (κ (g_{c} g_{m}))) (G)} \\ \Leftrightarrow R E (κ (g_{c} g_{m})) (G) & = \sqrt{2 n \sum_{c \sim m} \frac{1}{d_{(κ)} (g_{c}) d_{(κ)} (g_{m})}} . \end{matrix}$

In a similar manner, we can show that $\Leftrightarrow R E (λ (g_{c} g_{m})) (G) = \sqrt{2 n \sum_{c \sim m} \frac{1}{d_{(λ)} (g_{c}) d_{(λ)} (g_{m})}} .$ Hence, $R E (G) \leq \sqrt{2 n \sum_{c \sim m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})}} .$ Further, if $G = (U, V)$ is a q-ROFHG with only isolated vertices, then η_c = 0 for all c = 1, 2, …, n, and therefore $R E (κ (g_{c} g_{m})) (G) = 0 .$ Since $G = (U, V)$ is a q-ROFHG without edges, therefore $\sum_{c \sim m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})} = 0 .$ If $G$ is a q-ROFHG with only end vertices, that is, having degree one, then $η_{c} = \pm d_{(κ)} (g_{c}),$ then the variance of |η_c|=0, c = 1, 2, …, n . Thus $R E (G) = \sqrt{2 n \sum_{c \sim m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})}} .$ □

Theorem 4.2. Suppose a q-ROFHG $G = (U, V)$ on n vertices and at least one edge. Then $R E (G) \geq 2 \sum_{c \sim m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})} \sqrt{\frac{2 \sum_{c \sim m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})}}{\sum_{c = 1}^{n} (\sum_{c \sim m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})})^{2} + \sum_{c \neq m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})} (\sum_{r \sim c, r \sim m} \frac{1}{d_{G} (g_{r})})^{2}}} .$

Proof. By using the Hölder inequality $\sum_{c = 1}^{n} e_{c} f_{c} \leq (\sum_{c = 1}^{n} e_{c}^{p})^{\frac{1}{p}} (\sum_{c = 1}^{n} f_{c}^{q})^{\frac{1}{q}} .$ The above inequality holds for any real numbers e_c, f_c ≥ 0, c = 1, 2, …, n . Taking $e_{c} = | η_{c} |^{\frac{2}{3}},$ $f_{c} = | η_{c} |^{\frac{4}{3}},$ $p = \frac{3}{2},$ and q = 3, we get $\begin{matrix} (\sum_{c = 1}^{n} | η_{c} |^{2}) & = \sum_{c = 1}^{n} (| η_{c} |^{\frac{2}{3}}) (| η_{c} |^{4})^{\frac{1}{3}} \\ \leq (\sum_{c = 1}^{n} | η_{c} |)^{\frac{2}{3}} (\sum_{c = 1}^{n} | η_{c} |^{4})^{\frac{1}{3}} . \end{matrix}$ If $G$ contain atleast one edge, then all $η_{c}^{,} s$ are not equal to zero. Therefore $(\sum_{c = 1}^{n} | η_{c} |^{4}) \neq 0$ and $\begin{matrix} R E (κ (g_{c} g_{m})) (G) & = & \sum_{c = 1}^{n} (| η_{c} |) \\ \geq & \sqrt{\frac{(\sum_{c = 1}^{n} (| η_{c} |^{2}))^{3}}{(\sum_{c = 1}^{n} (| η_{c} |^{4}))}} \\ = & \sqrt{\frac{(\sum_{c = 1}^{n} (η_{c}^{2}))^{3}}{(\sum_{c = 1}^{n} (η_{c}^{4}))}} \\ = & \sqrt{\frac{tr (R^{2} (κ (g_{c} g_{m})) (G))^{3}}{tr (R^{4} (κ (g_{c} g_{m})) (G))}} \end{matrix}$ $R E (κ (g_{c} g_{m})) (G) \geq \sqrt{\frac{tr (R^{2} (κ (g_{c} g_{m})) (G))^{3}}{tr (R (κ (g_{c} g_{m})) (G))}}$ $\begin{matrix} = 2 \sum_{c \sim m} \frac{1}{d_{(κ)} (g_{c}) d_{(κ)} (g_{m})} \sqrt{\frac{2 \sum_{c \sim m} \frac{1}{d_{(κ)} (g_{c}) d_{(κ)} (g_{m})}}{\sum_{c = 1}^{n} (\sum_{c \sim m} \frac{1}{d_{(κ)} (g_{c}) d_{(κ)} (g_{m})})^{2} + \sum_{c \neq m} \frac{1}{d_{(κ)} (g_{c}) d_{(κ)} (g_{m})} (\sum_{r \sim c, r \sim m} \frac{1}{d_{(κ)} (g_{r})})^{2}}} . \end{matrix}$ Similarly,

$R E (λ (g_{c} g_{m})) (G) \geq 2 \sum_{c \sim m} \frac{1}{d_{(λ)} (g_{c}) d_{(λ)} (g_{m})} \sqrt{\frac{2 \sum_{c \sim m} \frac{1}{d_{(λ)} (g_{c}) d_{(λ)} (g_{m})}}{\sum_{c = 1}^{n} (\sum_{c \sim m} \frac{1}{d_{(λ)} (g_{c}) d_{(λ)} (g_{m})})^{2} + \sum_{c \neq m} \frac{1}{d_{(λ)} (g_{c}) d_{(λ)} (g_{m})} (\sum_{r \sim c, r \sim m} \frac{1}{d_{(λ)} (g_{r})})^{2}}} .$ Thus,

$R E (G) \geq 2 \sum_{c \sim m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})} \sqrt{\frac{2 \sum_{c \sim m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})}}{\sum_{c = 1}^{n} (\sum_{c \sim m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})})^{2} + \sum_{c \neq m} \frac{1}{d_{G} (g_{c}) d_{G} (g_{m})} (\sum_{r \sim c, r \sim m} \frac{1}{d_{G} (g_{r})})^{2}}} .$ □

Example 4.1. Let $G = (U, V)$ be a 4-ROFHG on G^* = (Y, E) , as shown in Fig. 6, where $Y = {g_{1}, g_{2}, g_{3}, g_{4}, g_{5}, g_{6}}$ and $E = {g_{1} g_{2}, g_{1} g_{6}, g_{2} g_{3}, g_{2} g_{4}, g_{2} g_{5}, g_{2} g_{6}, g_{3} g_{4}, g_{5} g_{6}} .$ The set of vertices $U$ and set of edges $V$ of $G$ are defined on Y and E, respectively.

$\begin{matrix} U = & (\frac{g_{1}}{(0.8, 0.7)}, \frac{g_{2}}{(0.9, 0.6)}, \frac{g_{3}}{(0.9, 0.6)}, \frac{g_{4}}{(0.5, 0.9)}, \\ \frac{g_{5}}{(0.9, 0.7)}, \frac{g_{6}}{(0.8, 0.8)}), \\ V = & (\frac{g_{1} g_{2}}{(0.70, 0.75)}, \frac{g_{1} g_{6}}{(0.65, 0.80)}, \frac{g_{2} g_{3}}{(0.81, 0.72)}, \frac{g_{2} g_{4}}{(0.43, 0.90)}, \\ \frac{g_{2} g_{5}}{(0.78, 0.69)}, \frac{g_{2} g_{6}}{(0.72, 0.82)}, \frac{g_{3} g_{4}}{(0.40, 0.91)}, \frac{g_{5} g_{6}}{(0.66, 0.86)}) . \end{matrix}$

Fig. 6

4-rung orthopair fuzzy Hamacher graph

The AM of the 4-ROFHG is given as follows: $A (G) = [\begin{matrix} (0, 0) & (0.70, 0.75) & (0, 0) & (0, 0) & (0, 0) & (0.65, 0.80) \\ (0.70, 0.75) & (0, 0) & (0.80, 0.72) & (0.43, 0.90) & (0.78, 0.69) & (0.72, 0.82) \\ (0, 0) & (0.80, 0.72) & (0, 0) & (0.40, 0.91) & (0, 0) & (0, 0) \\ (0, 0) & (0.43, 0.90) & (0.40, 0.91) & (0, 0) & (0, 0) & (0, 0) \\ (0, 0) & (0.78, 0.69) & (0, 0) & (0, 0) & (0, 0) & (0.66, 0.86) \\ (0.65, 0.80) & (0.72, 0.82) & (0, 0) & (0, 0) & (0.66, 0.86) & (0, 0) \end{matrix}] .$

The Randić matrix $(R M)$ of the 4-ROFHG is given as follows: $R (G) = [\begin{matrix} (0, 0) & (0.46, 0.40) & (0, 0) & (0, 0) & (0, 0) & (0.60, 0.51) \\ (0.46, 0.40) & (0, 0) & (0.49, 0.39) & (0.59, 0.37) & (0.44, 0.40) & (0.37, 0.32) \\ (0, 0) & (0.49, 0.39) & (0, 0) & (0.99, 0.58) & (0, 0) & (0, 0) \\ (0, 0) & (0.59, 0.37) & (0.99, 0.58) & (0, 0) & (0, 0) & (0, 0) \\ (0, 0) & (0.44, 0.40) & (0, 0) & (0, 0) & (0, 0) & (0.58, 0.51) \\ (0.60, 0.51) & (0.37, 0.32) & (0, 0) & (0, 0) & (0.58, 0.51) & (0, 0) \end{matrix}] .$

The Randić spectrum and the $RE$ of a 4-ROFHG $G$ are given as follows: $Spec (G)$ = {(-1.0106, - 0.8090) , (-0.9226, -0.5808) , (-0.5790, - 0.4760) , (0.0000, - 0.0000) , (0.9020, 0.6356) , (1.6101, 1.2303)} Thus, $R E (G) = (5.0243, 3.7317) .$

Definition 4.4. Suppose a q-ROFHDG, $\vec{G} = (U, \vec{V})$ on n vertices. An n × n matrix $R (\vec{G}) = (R (κ_{\vec{G}} (g_{c} g_{m}), λ_{\vec{G}} (g_{c} g_{m}))) = [g_{cm}]$ of $\vec{G}$ is defined to be Randić matrix with the following conditions:

$g_{cm} = {\begin{matrix} 0, & if c = m; \\ \frac{1}{\sqrt{d_{\vec{G}} (g_{c}) d_{\vec{G}} (g_{m})}}, & if the vertices g_{c} and g_{m} of G are adjacent; \\ 0, & if the vertices g_{c} and g_{m} of G are not adjacent . \end{matrix}$

Definition 4.5. The spectrum of Randić matrix of a q-ROFHDG $R (\vec{G})$ is defined as $(C_{R}, D_{R}),$ where $C_{R}$ and $D_{R}$ are the sets of eigenvalues of $R (κ_{\vec{V}} (g_{c} g_{m}))$ and $R (λ_{\vec{V}} (g_{c} g_{m})),$ respectively.

Definition 4.6. The Randić energy of a q-ROFHDG $\vec{G} = (\vec{U}, \vec{V})$ is defined as: $\begin{matrix} R E (\vec{G}) & = & (R E (κ_{\vec{V}} (g_{c} g_{m})), R E (λ_{\vec{V}} (g_{c} g_{m}))) \\ = & (\sum_{c = 1}^{n} | η_{c} |, \sum_{c = 1}^{n} | ρ_{c} |), \end{matrix}$ where $η_{c} \in C_{R}$ and $ρ_{c} \in D_{R} .$

5 Application to decision making

In this section, we provide an application related to the selection of housing society for investment to illustrate the applicability of q-ROFHGs. Further, we present a comparison analysis with existing methods.

5.1 Selection of housing society for investment

Suppose a person wants to invest money in a housing society. He hires a group of five decision-making experts {e₁, e₂, e₃, e₄, e₅} to select the housing society among the four housing societies {H₁, H₂, H₃, H₄} . The experts make pairwise comparison among the four housing societies keeping the following parameters:

Variety of plots, cutting and map

Payment plans and location

Construction approval

Development level and security

Shopping malls and other necessities

All the five experts individually compare each pair of housing society H_c (c = 1, 2, 3, 4) and H_m (m = 1, 2, 3, 4) , and give his q-ROFN

g_{cm}^{a} = (κ_{cm}^{a}, λ_{cm}^{a}) (a = 1, 2, 3, 4, 5),

with the MD

κ_{cm}^{a}

to which H_c is preferred to H_m and the NMD

λ_{cm}^{a}

to which H_c is not-preferred to H_m . The algorithm of our proposed model for the selection of housing society for investment is given below:

Algorithm. INPUT: A discrete set of housing societies Y = {H₁, H₂, ⋯ , H_n} , a set of decision making experts e = {e₁, e₂, ⋯ , e_n} in order to achieve the target and construction of q-ROFPR $Q_{a} = (b_{cm}^{a})_{n \times n}$ for each expert. OUTPUT: The selection of housing society for investment.

Calculate the energy and Randić energy of each q-ROFHDG D_a (a = 1, 2, ⋯ , t) .

Calculate the weight vector of each decision maker on the basis of energy and Randić energy of q-ROFHDGs by using $W_{a} = (\frac{E ((D_{κ})_{a})}{{t \sum}_{p = 1} E ((D_{κ})_{p})}, \frac{E ((D_{λ})_{a})}{\sum_{p = 1}^{t} E ((D_{κ})_{p})}),$ and $W_{a} = (\frac{R E ((D_{κ})_{a})}{\sum_{p = 1}^{t} R E ((D_{κ})_{p})}, \frac{R E ((D_{λ})_{a})}{\sum_{p = 1}^{t} R E ((D_{κ})_{p})}),$ respectively.

By using q-ROFA operator and q-ROFGA operator, aggregate all $b_{cm}^{(a)} (m = 1, 2, \dots, n)$ corresponding to the industry H_c, and obtain the q-rung orthopair fuzzy element (q-ROFE) $b_{c}^{a}$ of the society H_c over all other societies for the decision maker e_a .

Calculate a collective q-ROFE $b_{c}$ for the housing society H_c by aggregating all $b_{cm}^{a} (a = 1, 2, \dots, t),$ utilizing q-ROFWA operator and q-ROFWGA operator.

Determine the score functions $S (b_{c})$ of $b_{c} (c = 1, 2, \dots, n) .$

Rank all the housing societies H_c (c = 1, 2, ⋯ , n) according to $S (b_{c}) (c = 1, 2, \dots, n) .$

Output the suitable housing society.

The q-ROFPRs

Q_{a} = (g_{cm}^{a})_{4 \times 4} (a = 1, 2, 3, 4, 5)

are given as follows:

The q-ROFHDGs D_a of the q-ROFPRs given in Tables 1-5, are displayed in Fig. 7.

Table 1
q-ROFPRs of the first decision making expert

Q ₁ H ₁ H ₂ H ₃ H ₄

H ₁ (0.5,0.5) (0.5,0.9) (0.2,0.8) (0.7,0.8)

H ₂ (0.9,0.5) (0.5,0.5) (0.7,0.7) (0.9,0.4)

H ₃ (0.8,0.2) (0.7,0.7) (0.5,0.5) (0.3,0.9)

H ₄ (0.8,0.7) (0.4,0.9) (0.9,0.3) (0.5,0.5)

Q ₁	H ₁	H ₂	H ₃	H ₄
H ₁	(0.5,0.5)	(0.5,0.9)	(0.2,0.8)	(0.7,0.8)
H ₂	(0.9,0.5)	(0.5,0.5)	(0.7,0.7)	(0.9,0.4)
H ₃	(0.8,0.2)	(0.7,0.7)	(0.5,0.5)	(0.3,0.9)
H ₄	(0.8,0.7)	(0.4,0.9)	(0.9,0.3)	(0.5,0.5)

Table 2

q-ROFPRs of the second decision making expert

Q ₂	H ₁	H ₂	H ₃	H ₄
H ₁	(0.5,0.5)	(0.9,0.6)	(0.8,0.7)	(0.5,0.9)
H ₂	(0.6,0.9)	(0.5,0.5)	(0.6,0.8)	(0.2,0.8)
H ₃	(0.7,0.8)	(0.8,0.6)	(0.5,0.5)	(0.8,0.6)
H ₄	(0.9,0.5)	(0.8,0.2)	(0.6,0.8)	(0.5,0.5)

Table 3

q-ROFPRs of the third decision making expert

Q ₃	H ₁	H ₂	H ₃	H ₄
H ₁	(0.5,0.5)	(0.9,0.6)	(0.6,0.7)	(0.4,0.9)
H ₂	(0.6,0.9)	(0.5,0.5)	(0.8,0.7)	(0.7,0.7)
H ₃	(0.7,0.6)	(0.7,0.8)	(0.5,0.5)	(0.6,0.8)
H ₄	(0.9,0.4)	(0.7,0.7)	(0.8,0.6)	(0.5,0.5)

Table 4

q-ROFPRs of the fourth decision making expert

Q ₄	H ₁	H ₂	H ₃	H ₄
H ₁	(0.5,0.5)	(0.9,0.5)	(0.8,0.6)	(0.5,0.8)
H ₂	(0.5,0.9)	(0.5,0.5)	(0.3,0.9)	(0.8,0.7)
H ₃	(0.6,0.8)	(0.9,0.3)	(0.5,0.5)	(0.6,0.7)
H ₄	(0.8,0.5)	(0.7,0.8)	(0.7,0.6)	(0.5,0.5)

Table 5

q-ROFPRs of the fifth decision making expert

Q ₅	H ₁	H ₂	H ₃	H ₄
H ₁	(0.5,0.5)	(0.4,0.9)	(0.5,0.8)	(0.5,0.9)
H ₂	(0.9,0.4)	(0.5,0.5)	(0.7,0.7)	(0.7,0.8)
H ₃	(0.8,0.5)	(0.7,0.7)	(0.5,0.5)	(0.6,0.5)
H ₄	(0.9,0.5)	(0.8,0.7)	(0.5,0.6)	(0.5,0.5)

Fig. 7

3-rung orthopair fuzzy digraphs

The energy of each q-ROFHDG is given as follows:

$\begin{matrix} E (D_{1}) = (2.8102, 2.8102), E (D_{2}) = (3.0182, 3.0182), \\ E (D_{3}) = (3.1740, 3.1740), \\ E (D_{4}) = (3.0030, 3.0030), E (D_{5}) = (2.9112, 2.9112) . \end{matrix}$

The weight vector of each expert can be computed as: $\begin{matrix} W_{a} & = & (\frac{E ((D_{κ})_{a})}{\sum_{p = 1}^{t} E ((D_{κ})_{p})}, \frac{E ((D_{λ})_{a})}{\sum_{p = 1}^{t} E ((D_{κ})_{p})}), \\ a = 1, 2, \dots, t, \end{matrix}$ $\begin{matrix} W_{1} = (0.1884, 0.1884), W_{2} = (0.2023, 0.2023), \\ W_{3} = (0.2128, 0.2128), \\ W_{4} = (0.2013, 0.2013), W_{5} = (0.1951, 0.1951) . \end{matrix}$ The weight vector of five experts e_a (a = 1, 2, 3, 4, 5) is: $\begin{matrix} W = ((0.1884, 0.1884), (0.2023, 0.2023), (0.2128, \\ 0.2128), (0.2013, 0.2013), (0.1951, 0.1951))^{T} . \end{matrix}$ The q-rung orthopair fuzzy Hamacher averaging (q-ROFHA) operator is given as:

$q - ROFHA (b_{c 1}^{(a)}, b_{c 2}^{(a)}, \dots, b_{cn}^{(a)}) = \oplus_{m = 1}^{n} (\frac{1}{n} (b_{m}))$ $\begin{matrix} = (\sqrt[q]{\frac{\prod_{m = 1}^{n} (1 + (ϱ - 1) (κ_{m})^{q})^{\frac{1}{n}} - \prod_{m = 1}^{n} (1 - (κ_{m})^{q})^{\frac{1}{n}}}{\prod_{m = 1}^{n} (1 + (ϱ - 1) (κ_{m})^{q})^{\frac{1}{n}} + (ϱ - 1) \prod_{m = 1}^{n} (1 - (κ_{m})^{q})^{\frac{1}{n}}}}, \\ \frac{\sqrt[q]{ϱ} \prod_{m = 1}^{n} (λ_{m})^{\frac{1}{n}}}{\sqrt[q]{\prod_{m = 1}^{n} (1 + (ϱ - 1) (1 - (λ_{m})^{q}))^{\frac{1}{n}} + (ϱ - 1) \prod_{m = 1}^{n} (λ_{m})^{q \frac{1}{n}}}}) \end{matrix}$

For ϱ = 1, the q-ROFHA operator is:

$\begin{matrix} b_{c}^{(a)} & = q - ROFHA (b_{c 1}^{(a)}, b_{c 2}^{(a)}, \dots, b_{cn}^{(a)}) \\ = (\sqrt[q]{1 - (\prod_{m = 1}^{n} (1 - (κ_{cm})^{q}))^{\frac{1}{n}}}, (\prod_{m = 1}^{n} λ_{cm})^{\frac{1}{n}}) . \end{matrix}$

The aggregation results are calculated in Table 6.

Table 6

Aggregation results of the decision-making experts

Experts		Aggregated Results of the Experts
e ₁	$b_{1}^{(1)}$	(0.5423, 0.7325)
	$b_{2}^{(1)}$	(0.8176, 0.5144)
	$b_{3}^{(1)}$	(0.6520, 0.5009)
	$b_{4}^{(1)}$	(0.7526, 0.5544)
e ₂	$b_{1}^{(2)}$	(0.7582, 0.6593)
	$b_{2}^{(2)}$	(0.5257, 0.7325)
	$b_{3}^{(2)}$	(0.7316, 0.6160)
	$b_{4}^{(2)}$	(0.7669, 0.4472)
e ₃	$b_{1}^{(3)}$	(0.7074, 0.6593)
	$b_{2}^{(3)}$	(0.6805, 0.6852)
	$b_{3}^{(3)}$	(0.6401, 0.6619)
	$b_{4}^{(3)}$	(0.7801, 0.5383)
e ₄	$b_{1}^{(4)}$	(0.7582, 0.5885)
	$b_{2}^{(4)}$	(0.6068, 0.7296)
	$b_{3}^{(4)}$	(0.7256, 0.5383)
	$b_{4}^{(4)}$	(0.7012, 0.5885)
e ₅	$b_{1}^{(5)}$	(0.4793, 0.7544)
	$b_{2}^{(5)}$	(0.7573, 0.5785)
	$b_{3}^{(5)}$	(0.6805, 0.5438)
	$b_{4}^{(5)}$	(0.7582, 0.5692)

The q-ROFHWA operator is given as:

$q - {ROFHWA}_{W} (b_{c 1}^{(a)}, b_{c 2}^{(a)}, \dots, b_{cn}^{(a)}) = \oplus_{m = 1}^{n} (W_{m} b_{m})$ $\begin{matrix} = (\sqrt[q]{\frac{\prod_{m = 1}^{n} (1 + (ϱ - 1) (κ_{m})^{q})^{W_{m}} - \prod_{m = 1}^{n} (1 - (κ_{m})^{q})^{W_{m}}}{\prod_{m = 1}^{n} (1 + (ϱ - 1) (κ_{m})^{q})^{W_{m}} + (ϱ - 1) \prod_{m = 1}^{n} (1 - (κ_{m})^{q})^{W_{m}}}}, \\ \frac{\sqrt[q]{ϱ} \prod_{m = 1}^{n} (λ_{m})^{W_{m}}}{\sqrt[q]{\prod_{m = 1}^{n} (1 + (ϱ - 1) (1 - (λ_{m})^{q}))^{W_{m}} + (ϱ - 1) \prod_{j = 1}^{n} (λ_{m})^{q W_{m}}}}) \end{matrix}$

For ϱ = 1, the q-ROFHWA operator transforms to the q-rung orthopair fuzzy weighted average (q-ROFWA) operator as follows: $\begin{matrix} q - ROFWA (b_{c 1}^{(a)}, b_{c 2}^{(a)}, \dots, b_{cn}^{(a)}) \\ = (\sqrt[q]{1 - (\prod_{m = 1}^{n} (1 - (κ_{cm})^{q})^{W_{m}})}, (\prod_{m = 1}^{n} λ_{cm}^{W_{m}})) . \end{matrix}$ The collective q-rung orthopair fuzzy elements (q-ROFE) using q-ROFHWA operator with ϱ = 1 are given as follows: $3 b_{1} = (0.6810, 0.6748), b_{2} = (0.7015, 0.6447), b_{3} = (0.6895, 0.5714), b_{4} = (0.7541, 0.5366) .$ The score functions by using Eq. 2.1, $S (b_{c})$ of $b_{c} (c = 1, 2, 3, 4)$ are computed below: $3 S (b_{1}) = 0.5043, S (b_{2}) = 0.5386, S (b_{3}) = 0.5706, S (b_{4}) = 0.6372 .$ Then, $H_{4} ≻ H_{3} ≻ H_{2} ≻ H_{1} .$ From the ranking, H₄ is the best housing society for investment among all. The q-rung orthopair fuzzy Hamacher geometric (q-ROFHG) operator is given as:

$q - ROFHG (b_{c 1}^{(a)}, b_{c 2}^{(a)}, \dots, b_{cn}^{(a)}) = ⨂_{m = 1}^{n} (b_{m})^{\frac{1}{n}}$ $\begin{matrix} = (\frac{\sqrt[q]{ϱ} \prod_{m = 1}^{n} (κ_{m})^{\frac{1}{n}}}{\sqrt[q]{\prod_{m = 1}^{n} (1 + (ϱ - 1) (1 - (κ_{m})^{q}))^{\frac{1}{n}} + (ϱ - 1) \prod_{m = 1}^{n} (κ_{m})^{q \frac{1}{n}}}}, \\ \sqrt[q]{\frac{\prod_{m = 1}^{n} (1 + (ϱ - 1) (λ_{m})^{q})^{\frac{1}{n}} - \prod_{m = 1}^{n} (1 - (λ_{m})^{q})^{\frac{1}{n}}}{\prod_{m = 1}^{n} (1 + (ϱ - 1) (λ_{m})^{q})^{\frac{1}{n}} + (ϱ - 1) \prod_{m = 1}^{n} (1 - (λ_{m})^{q})^{\frac{1}{n}}}}) . \end{matrix}$

For ϱ = 1, the q-ROFHG operator is:

$\begin{matrix} b_{c}^{(a)} & = q - ROFHG (b_{c 1}^{(a)}, b_{c 2}^{(a)}, \dots, b_{cn}^{(a)}) \\ = ((\prod_{m = 1}^{n} κ_{cm})^{\frac{1}{n}}, \sqrt[q]{1 - (\prod_{m = 1}^{n} (1 - (λ_{cm})^{q}))^{\frac{1}{n}}}) . \end{matrix}$

The aggregation results are calculated in Table 7.

Table 7

Aggregation results of the decision-making experts

Experts		Aggregated Results of the Experts
e ₁	$b_{1}^{(1)}$	(0.4325, 0.8003)
	$b_{2}^{(1)}$	(0.7297, 0.5557)
	$b_{3}^{(1)}$	(0.5383, 0.7198)
	$b_{4}^{(1)}$	(0.6160, 0.7218)
e ₂	$b_{1}^{(2)}$	(0.6513, 0.7422)
	$b_{2}^{(2)}$	(0.4356, 0.8003)
	$b_{3}^{(2)}$	(0.6879, 0.6575)
	$b_{4}^{(2)}$	(0.6817, 0.6034)
e ₃	$b_{1}^{(3)}$	(0.5733, 0.7422)
	$b_{2}^{(3)}$	(0.6402, 0.7573)
	$b_{3}^{(3)}$	(0.6192, 0.7142)
	$b_{4}^{(3)}$	(0.7085, 0.5789)
e ₄	$b_{1}^{(4)}$	(0.6513, 0.6418)
	$b_{2}^{(4)}$	(0.4949, 0.8177)
	$b_{3}^{(4)}$	(0.6344, 0.6520)
	$b_{4}^{(4)}$	(0.6654, 0.6417)
e ₅	$b_{1}^{(5)}$	(0.4729, 0.8336)
	$b_{2}^{(5)}$	(0.6852, 0.6574)
	$b_{3}^{(5)}$	(0.6402, 0.5703)
	$b_{4}^{(5)}$	(0.6513, 0.5921)

The q-ROFHWG operator is given as:

$q - {ROFHWG}_{W} (b_{c 1}^{(a)}, b_{c 2}^{(a)}, \dots, b_{cn}^{(a)}) = ⨂_{m = 1}^{n} (b_{m})^{W_{m}}$ $\begin{matrix} = (\frac{\sqrt[q]{ϱ} \prod_{m = 1}^{n} (κ_{m})^{W_{m}}}{\prod_{m = 1}^{n} (1 + (ϱ - 1) (1 - (κ_{m})^{q}))^{W_{m}} + (ϱ - 1) \prod_{m = 1}^{n} (κ_{m})^{q W_{m}} q}, \\ \sqrt[q]{\frac{\prod_{m = 1}^{n} (1 + (ϱ - 1) (λ_{m})^{q})^{W_{m}} - \prod_{m = 1}^{n} (1 - (λ_{m})^{q})^{W_{m}}}{\prod_{m = 1}^{n} (1 + (ϱ - 1) (λ_{m})^{q})^{W_{m}} + (ϱ - 1) \prod_{m = 1}^{n} (1 - (λ_{m})^{q})^{W_{m}}}}) . \end{matrix}$

For ϱ = 1, the q-ROFHWGA operator transforms to the q-rung orthopair fuzzy weighted geometric (q-ROFWG) operator as follows:

$\begin{matrix} q - ROFWA (b_{c 1}^{(a)}, b_{c 2}^{(a)}, \dots, b_{cn}^{(a)}) \\ = ((\prod_{m = 1}^{n} κ_{cm}^{W_{m}}), \sqrt[q]{1 - (\prod_{m = 1}^{n} (1 - (λ_{cm})^{q})^{W_{m}})}) . \end{matrix}$

The collective q-ROFE using q-ROFHWG operator with ϱ = 1 are given as follows: $3 b_{1} = (0.5513, 0.7620), b_{2} = (0.5814, 0.7426), b_{3} = (0.6231, 0.6698), b_{4} = (0.6651, 0.6328) .$ The score functions $S (b_{c})$ of $b_{c} (c = 1, 2, 3, 4)$ are computed below: $3 S (b_{1}) = 0.3625, S (b_{2}) = 0.3935, S (b_{3}) = 0.4707, S (b_{4}) = 0.5204 .$ Then, $H_{4} ≻ H_{3} ≻ H_{2} ≻ H_{1} .$ From the ranking, H₄ is the best housing society for investment among all.

The Randić matrices of the q-ROFHDGs $Q_{a}^{R} (a = 1, 2, 3, 4, 5),$ displayed in Fig. 7 are given in Tables 8-12:

Table 8

$R M$ of the q-ROFHDG D₁

$Q_{1}^{R}$	H ₁	H ₂	H ₃	H ₄
H ₁	(0.50,0.50)	(0.53,0.50)	(0.63,0.47)	(0.58,0.46)
H ₂	(0.50,0.53)	(0.50,0.50)	(0.47,0.59)	(0.44,0.57)
H ₃	(0.47,0.63)	(0.59,0.47)	(0.50,0.50)	(0.51,0.63)
H ₄	(0.45,0.58)	(0.57,0.44)	(0.63,0.51)	(0.50,0.50)

Table 9

$R M$ of the q-ROFHDG D₂

$Q_{2}^{R}$	H ₁	H ₂	H ₃	H ₄
H ₁	(0.50,0.50)	(0.57,0.43)	(0.44,0.48)	(0.44,0.55)
H ₂	(0.43,0.57)	(0.50,0.50)	(0.56,0.45)	(0.56,0.52)
H ₃	(0.48,0.44)	(0.45,0.56)	(0.50,0.50)	(0.43,0.58)
H ₄	(0.55,0.44)	(0.52,0.56)	(0.58,0.43)	(0.50,0.50)

Table 10

$R M$ of the q-ROFHDG D₃

$Q_{3}^{R}$	H ₁	H ₂	H ₃	H ₄
H ₁	(0.50,0.50)	(0.50,0.44)	(0.51,0.45)	(0.47,0.52)
H ₂	(0.44,0.50)	(0.50,0.50)	(0.49,0.44)	(0.46,0.51)
H ₃	(0.45,0.51)	(0.44,0.49)	(0.50,0.50)	(0.46,0.52)
H ₄	(0.52,0.47)	(0.51,0.46)	(0.52,0.46)	(0.50,0.50)

Table 11

$R M$ of the q-ROFHDG D₄

$Q_{4}^{R}$	H ₁	H ₂	H ₃	H ₄
H ₁	(0.50,0.50)	(0.53,0.46)	(0.47,0.54)	(0.45,0.53)
H ₂	(0.46,0.53)	(0.50,0.50)	(0.56,0.47)	(0.53,0.46)
H ₃	(0.54,0.47)	(0.47,0.56)	(0.50,0.50)	(0.47,0.54)
H ₄	(0.53,0.45)	(0.46,0.53)	(0.54,0.47)	(0.50,0.50)

Table 12

$R M$ of the q-ROFHDG D₅

$Q_{5}^{R}$	H ₁	H ₂	H ₃	H ₄
H ₁	(0.50,0.50)	(0.56,0.45)	(0.58,0.48)	(0.57,0.46)
H ₂	(0.45,0.56)	(0.50,0.50)	(0.46,0.56)	(0.44,0.54)
H ₃	(0.48,0.58)	(0.56,0.46)	(0.50,0.50)	(0.47,0.57)
H ₄	(0.46,0.57)	(0.54,0.44)	(0.57,0.47)	(0.50,0.50)

The Randić energy of each q-ROFHDG is given as follows:

$\begin{matrix} E (D_{1}) = (2.1730, 2.1730), E (D_{2}) = (2.0025, 2.0025), \\ E (D_{3}) = (2.0000, 2.0000), E (D_{4}) = (2.0055, 2.0055), \\ E (D_{5}) = (2.0572, 2.0572) . \end{matrix}$

The weight vector of each expert can be computed as: $\begin{matrix} W_{a} & = & (\frac{R E ((D_{κ})_{a})}{\sum_{p = 1}^{t} R E ((D_{κ})_{p})}, \frac{R E ((D_{λ})_{a})}{\sum_{p = 1}^{t} R E ((D_{κ})_{p})}), \\ a = 1, 2, \dots, t, \end{matrix}$ $\begin{matrix} W_{1} = (0.2122, 0.2122), W_{2} = (0.1956, 0.1956), \\ W_{3} = (0.1953, 0.1953), W_{4} = (0.1959, 0.1959), \\ W_{5} = (0.2009, 0.2009) . \end{matrix}$ The weight vector of five experts e_a (a = 1, 2, 3, 4, 5) is:

$\begin{matrix} W = ((0.2122, 0.2122), (0.1956, 0.1956), (0.1953, 0.1953), \\ (0.1959, 0.1959), (0.2009, 0.2009))^{T} . \end{matrix}$

For ϱ = 1, the q-ROFHA operator is:

$\begin{matrix} b_{c}^{(a)} & = & q - ROFHA (b_{c 1}^{(a)}, b_{c 2}^{(a)}, \dots, b_{cn}^{(a)}) \\ = & (\sqrt[q]{1 - (\prod_{m = 1}^{n} (1 - (κ_{cm})^{q}))^{\frac{1}{n}}}, (\prod_{m = 1}^{n} λ_{cm})^{\frac{1}{n}}) . \end{matrix}$

The aggregation results are calculated in Table 13.

Table 13

Aggregation results of the decision-making experts

Experts		Aggregated Results of the Experts
e ₁	$b_{1}^{(1)}$	(0.5658, 0.4795)
	$b_{2}^{(1)}$	(0.4790, 0.5464)
	$b_{3}^{(1)}$	(0.5224, 0.5526)
	$b_{4}^{(1)}$	(0.5485, 0.5051)
e ₂	$b_{1}^{(2)}$	(0.4947, 0.4881)
	$b_{2}^{(2)}$	(0.5191, 0.5082)
	$b_{3}^{(2)}$	(0.4668, 0.5170)
	$b_{4}^{(2)}$	(0.5397, 0.4797)
e ₃	$b_{1}^{(3)}$	(0.4955, 0.4763)
	$b_{2}^{(3)}$	(0.4739, 0.4867)
	$b_{3}^{(3)}$	(0.4638, 0.5049)
	$b_{4}^{(3)}$	(0.5126, 0.4722)
e ₄	$b_{1}^{(4)}$	(0.4897, 0.5065)
	$b_{2}^{(4)}$	(0.5157, 0.4892)
	$b_{3}^{(4)}$	(0.5051, 0.5163)
	$b_{4}^{(4)}$	(0.5097, 0.4865)
e ₅	$b_{1}^{(5)}$	(0.5547, 0.4721)
	$b_{2}^{(5)}$	(0.4638, 0.5394)
	$b_{3}^{(5)}$	(0.5056, 0.5251)
	$b_{4}^{(5)}$	(0.5216, 0.4927)

The collective q-ROFE using q-ROFHWA operator with ϱ = 1 are given as follows: $3 b_{1} = (0.5238, 0.4843), b_{2} = (0.4913, 0.5139), b_{3} = (0.4930, 0.5233), b_{4} = (0.5274, 0.4874) .$ The score functions $S (b_{c})$ of $b_{c} (c = 1, 2, 3, 4)$ are computed below: $3 S (b_{1}) = 0.5151, S (b_{2}) = 0.4914, S (b_{3}) = 0.4883, S (b_{4}) = 0.5154 .$ Then, $H_{4} ≻ H_{1} ≻ H_{2} ≻ H_{3} .$ From the ranking, H₄ is the best housing society among all. For ϱ = 1, the q-ROFHG operator is:

$\begin{matrix} b_{c}^{(a)} & = & q - ROFHG (b_{c 1}^{(a)}, b_{c 2}^{(a)}, \dots, b_{cn}^{(a)}) \\ = & ((\prod_{m = 1}^{n} κ_{cm})^{\frac{1}{n}}, \sqrt[q]{1 - (\prod_{m = 1}^{n} (1 - (λ_{cm})^{q}))^{\frac{1}{n}}}) . \end{matrix}$

The aggregation results are calculated in Table 14.

Table 14

Aggregation results of the decision-making experts

Experts		Aggregated Results of the Experts
e ₁	$^{b} {(1)}_{1}$	(0.5578, 0.4811)
	$b_{2}^{(1)}$	(0.4768, 0.5503)
	$b_{3}^{(1)}$	(0.5156, 0.5698)
	$b_{4}^{(1)}$	(0.5331, 0.5135)
e ₂	$b_{1}^{(2)}$	(0.4847, 0.4945)
	$b_{2}^{(2)}$	(0.5096, 0.5144)
	$b_{3}^{(2)}$	(0.4642, 0.5269)
	$b_{4}^{(2)}$	(0.5366, 0.4893)
e ₃	$b_{1}^{(3)}$	(0.4947, 0.4803)
	$b_{2}^{(3)}$	(0.4719, 0.4893)
	$b_{3}^{(3)}$	(0.4619, 0.5053)
	$b_{4}^{(3)}$	(0.5124, 0.4732)
e ₄	$b_{1}^{(4)}$	(0.4866, 0.5098)
	$b_{2}^{(4)}$	(0.5111, 0.4918)
	$b_{3}^{(4)}$	(0.4942, 0.5204)
	$b_{4}^{(4)}$	(0.5065, 0.4897)
e ₅	$b_{1}^{(5)}$	(0.5516, 0.4734)
	$b_{2}^{(5)}$	(0.4619, 0.5414)
	$b_{3}^{(5)}$	(0.5013, 0.5332)
	$b_{4}^{(5)}$	(0.5158, 0.5008)

The collective q-ROFE using q-ROFHWG operator with ϱ = 1 are given as follows: $3 b_{1} = (0.5149, 0.4882), b_{2} = (0.4856, 0.5197), b_{3} = (0.4876, 0.5329), b_{4} = (0.5209, 0.4941) .$ The score functions $S (b_{c})$ of $b_{c} (c = 1, 2, 3, 4)$ are computed below: $3 S (b_{1}) = 0.5101, S (b_{2}) = 0.4871, S (b_{3}) = 0.4823, S (b_{4}) = 0.5104 .$ Then, $H_{4} ≻ H_{1} ≻ H_{2} ≻ H_{3} .$ From the ranking, H₄ is the best housing society among all. The final score values and ranking of alternatives for energy and Randić energy is presented in Table 15 and 16, respectively.

Table 15

Ranking results using proposed method (for energy)

Operators	S (H₁)	S (H₂)	S (H₃)	S (H₄)	Ranking
q-ROFHWA operator	0.5043	0.5386	0.5706	0.6372	H₄ ≻ H₃ ≻ H₂ ≻ H₁
q-ROFHWG operator	0.3625	0.3935	0.4707	0.5204	H₄ ≻ H₃ ≻ H₂ ≻ H₁

Table 16

Ranking results using proposed method (for Randić energy)

Operators	S (H₁)	S (H₂)	S (H₃)	S (H₄)	Ranking
q-ROFHWA operator	0.5151	0.4914	0.4883	0.5154	H₄ ≻ H₁ ≻ H₂ ≻ H₃
q-ROFHWG operator	0.5105	0.4871	0.4823	0.5104	H₄ ≻ H₁ ≻ H₂ ≻ H₃

5.2 Comparison analysis

In this subsection, we discuss the results of the above numerical example by applying some existing operators. We compare the results of the presented model with q-rung orthopair fuzzy Yager weighted arithmetic (q-ROFYWA) [3] and q-rung orthopair fuzzy Yager weighted geometric (q-ROFYWG) [3] operators. The ranking results obtained by utilizing the aforesaid operators are summarized in Table 17 and 18.

Table 17
Ranking results using existing operators (for energy)

Operators S (H₁) S (H₂) S (H₃) S (H₄) Ranking

q-ROFYWA operator [3] 0.4369 0.4709 0.5079 0.5761 H₄ ≻ H₃ ≻ H₂ ≻ H₁

q-ROFYWG operator [3] 0.4730 0.4711 0.5172 0.5763 H₄ ≻ H₃ ≻ H₂ ≻ H₁

Operators	S (H₁)	S (H₂)	S (H₃)	S (H₄)	Ranking
q-ROFYWA operator [3]	0.4369	0.4709	0.5079	0.5761	H₄ ≻ H₃ ≻ H₂ ≻ H₁
q-ROFYWG operator [3]	0.4730	0.4711	0.5172	0.5763	H₄ ≻ H₃ ≻ H₂ ≻ H₁

Table 18

Ranking results using existing operators (for Randić energy)

Operators	S (H₁)	S (H₂)	S (H₃)	S (H₄)	Ranking
q-ROFYWA operator [3]	0.5132	0.4891	0.4845	0.5127	H₁ ≻ H₄ ≻ H₂ ≻ H₃
q-ROFYWG operator [3]	0.5134	0.4888	0.4848	0.5131	H₁ ≻ H₄ ≻ H₂ ≻ H₃

From Table 17, it is clear that the ranking results are similar with our proposed method, which depicts the authenticity of the method. However, the decision results in Table 18 are different from our method. This is due to applying different aggregation operators or due to a slight difference in the values obtained by the score function. The comparison among the proposed and existing methods by finding the energy and Randić energy of q-ROFHGs is graphically displayed in Fig. 8 and 9, respectively.

Fig. 8

Graphical representation of the results of four operators (for energy)

Fig. 9

Graphical representation of the results of four operators (for Randić energy)

The Hamacher operations, as a generalization of algebraic operations and Einstein t-norm and t-conorm have a great significance. The Hamacher operations transforms into the algebraic t-norm and t-conorm for the value of operational parameter one and, if we take the value of operational parameter two, then the Hamacher operations transforms into the Einstein t-norm and t-conorm. The comparison among the different aggregation operators of q-ROFNs is given in Table 19.

Table 19

Comparison of different aggregation operators of q-ROFNs

Aggregation operators	Whether the operator can capture the interrelationship between q-ROFNs	Parameterized
q-ROFWA [27]	No	No
q-ROFWG [27]	No	No
q-ROFGHM [37]	Yes	Yes
q-ROFGWHM [37]	Yes	Yes
q-ROFWGHM [37]	Yes	Yes
q-ROFHA [9]	Yes	Yes
q-ROFWHA [9]	Yes	Yes
q-ROFHMSM [9]	Yes	Yes
q-ROFYWA [3]	Yes	Yes
q-ROFYOWA [3]	Yes	Yes
q-ROFYHWA [3]	Yes	Yes
q-ROFYWG [3]	Yes	Yes
q-ROFYOWG [3]	Yes	Yes
q-ROFYHWG [3]	Yes	Yes

5.3 Discussion

q-ROFS as a generalization of IFS and PFS is more flexible as it accommodates greater space for orthopairs. The constraint of q-ROFS, κ^q + λ^q ≤ 1 (q ≥ 1) has made it easier to express the values of orthopairs more accurately, where, κ and λ represents the MD and NMD. The following Tables represents ranking of the above example for energy and Randić energy using IF, PF and q-ROF models:

From Table 20 and 21, it can be observe that the ranking results cannot be calculated as it is not possible to find the energy and Randić energy by using HWA and HWG operators of the example for IF model, since the sum of MD and NMD of orthopairs exceeds one. Similarly, using PF model it is impossible to evaluate the results as the sum of squares for MD and NMD is greater than one. However, for q = 3, the results can be calculated. For q = 1, the q-ROFN becomes IFN and for q = 2 it becomes PFN. As the value of q increases, the space for admissible orthopairs also increases. For instance, consider the orthopair (0.5, 0.9) _q, which is a 3-ROFN. For q=3, (0.5) ³ + (0.9) ³ = 0.854 ≤ 1, but (0.5) ² + (0.9) ² = 1.06 > 1 and 0.5 + 0.9 = 1.4 > 1 . Here IF and PF models fails to manage this information. So, (0.5,0.9) is not an IFN as well as PFN. Thus, every IF grade is PF grade and q-ROF grade. But the q-ROF grades are not IF and PF grades. Hence the best housing society for investment by finding the energy and Randić energy of q-ROFHDGs is H₄ .

Table 20
Ranking results different models (for energy)

Models Ranking(HWA) Ranking(HWG)

IFN(1-ROFN) impossible impossible

PFN(2-ROFN) impossible impossible

q-ROFN(q=3) H₄ ≻ H₃ ≻ H₂ ≻ H₁ H₄ ≻ H₃ ≻ H₂ ≻ H₁

Models	Ranking(HWA)	Ranking(HWG)
IFN(1-ROFN)	impossible	impossible
PFN(2-ROFN)	impossible	impossible
q-ROFN(q=3)	H₄ ≻ H₃ ≻ H₂ ≻ H₁	H₄ ≻ H₃ ≻ H₂ ≻ H₁

Table 21

Ranking results using different models (for Randić energy)

Models	Ranking(HWA)	Ranking(HWG)
IFN(1-ROFN)	impossible	impossible
PFN(2-ROFN)	impossible	impossible
q-ROFN(q=3)	H₄ ≻ H₁ ≻ H₂ ≻ H₃	H₄ ≻ H₁ ≻ H₂ ≻ H₃

In the article [25], the authors introduced the concept of degree and total degree of q-ROFGs by applying min - max operator. Further, some product operations on q-ROFGs, including direct product, cartesian product, semi-strong product, strong product and lexicographic product, are presented and discussed with examples by utilizing min - max operator. However, in our proposed work, we have studied and discussed the energy and Randić energy of q-ROFGs using Hamacher operators.

6 Conclusions

The decision making situations in real life are generally incomprehensible and imprecise in nature. Due to insufficient information, the future results might not be known properly. The concept of FS is capable to provide a framework to handle the imprecise information. In addition, the q-ROFS is more effective than IFS and PFS as it accommodates a large space of admissible orthopairs. In this paper, we have proposed a novel concept of q-ROFGs using Hamacher operator, namely q-ROFHGs. We have illustrated the energy and Randić energy of q-ROFHGs and provide some crucial results. Moreover, the energy of a splitting q-ROFHG and shadow q-ROFHG has introduced. We have also discussed the energy and Randić energy of q-ROFHDGs. Finally, a decision making example related to the housing society selection and an algorithm has quoted by to enhance the validity of our method. In the decision making example, we have find the energy and Randić energy of q-ROFGs by using q-rung orthopair fuzzy Hamacher operators. It is to be noted that by finding energy and Randić energy of q-ROFHDGs, the ranking of the alternatives is changed. However, the best alternative, which is the housing society H₄, remains same. Thus, the presented work gives a new platform to the information measure theory and provide a new track to tackle the ambiguities throughout the decision making procedure. We are planning to extend our work to: (1) Interval-valued q-rung orthopair fuzzy graphs; (2) simplified interval-valued q-rung orthopair fuzzy graphs; and (3) hesitant q-rung orthopair fuzzy graphs.

Conflict of Interest: The authors declare no conflict of interest.

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q -Rung orthopair fuzzy graphs under Hamacher operators

Abstract

Keywords

1 Introduction

2 Preliminaries

3 q-Rung orthopair fuzzy hamacher graphs

5.1 Selection of housing society for investment

Table 1 q-ROFPRs of the first decision making expert Q 1 H 1 H 2 H 3 H 4 H 1 (0.5,0.5) (0.5,0.9) (0.2,0.8) (0.7,0.8) H 2 (0.9,0.5) (0.5,0.5) (0.7,0.7) (0.9,0.4) H 3 (0.8,0.2) (0.7,0.7) (0.5,0.5) (0.3,0.9) H 4 (0.8,0.7) (0.4,0.9) (0.9,0.3) (0.5,0.5)

Table 17 Ranking results using existing operators (for energy) Operators S (H1) S (H2) S (H3) S (H4) Ranking q-ROFYWA operator [3] 0.4369 0.4709 0.5079 0.5761 H4 ≻ H3 ≻ H2 ≻ H1 q-ROFYWG operator [3] 0.4730 0.4711 0.5172 0.5763 H4 ≻ H3 ≻ H2 ≻ H1

Table 20 Ranking results different models (for energy) Models Ranking(HWA) Ranking(HWG) IFN(1-ROFN) impossible impossible PFN(2-ROFN) impossible impossible q-ROFN(q=3) H4 ≻ H3 ≻ H2 ≻ H1 H4 ≻ H3 ≻ H2 ≻ H1

References

Table 1
q-ROFPRs of the first decision making expert

Q ₁ H ₁ H ₂ H ₃ H ₄

H ₁ (0.5,0.5) (0.5,0.9) (0.2,0.8) (0.7,0.8)

H ₂ (0.9,0.5) (0.5,0.5) (0.7,0.7) (0.9,0.4)

H ₃ (0.8,0.2) (0.7,0.7) (0.5,0.5) (0.3,0.9)

H ₄ (0.8,0.7) (0.4,0.9) (0.9,0.3) (0.5,0.5)

Table 17
Ranking results using existing operators (for energy)

Operators S (H₁) S (H₂) S (H₃) S (H₄) Ranking

q-ROFYWA operator [3] 0.4369 0.4709 0.5079 0.5761 H₄ ≻ H₃ ≻ H₂ ≻ H₁

q-ROFYWG operator [3] 0.4730 0.4711 0.5172 0.5763 H₄ ≻ H₃ ≻ H₂ ≻ H₁

Table 20
Ranking results different models (for energy)

Models Ranking(HWA) Ranking(HWG)

IFN(1-ROFN) impossible impossible

PFN(2-ROFN) impossible impossible

q-ROFN(q=3) H₄ ≻ H₃ ≻ H₂ ≻ H₁ H₄ ≻ H₃ ≻ H₂ ≻ H₁