Geometric-arithmetic energy and atom bond connectivity energy of dual hesitant q -rung orthopair fuzzy graphs

Abstract

q-Rung orthopair fuzzy sets (q-ROFSs), originally proposed by Yager, can powerfully modify the range of indication of decision information by changing a parameter q based on the different hesitation degree, and the dual hesitant q-rung orthopair fuzzy set (DHq-ROFS), a new technique to consider human’s hesitance, can be more substantial of dealing with real multi-attribute decision making (MADM) problems. Inspired by DHq-ROFSs, in this article, we extend the concept of q-rung orthopair fuzzy graphs to dual hesitant q-rung orthopair fuzzy context and introduce the innovative concept of a dual hesitant q-rung orthopair fuzzy graphs based on Hamacher operator called dual hesitant q-rung orthopair fuzzy Hamacher graphs (DHq-ROFHGs). We propose the new concepts of geometric-arithmetic energy and atom bond connectivity energy of a DHq-ROFHG and determine its upper and lower bounds. Moreover, on the basis of the proposed concept of DHq-ROFHGs, we introduce a new approach to solve the MADM problems with dual hesitant q-rung orthopair fuzzy information. At the end, we give a numerical model related to the selection of most significant defensive factor to illustrate the applicability of the developed approach, and exhibit its viability. Comparative analysis is conducted and the superiorities are illustrated.

Keywords

Dual hesitant q-rung orthopair fuzzy graph Hamacher operator Geometric-arithmetic energy Atom bond connectivity energy Multi-attribute decision making

1 Introduction

Yager [31] introduced the concept of q-ROFS as an effective and powerful tool for describing fuzziness and uncertainty. It is one of the useful extensions of fuzzy set theory [34], characterized by a membership degree (MD) and a non-membership degree (NMD), with a remarkable characteristic that the qth power sum of the MD and the NMD is not greater than one, that is, Q = 〈r, (φ_Q (r) , ϑ_Q (r)) |r ∈ U〉, where $0 \leq φ_{Q}^{q} (r) + ϑ_{Q}^{q} (r) \leq 1 .$ If q = 1, q = 2 and q = 3, the q-ROFS is reduced to the intuitionistic fuzzy set [6], Pythagorean fuzzy set [32] and Fermatean fuzzy set [22], respectively. Expanding estimation of q represents the more substantial orthopairs, which makes the decision maker’s assessments more reliably about the membership and non-membership of an element to a set. The q-ROFS has larger opportunity for decision makers in portraying their preferences on alternatives. In any case, when deciding the MD of an element to a set, the decision makers encounter the hesitation among a few different values. Torra [24] presented the idea of hesitant fuzzy sets (HFSs) to manage such circumstances. HFSs grant the MD of an element to a set of a few possible values between [0, 1], which is progressively objective and powerful in expressing decision makers inherent hesitancy. Hesitant fuzzy elements (HFEs) just express the decision maker’s conviction assessments, but ignore their uncertainty opinions. So to illuminate such cases, Zhu et al. [38] gave the concept of dual hesitant fuzzy sets (DHFSs), whose membership hesitancy function and non-membership hesitancy function are represented by a set of a few possible values between [0, 1]. Since its appearance, more and more researchers have started to apply this idea to manage MADM problems under DHF environment. Wang et al. [27] incorporated the DHFSs into q-ROFSs, and set forward another tool to manage ambiguity, called DHq-ROFSs and developed the weighting average operator and weighting geometric operator in DHq-ROF environment, using Hamacher operator. The q-ROFSs are more useful to express more fuzzy and ambiguous information. However, to consider human’s hesitance, the DHq-ROFSs can be more valid in handling real MADM problems. Xu et al. [30] presented a group of DHq-ROF Heronian mean operators and built up a MADM approach utilizing the proposed aggregation operators. Wang et al. [26] developed some DHq-ROF Muirhead mean operators to combine the DHq-ROFSs information more successfully. Liu et al. [16] presented the distance measures between DHq-ROFSs dependent on the idea of “multiple fuzzy sets”, and build up the TOPSIS technique to the introduced distance measures. Recently, many researchers developed several decision making methods in extended fuzzy context [12–15 , 35–37].

To understand and design our world, we need to model and examine connections between objects. Objects and connections can be demonstrated by a graph, whose vertices and edges represent the objects and the connections, respectively. Spectral graph theory is the study of the graph properties in relationship to the characteristic polynomial and eigenvalues of matrices related to the graph, such as its adjacency matrix, Randić matrix, Harmonic matrix, Zagreb matrix, Laplacian matrix or signless Laplacian matrix. Gutman [8] proposed the idea of the graph energy in chemistry and discovered lower and upper limits for the graphs energy. The invariants, which are based on the distance or degree of vertices in molecules are called topological indices. In theoretical chemistry, physics and graph theory, topological indices are the molecular descriptors that depict the structures of chemical compounds, and they help us to predict certain physico-chemical properties. The well-known molecular descriptors are the geometric arithmetic (GA) index and the atom bond connectivity (ABC) index. $Vuki \overset{ˇ}{c} evi \overset{ˇ}{c}$ et al. [25] defined geometric-arithmetic index of a graph G, represented by GA (G). The geometric-arithmetic matrix GA (G) = (a_ij) of a graph G whose vertex f_i has degree d_i is defined by $a_{ij} = \sum_{i j \in E (G)} \frac{2 \sqrt{d_{i} . d_{j}}}{d_{i} + d_{j}}$ if the vertices f_i and f_j are adjacent and a_ij = 0 otherwise. The geometric-arithmetic energy is the sum of absolute values of the eigenvalues of GA (G). Estrada et al. [7] proposed the atom-bond connectivity (ABC) index. The atom bond connectivity matrix ABC (G) = (r_ij) of a graph G whose vertex f_i has degree d_i is defined by $r_{ij} = \sum_{i j \in E (G)} \sqrt{\frac{d_{i} + d_{j} - 2}{d_{i} . d_{j}}}$ if the vertices f_i and f_j are adjacent and r_ij = 0 otherwise. The atom bond connectivity energy is the sum of absolute values of the eigenvalues of ABC (G).

Uncertainties in graph vertices and edges are common in this context, due to a variety of reasons such as noisy measurements and inconsistent information sources. To deal such uncertainties in objects and connections, Rosenfeld [21] proposed the concept of fuzzy graphs (FGs) and developed its structure. Anjali and Mathew [5] determine the energy of a FG. Naz et al. proposed the concepts of Pythagorean fuzzy graphs (PFG) [19] and complex PFG [4] as well as its pertinent applications in decision making. Karaaslan [11] proposed hesitant FG and discussed some of its properties. To deal with the MADM problems, Naz and Akram [18] developed a new decision making approach based on graph theory in which the decision information is expressed by HFEs. Akram et al. [9] put forward the new concept of q-rung orthopair FGs along with its application in the soil ecosystem. Akram et al. [3, 23] introduced many new concepts of graphs in generalized fuzzy circumstances. Recently, Akram et al. [1, 2] defined trapezoidal picture fuzzy numbers along with its graphical representation and proposed a new approach of formation of granular structures based on fuzzy soft graphs.

A series of FG generalizations were introduced to deal with the complexity of the real-life complex problems. Since uncertainties are well represented using DHq-ROFS, dual hesitant q-rung orthopair fuzzy graph (DHq-ROFG) will be prominent research approach for modeling the uncertain problems of optimization. This inspired us to present DHq-ROFG’s new concept and discuss its properties. As the generalization of q-ROFGs and HFGs, the DHq-ROFGs are more useful to express more fuzzy and ambiguous information by considering human’s hesitance. Hamacher operations include Hamacher product and Hamacher sum, which are powerful alternatives to the algebraic product and algebraic sum, respectively. Hamacher t-conorm and t-norm, which are the algebraic and Einstein t-conorm and t-norm generalization, are more general and more versatile. There is important significance to research graph theoretical concepts based on Hamacher operations with DHq-ROF information, and introduce DHq-ROFHG. The key features of this research are:

To introduce the novel concept of DHq-ROFGs based on Hamacher operator, called DHq-ROFHG.

To determine the geometric-arithmetic energy of DHq-ROFHG.

To present the atom bond connectivity energy of a DHq-ROFHG.

To develop a novel MADM model which can deal with the problems.

To provide application of proposed concept of DHq-ROFHG in MADM.

Compared with many existing generalized fuzzy graph theories, the newly proposed DHq-ROFHGs show extraordinary effectiveness and flexibility and can successfully express the decision making opinions of decision experts in a very hesitant state.

The remainder of the paper is organized as follows. In Section 2, some basic concepts are briefly recalled. Section 3 introduces the concept of DHq-ROFHGs and determines its geometric-arithmetic energy, the atom bond connectivity energy, and provides its useful properties along with upper and lower bounds. In Section 4, we develop the graph-based DHq-ROF decision-making model. Section 5 provides a numerical example to demonstrate the validity and superiority of the proposed concept of DHq-ROFHGs in decision making. Finally, we summarize the paper in Section 6.

2 Preliminaries

In this section, the basic concepts of DHq-ROFSs and t-norms are provided to facilitate the next sections.

Definition 2.1. [30] Let U be an ordinary fixed set. A DHq-ROFS D defined on U is given by $D = {〈 r, h_{D} (r), g_{D} (r) 〉 | r \in U},$ in which h_D (r) and g_D (r) are two sets of values in [0, 1] denoting the possible MD and NMD of the element r ∈ U to the set D, respectively, with the conditions $φ^{q} + ϑ^{q} \leq 1 (q \geq 1),$ where φ ∈ h_D (r) , ϑ ∈ g_D (r) for all r ∈ U. For convenience, the pair ς (r) = (h_D (r) , g_D (r)) is called a DHq-ROFE denoted by ς = (h, g) with the conditions φ ∈ h, ϑ ∈ g, 0 ≤ φ, ϑ ≤ 1, 0 ≤ φ^q + ϑ^q ≤ 1. Evidently, when q = 1, then DHq-ROFS is reduced to Zhu et al.’s [38] DHFS and when q = 2, then DHq-ROFS is reduced to Wei and Lu’s [29] DHPFS.

Example 2.1. Suppose a fixed set U has only one element r, $h_{D} (r) = {0.1, 0.4, 0.6}, g_{D} (r) = {0.2, 0.7}$ . Then $D = 〈 r, {0.1, 0.4, 0.6}, {0.2, 0.7} 〉$ is a DH4-ROFN and denoted by $D = 〈 {0.1, 0.4, 0.6}, {0.2, 0.7} 〉_{4}$ for simplicity.

Definition 2.2. [27] Suppose ς = (h, g) is a DHq-ROFN. Let $S (ς) = \frac{1}{2} (1 + (\frac{1}{# h} \sum_{φ \in h} φ^{q}) - (\frac{1}{# g} \sum_{ϑ \in g} ϑ^{q}))$ be the score function of ς = (h, g) and $H (ς) = \frac{1}{# h} \sum_{φ \in h} φ^{q} + \frac{1}{# g} \sum_{ϑ \in g} ϑ^{q}$ be the accuracy function of ς = (h, g), where # h and # g indicate the number of elements in sets h and g, respectively. Let ς_i = (h_i, g_i) (i = 1, 2) be any two DHq-ROFNs, then

(1) if S (ς₁) > S (ς₂), then ς₁ ≻ ς₂;

(2) if S (ς₁) = S (ς₂), then: (a) if H (ς₁) = H (ς₁), then ς₁ = ς₂; (b) if H (ς₁) > H (ς₂), then ς₁ ≻ ς₂.

Definition 2.3. [30] Let ς = (h, g) , ς₁ = (h₁, g₁), and ς₂ = (h₂, g₂) be any three DHq-ROFEs and λ be a positive real number. Then,

(i) ς₁ ⊕ ς₂ = ∪ _{φ₁∈h₁,φ₂∈h₂,ϑ₁∈g₁,ϑ₂∈g₂}

${{(φ_{1}^{q} + φ_{2}^{q} - φ_{1}^{q} φ_{2}^{q})^{\frac{1}{q}}}, {ϑ_{1} ϑ_{2}}};$

(ii) ς₁ ⊗ ς₂ = ∪ _{φ₁∈h₁,φ₂∈h₂,ϑ₁∈g₁,ϑ₂∈g₂}

${{φ_{1} φ_{2}}, {(ϑ_{1}^{q} + ϑ_{2}^{q} - ϑ_{1}^{q} ϑ_{2}^{q})^{\frac{1}{q}}}};$

(iii) λς = ∪ _{φ∈h,ϑ∈g}

${{{(1 - (1 - φ^{q})^{λ})}^{\frac{1}{q}}}, {ϑ^{λ}}}, λ > 0;$

(iv) ς^λ = ∪ _{φ∈h,ϑ∈g}

${{φ^{λ}}, {{(1 - (1 - ϑ^{q})^{λ})}^{\frac{1}{q}}}}, λ > 0 .$

To extend the existing operations of t-norm and t-conorm, Hamacher [10] introduced the Hamacher product t-norm and Hamacher sum t-conorm as generalizations of t-norms and t-conorms, respectively, as follows: $T_{ϱ}^{H} (r, s) = {\begin{matrix} \frac{rs}{ϱ + (1 - ϱ) (r + s - rs)} & if ϱ > 0, \\ \frac{rs}{r + s - rs} & if ϱ = 0, \end{matrix}$

$(T^{*})_{ϱ}^{H} (r, s) = {\begin{matrix} \frac{r + s - rs - (1 - ϱ) rs}{1 - (1 - ϱ) rs} & if ϱ > 0, \\ \frac{r + s - 2 rs}{1 - rs} & if ϱ = 0, \end{matrix}$

Definition 2.4. [16] Let $L (D_{j} (r_{i}))_{j = 1}^{m}$ be the least common multiple (LCM) of l (h_{D
₁} (r_i)) , l (g_{D
₁} (r_i)) , l (h_{D
₂} (r_i)) , l (g_{D
₂} (r_i)) , …, l (h_{D
_m} (r_i)) , l (g_{D
_m} (r_i)), then the DHq-ROFSs $\bar{D} = {D_{1}, D_{2}, \dots, D_{m}}$ can be extended to multiple DHq-ROFSs $\bar{D^{'}} = {\underset{1}{D^{'}}, \underset{2}{D^{'}}, \dots, \underset{m}{D^{'}}}$ as follows: $\begin{matrix} \underset{1}{D^{'}} (r_{i}) = {〈 {φ_{D_{1}^{1}} (r_{i}), \dots, φ_{D_{1}^{2}} (r_{i}), \dots, φ_{D_{1}^{l (h_{D_{1}})}} (r_{i}), \dots}, {ϑ_{D_{1}^{1}} (r_{i}), \dots, ϑ_{D_{1}^{2}} (r_{i}), \dots, ϑ_{D_{1}^{l (g_{D_{1}})}} (r_{i}), \dots} 〉}_{q}; \\ \underset{2}{D^{'}} (r_{i}) = {〈 {φ_{D_{2}^{1}} (r_{i}), \dots, φ_{D_{2}^{2}} (r_{i}), \dots, φ_{D_{2}^{l (h_{D_{2}})}} (r_{i}), \dots}, {ϑ_{D_{2}^{1}} (r_{i}), \dots, ϑ_{D_{2}^{2}} (r_{i}), \dots, ϑ_{D_{2}^{l (g_{D_{2}})}} (r_{i}), \dots} 〉}_{q}; \\ \dots & \dots & \dots & \dots \\ \underset{m}{D^{'}} (r_{i}) = {〈 {φ_{D_{m}^{1}} (r_{i}), \dots, φ_{D_{m}^{2}} (r_{i}), \dots, φ_{D_{m}^{l (h_{D_{m}})}} (r_{i}), \dots}, {ϑ_{D_{m}^{1}} (r_{i}), \dots, ϑ_{D_{m}^{2}} (r_{i}), \dots, ϑ_{D_{m}^{l (g_{D_{m}})}} (r_{i}), \dots} 〉}_{q} . \end{matrix}$ where the numbers of $φ_{D_{j}^{k_{1}}} (r_{i})$ and $ϑ_{D_{j}^{k_{2}}} (r_{i}) (j = 1, 2, \dots, m; k_{1} = 1, 2, \dots, l (h_{D_{j}}); k_{2} = 1, 2, \dots, l (g_{D_{j}}))$ are determined by $N (φ_{D_{j}^{k_{1}}} (r_{i})) = \frac{L (D_{j} (r_{i}))_{j = 1}^{m}}{l (h_{D_{j}} (r_{i}))}$ and $N (ϑ_{D_{j}^{k_{2}}} (r_{i})) = \frac{L (D_{j} (r_{i}))_{j = 1}^{m}}{l (g_{D_{j}} (r_{i}))}$ , respectively.

Definition 2.5. [16] Let ς₁ = (h₁, g₁) and ς₂ = (h₂, g₂) be two dual hesitant q-rung orthopair fuzzy normalized Euclidean (DHq-ROFNE) distance measure between ς₁ and ς₂ can be defined as follows: $\begin{matrix} d_{DHq - ROFNE} (ς_{1}, ς_{2}) = {[\sum_{i = 1}^{n} \frac{1}{n} \frac{1}{2 L (ς_{j} (r_{i}))_{j = 1}^{2}} (\sum_{j = 1}^{\underset{j = 1}{L (ς_{j} (r_{i}))^{2}}} (| (φ_{ς_{1}^{σ (j)}} (r_{i}))^{q} - (φ_{ς_{2}^{σ (j)}} (r_{i}))^{q} |^{2} + | (ϑ_{ς_{1}^{σ (j)}} (r_{i}))^{q} - (ϑ_{ς_{2}^{σ (j)}} (r_{i}))^{q} |^{2}))]}^{\frac{1}{2}} . \end{matrix}$

Theorem 2.1. [20] If p ≥ 1 is an integer and 0 ≤ x₁, x₂, …, x_n ≤ n - 1, then $\begin{matrix} {(\sum_{j = 1}^{n} x_{j}^{\frac{1}{p}})}^{p} \geq (n - 1)^{1 - p} \sum_{j = 1}^{n} x_{j}^{p} . (2.1) \end{matrix}$ (2.1)

Theorem 2.2. [20] If f is convex function in $ℝ_{+}$ and x₁, x₂, …, x_n ≥ 0, then $\begin{matrix} f (\frac{x_{1} + x_{2} + \dots + x_{n}}{n}) \\ \leq \frac{1}{n} (f (x_{1}) + f (x_{2}) + \dots + f (x_{n})) . (2.2) \end{matrix}$ (2.2)

3 Dual hesitant q-rung orthopair fuzzy Hamacher graphs

In this section, we introduce the new concepts of DHq-ROF relations, dual hesitant q-rung orthopair fuzzy preference relations and DHq-ROFHGs. Further we determine the geometric-arithmetic energy and atom bond connectivity energy of a DHq-ROFHG.

Definition 3.1. Let U be the universe of discourse. A DHq-ROFS $E$ in U × U is said to be a dual hesitant q-rung orthopair fuzzy relation (DHq-ROFR) in U, denoted by $E = {〈 rs, h_{E} (rs), g_{E} (rs) 〉 | rs \in U \times U},$ where $h_{E} : U \times U \to [0, 1]$ and $g_{E} : U \times U \to [0, 1]$ represent the membership and non-membership function of $E$ , respectively, such that $0 \leq h_{E}^{q} (rs) + g_{E}^{q} (rs) \leq 1$ for all rs ∈ U × U .

Definition 3.2. A dual hesitant q-rung orthopair fuzzy preference relation (DHq-ROFPR) H on U is represented by a matrix H = [ς_ij] _n×n ⊂ U × U, where ς_ij = (h_ij, g_ij) (i, j = 1, 2, …, n) is a DHq-ROFE with h_ij representing all the possible degrees to which r_i is preferred to r_j and g_ij representing all the possible degrees to which r_i is non-preferred to r_j. Further, ς_ij should satisfy the following conditions:

if h_ij ≠ φ and g_ij ≠ φ, then h_ji = g_ij and g_ji = h_ij;

if h_ij ≠ φ and g_ij = φ, then $φ_{ij}^{σ (t)} + φ_{ji}^{σ (# h_{ji} - t + 1)} = 1$ , # h_ji = # h_ij, g_ji = φ, where $φ_{ij}^{σ (t)}$ is the t-th smallest element in h_ij and $φ_{ji}^{σ (# h_{ji} - t + 1)}$ is the # (h_ji - t + 1) th smallest element in h_ji;

if h_ij = φ and g_ij ≠ φ, then h_ji = φ, $ϑ_{ij}^{σ (t)} + ϑ_{ji}^{σ (# g_{ji} - t + 1)} = 1$ , # g_ji = # g_ij, , where $ϑ_{ij}^{σ (t)}$ is the t-th smallest element in g_ij and $ϑ_{ji}^{σ (# h_{ji} - t + 1)}$ is the # (g_ji - t + 1) th smallest element in g_ji;

and h_ii = {0.5}, g_ii = {0.5} for all i, j = 1, 2, …, n .

Definition 3.3. A dual hesitant q-rung orthopair fuzzy Hamacher graph (DHq-ROFHG) on a non-empty set U is a pair $G = (D, E)$ , where $D$ is a DHq-ROFS on U and $E$ is a DHq-ROFR on U such that:

$\begin{matrix} S (h_{E} (rs)) \leq \frac{S (h_{D} (r) S (h_{D} (s))}{S (h_{D} (r)) + S (h_{D} (s)) - S (h_{D} (r)) S (h_{D} (s))}; \\ S (g_{E} (rs)) \leq \frac{S (g_{D} (r)) + S (g_{D} (s)) - 2 S (g_{D} (r)) S (g_{D} (s))}{1 - S (g_{D} (r)) S (g_{D} (s))} . \end{matrix}$ $0 \leq S (h_{E}^{q} (rs)) + S (g_{E}^{q} (rs)) \leq 1$ for all r, s ∈ U . We call $D$ and $E$ the DHq-ROFS of vertices and the DHq-ROFS of edges in $G$ , respectively. Here, $E$ is a symmetric DHq-ROFR on $D$ . If $E$ is not symmetric on $D$ , then $D = (D, \vec{E})$ is called dual hesitant q-rung orthopair fuzzy Hamacher digraph (DHq-ROFHDG).

Example 3.1. Consider a graph G = (V, E), with V = {f₁, f₂, f₃, f₄, f₅, f₆, f₇, f₈} and E = {f₁f₂, f₂f₃, f₃f₄, f₄f₁, f₅f₆, f₅f₈, f₆f₇, f₇f₈, f₁f₆, f₁f₈, f₂f₅, f₂f₇, f₃f₆, f₃f₈, f₄f₅, f₄f₇} as the set of vertices and edges, respectively. Let $G = (D, E)$ be a DH3-ROFHG on V, as in Fig. 1:

Fig. 1

Dual hesitant 3-rung orthopair fuzzy Hamacher graph.

Clearly, $G = (D, E)$ shown in Fig. 1, is a DH3-ROFHG. Tabular representation of DH3-ROFHG is given in Table 1.

Table 1

Tabular representation of a DH3-ROFHG

	f ₁	f ₂	f ₃	f ₄	f ₅	f ₆	f ₇	f ₈
$h_{D}$	{0.3, 0.4}	{0.2, 0.5}	{0.3, 0.5, 0.6}	{0.2, 0.5, 0.6}	{0.1, 0.7, 0.8}	{0.2, 0.4, 0.5}	{0.9, 0.7}	{0.4, 0.6, 0.7}
$S (h_{D})$	{0.35}	{0.35}	{0.47}	{0.65}	{0.53}	{0.37}	{0.80}	{0.57}
$g_{D}$	{0.8, 0.9}	{0.4, 0.9}	{0.6, 0.7}	{0.5, 0.6}	{0.4, 0.6}	{0.3, 0.9}	{0.4, 0.6}	{0.7, 0.8}
$S (g_{D})$	{0.85}	{0.65}	{0.65}	{0.55}	{0.50}	{0.40}	{0.50}	{0.75}
	f ₆ f ₇	f ₇ f ₈	f ₁ f ₆	f ₁ f ₈	f ₂ f ₅	f ₂ f ₇	f ₃ f ₆
$h_{E}$	{0.16, 0.27, 0.31}	{0.37, 0.46, 0.54}	{0.16, 0.28}	{0.25, 0.27}	{0.22, 0.19}	{{0.19, 0.29, 0.32}	{0.09, 0.25, 0.30}
$S (h_{E})$	{0.25}	{0.46}	{0.22}	{0.26}	{0.21}	{0.27}	{0.21}
$g_{E}$	{0.58, 0.61}	{0.63, 0.88}	{0.73, 0.83}	{0.83, 0.90}	{0.68, 0.80}	{0.68, 0.80}	{0.68, 0.73}
$S (g_{E})$	{0.60}	{0.76}	{0.78}	{0.87}	{0.74}	{0.74}	{0.71}
f ₃ f ₈	f ₄ f ₅	f ₄ f ₇
{0.19, 0.26, 0.35}	{0.45, 0.36}	{0.54, 0.45}
{0.27}	{0.41}	{0.50}
{{0.81, 0.78}	{0.55, 0.60}	{0.59, 0.61}
{0.80}	{0.58}	{0.60}

3.1 Geometric-arithmetic energy of a DHq-ROFHG

This subsection defines and investigates the geometric-arithmetic energy of a DHq-ROFHG and provides its properties in detail.

Definition 3.4. Let $G = (D, E)$ be a DHq-ROFHG on n vertices. The geometric-arithmetic matrix, $GA (G) = (GA (h_{E} (r_{i} r_{j})), GA (g_{E} (r_{i} r_{j}))) = [a_{ij}]$ of $G$ is a n × n matrix defined as: $a_{ij} = {\begin{matrix} 0 & if i = j, \\ \sum_{r_{i} r_{j} \in E (G)} \frac{2 \sqrt{d_{G} (r_{i}) . d_{G} (r_{j})}}{d_{G} (r_{i}) + d_{G} (r_{j})} & if the vertices r_{i} and r_{j} of the DHq - ROFHG G are adjacent, \\ 0 & if the vertices r_{i} and r_{j} of the DHq - ROFHG G are non - adjacent . \end{matrix}$

Definition 3.5. The geometric-arithmetic energy of a DHq-ROFHG $G = (D, E)$ is defined as: $\begin{matrix} GAE (G) = (GAE (h_{E} (r_{i} r_{j})), GAE (g_{E} (r_{i} r_{j})) \\ = (\sum_{j = 1 ϒ_{j} \in Y_{GA}}^{n} | ϒ_{j} |, \sum_{j = 1 Θ_{j} \in Z_{GA}}^{n} | Θ_{j} |), \end{matrix}$ where Y_GA and Z_GA are the sets of geometric-arithmetic eigenvalues of $GA (h_{E} (r_{i} r_{j}))$ and $GA (g_{E} (r_{i} r_{j}))$ , respectively.

Example 3.2. Let $G = (D, E)$ be a DHq-ROFHG on V = {f₁, f₂, f₃,f₄,f₅} and E = {f₁f₂, f₂f₃, f₃f₄, f₄f₅, f₁f₅, f₁f₃, f₁f₄} as in Fig. 2, defined by:

Fig. 2

A dual hesitant 3-rung orthopair fuzzy Hamacher graph.

Therefore $Spec (GA (S (G)))$ = {(-1.0000, -1.0000), (-1.0000, -1.0000), (-0.9997, -0.9986), (-0.9279, -0.9017), (3.9275, 3.9002)}. Also, $GAE (G)$ = (7.8551, 7.8005) .

Now we firstly determine the trace of the geometric-arithmetic matrices $GA (G)$ , ${GA}^{2} (G)$ , ${GA}^{3} (G),$ and ${GA}^{4} (G)$ , i.e., $tr (GA (G))$ , $tr ({GA}^{2} (G))$ , $tr ({GA}^{3} (G))$ , and $tr ({GA}^{4} (G))$ . Moreover, using these equalities the upper and lower bounds for geometric-arithmetic energy are obtained.

Lemma 3.1. Let $G = (D, E)$ be a DHq-ROFHG on n vertices and $GA (G) = (GA (h_{E} (r_{i} r_{j})), GA (g_{E} (r_{i} r_{j})))$ be the geometric-arithmetic matrix of $G$ . Then

$tr (GA (G)) = 0$ ,

$tr ({GA}^{2} (G)) = 2 \sum_{r_{i} r_{j} \in E (G)} \frac{4 d_{G} (r_{i}) . d_{G} (r_{j})}{(d_{G} (r_{i}) + d_{G} (r_{j}))^{2}}$ ,

$tr ({GA}^{3} (G)) = 2 \sum_{r_{i} r_{j} \in E (G)} \frac{4 d_{G} (r_{i}) . d_{G} (r_{j})}{d_{G} (r_{i}) + d_{G} (r_{j})}$ $\sum_{r_{k} \in D (G) r_{k} \sim r_{i}, r_{k} \sim j} \frac{2 d_{G} (r_{k})}{(d_{G} (r_{i}) + d_{G} (r_{k})) (d_{G} (r_{j}) + d_{G} (r_{k}))}$ ,

$tr ({GA}^{4} (G))$ = $\sum_{r_{i} \in D (G)} {(\sum_{r_{i} \in D (G) i \sim j} \frac{4 (d_{G} (r_{i}) . d_{G} (r_{j}))}{(d_{G} (r_{i}) d_{G} (r_{j}))^{2}})}^{2} + \sum_{r_{i} r_{j} \in D (G) r_{i} \neq r_{j}} 4 (d_{G} (r_{i}) . d_{G} (r_{j}))$ ${(\frac{2 d_{G} (r_{k})}{(d_{G} (r_{i}) + d_{G} (r_{k})) (d_{G} (r_{j}) + d_{G} (r_{k}))})}^{2}$ .

Proof.

Obvious.

For matrix ${GA}^{2} (G)$ . If i = j $\begin{matrix} {GA}^{2} (h_{E} (r_{i} r_{i})) \\ = \sum_{j = 1}^{n} (GA (h_{E} (r_{i} r_{j}))) (GA (h_{E} (r_{j} r_{i}))) \\ = \sum_{j = 1}^{n} (GA (h_{E} (r_{i} r_{j})))^{2} \\ = \sum_{i \sim j} (GA (h_{E} (r_{i} r_{j})))^{2} \\ = \sum_{i \sim j} \frac{4 d_{h} (r_{i}) . d_{h} (r_{j})}{(d_{h} (r_{i}) + d_{h} (r_{j}))^{2}} . \end{matrix}$ Whereas if i ≠ j $\begin{matrix} {GA}^{2} (h_{E} (r_{i} r_{j})) \\ = \sum_{k = 1}^{n} (GA (h_{E} (r_{i} r_{k}))) (GA (h_{E} (r_{k} r_{j}))) \\ = (GA (h_{E} (r_{i} r_{i}))) (GA (h_{E} (r_{i} r_{j}))) \\ + (GA (h_{E} (r_{i} r_{j}))) ({GA}_{h_{E}} (r_{j} r_{j})) \\ + \sum_{k \sim i k \sim j} (GA (h_{E} (r_{i} r_{k}))) (GA (h_{E} (r_{k} r_{j}))) \\ = \sum_{r_{i} r_{j} \in E (G)} \frac{2 \sqrt{d_{G} (r_{i}) . d_{G} (r_{j})}}{d_{G} (r_{i}) + d_{G} (r_{j})} \\ \sum_{r_{k} \sim D (G) r_{k} \sim r_{i}, r_{k} \sim r_{j}} \frac{2 d (r_{k})}{(d_{G} (r_{i}) + d_{G} (r_{k})) (d_{G} (r_{j}) + d_{G} (r_{k}))} . \end{matrix}$ Therefore $\begin{matrix} tr ({GA}^{2} (h_{E} (r_{i} r_{j}))) \\ = \sum_{i = 1}^{n} \sum_{i \sim j} \frac{4}{{(d_{h} (r_{i}) + d_{h (r_{j})})}^{2}} \\ = 2 \sum_{i \sim j} \frac{4 d_{h} (r_{i}) . d_{h} (r_{j})}{(d_{h} (r_{i}) + d_{h} (r_{j}))^{2}} . \end{matrix}$ Similarly, we can show that $tr ({GA}^{2} (g_{E} (r_{i} r_{j}))) = 2 \sum_{i \sim j} \frac{4 d_{g} (r_{i}) . d_{g} (r_{j})}{(d_{g} (r_{i}) + d_{g} (r_{j}))^{2}} .$ Hence $tr ({GA}^{2} (G)) = 2 \sum_{r_{i} r_{j} \in E (G)} \frac{4 d_{G} (r_{i}) . d_{G} (r_{j})}{(d_{G} (r_{i}) d_{G} (r_{j}))^{2}}$ .

Now, we determine the matrix ${GA}^{3} (G)$ . The diagonal elements are $\begin{matrix} {GA}^{3} (h_{E} (r_{i} r_{i})) \\ = \sum_{j = 1}^{n} (GA (h_{E} (r_{i} r_{j}))) ({GA}^{2} (h_{E} (r_{j} r_{k}))) \\ = & \sum_{r_{i} r_{j} \in E (G)} \frac{2 \sqrt{d_{G} (r_{i}) . d_{G} (r_{j})}}{d_{G} (r_{i}) + d_{G} (r_{j})} {GA}^{2} (h_{E} (r_{i} r_{j})) \\ = 2 \sum_{r_{i} r_{j} \in E (G)} \frac{4 d_{h} (r_{i}) . d_{h} (r_{j})}{d_{h} (r_{i}) + d_{h} (r_{j})} \\ \sum_{k \in D (G) r_{k} \sim r_{i}, r_{k} \sim r_{j}} \frac{2 d_{h} (r_{k})}{(d_{h} (r_{k}) + d_{h} (r_{i})) (d_{h} (r_{k}) + d_{h} (r_{j}))} . \end{matrix}$ Therefore $\begin{matrix} tr ({GA}^{3} (h_{E} (r_{i} r_{j}))) \\ = 2 \sum_{r_{i} r_{j} \in E (G)} \frac{4 d_{h} (r_{i}) . d_{h} (r_{j})}{d_{h} (r_{i}) + d_{h} (r_{j})} \\ \sum_{r_{k} \in D (G) r_{k} \sim r_{i}, r_{k} \sim r_{j}} \frac{2 d_{h} (r_{k})}{(d_{h} (r_{k}) + d_{h} (r_{i})) (d_{h} (r_{k}) + d_{h} (r_{j}))} . \end{matrix}$

Similarly, we can show that

$\begin{matrix} tr ({GA}^{3} (g_{E} (r_{i} r_{j}))) \\ = 2 \sum_{r_{i} r_{j} \in E (G)} \frac{4 d_{g} (r_{i}) . d_{g} (r_{j})}{d_{g} (r_{i}) + d_{g} (r_{j})} \\ \sum_{r_{k} \in D (G) r_{k} \sim r_{i}, r_{k} \sim r_{j}} \frac{2 d_{g} (r_{k})}{(d_{g} (r_{k}) + d_{g} (r_{i})) (d_{g} (r_{k}) + d_{g} (r_{j}))} . \end{matrix}$ Hence $tr ({GA}^{3} (G)) = 2 \sum_{r_{i} r_{j} \in E (G)} \frac{4 d_{G} (r_{i}) . d_{G} (r_{j})}{d_{G} (r_{i}) + d_{G} (r_{j})} \sum_{r_{k} \in D (G) r_{k} \sim r_{i}, r_{k} \sim r_{j}} \frac{2 d_{G} (r_{k})}{(d_{G} (r_{k}) + d_{G} (r_{i})) (d_{G} (r_{k}) + d_{G} (r_{j}))} .$

Since $tr ({GA}^{4} (h_{E} (r_{i} r_{j})))$ = $∥ {GA}^{2} (h_{E} (r_{i} r_{j})) ∥_{F}^{2}$ , where $∥ {GA}^{2} (h_{E} (r_{i} r_{j})) ∥_{F}$ denotes the Frobenius norm of ${GA}^{2} (h_{E} (r_{i} r_{i}))$ , we obtain $\begin{matrix} tr ({GA}^{4} (h_{E} (r_{i} r_{j}))) \\ = \sum_{i, j = 1}^{n} | {GA}^{2} (h_{E} (r_{i} r_{j})) |^{2} \\ = \sum_{i = j} | {GA}^{2} (h_{E} (r_{i} r_{j})) |^{2} \\ + \sum_{i \neq j} | {GA}^{2} (h_{E} (r_{i} r_{j})) |^{2} \\ = 2 \sum_{r_{i} \in D (G)} {(\sum_{r_{j} \in D (G) r_{i} \sim r_{j}} \frac{4 d_{h} (r_{i}) . d_{h} (r_{j})}{(d_{h} (r_{i}) + d_{h} (r_{j}))^{2}})}^{2} \\ + \sum_{r_{i}, r_{j} \in D (G) r_{i} \neq r_{j}} 4 d_{h} (r_{i}) . d_{h} (r_{j}) \\ {(\sum_{r_{k} \in D (G) r_{k} \sim r_{i}, r_{k} \sim r_{j}} \frac{2 d_{h} (r_{k})}{(d_{h} (r_{k}) + d_{h} (r_{i})) (d_{h} (r_{k}) + d_{h} (r_{j}))})}^{2} . \end{matrix}$ Similarly, we can show that

$\begin{matrix} tr ({GA}^{4} (g_{E} (r_{i} r_{j}))) \\ = 2 \sum_{r_{i} \in D (G)} {(\sum_{r_{j} \in D (G) r_{i} \sim r_{j}} \frac{4 d_{g} (r_{i}) . d_{g} (r_{j})}{(d_{g} (r_{i}) + d_{g} (r_{j}))^{2}})}^{2} \\ + \sum_{r_{i}, r_{j} \in D (G) r_{i} \neq r_{j}} 4 d_{g} (r_{i}) . d_{g} (r_{j}) \\ {(\sum_{r_{k} \in D (G) r_{k} \sim r_{i}, r_{k} \sim r_{j}} \frac{2 d_{g} (r_{k})}{(d_{g} (r_{k}) + d_{g} (r_{i})) (d_{g} (r_{k}) + d_{g} (r_{j}))})}^{2} . \end{matrix}$ Hence

$\begin{matrix} tr ({GA}^{4} (G)) \\ = 2 \sum_{r_{i} \in D (G)} {(\sum_{r_{j} \in D (G) r_{i} \sim r_{j}} \frac{4 d_{G} (r_{i}) . d_{G} (r_{j})}{(d_{G} (r_{i}) + d_{G} (r_{j}))^{2}})}^{2} \\ + \sum_{r_{i}, r_{j} \in D (G) r_{i} \neq r_{j}} 4 d_{G} (r_{i}) . d_{G} (r_{j}) \\ {(\sum_{r_{k} \in D (G) r_{k} \sim r_{i}, r_{k} \sim r_{j}} \frac{2 d_{G} (r_{k})}{(d_{G} (r_{k}) + d_{G} (r_{i})) (d_{G} (r_{k}) + d_{G} (r_{j}))})}^{2} . \end{matrix}$

□

Theorem 3.1. Let $G = (D, E)$ be a DHq-ROFHG on n vertices. Then for any graph $G$ $GAE (G) \geq \frac{tr ({GA}^{2} (G))}{n (n - 1)} .$

Proof. According to Equation (2.2) to the convex function f (x) = x², we obtain ${(\frac{1}{n} \sum_{j = 1}^{n} | ϒ_{j} |^{\frac{1}{2}})}^{2} \leq \frac{1}{n} \sum_{j = 1}^{n} | ϒ_{j} |,$ ${GAE}_{h_{E}} (r_{i} r_{j}) \geq \frac{1}{n} {(\sum_{j = 1}^{n} | ϒ_{j} |^{\frac{1}{2}})}^{2} .$ Since |ϒ_j| ≤ ϒ₁ ≤ n - 1, so Equation (2.1) with p = 2 gives ${(\sum_{j = 1}^{n} | ϒ_{j} |^{\frac{1}{2}})}^{2} \geq (n - 1)^{- 1} \sum_{j = 1}^{n} ϒ_{j}^{2} = \frac{tr ({GA}_{h_{E}}^{2} (r_{i} r_{j}))}{n - 1},$

${GAE}_{h_{E}} (r_{i} r_{j}) \geq \frac{tr ({GA}_{h_{E}}^{2} (r_{i} r_{j}))}{n - 1} .$ Similarly, we can show that ${GAE}_{g_{E}} (r_{i} r_{j}) \geq \frac{tr ({GA}_{g_{E}}^{2} (r_{i} r_{j}))}{n - 1} .$ Hence $GAE (G) \geq \frac{tr ({GA}^{2} (G))}{n (n - 1)} .$ □

Theorem 3.2. Let $G = (D, E)$ be a DHq-ROFHG on n vertices. Then for any graph $G$ , $GAE (G) \geq \sqrt{tr ({GA}^{2} (G))} .$

Proof. Since the inequality $\sum_{j = 1}^{n} x_{j}^{2} \leq {(\sum_{j = 1}^{n} x_{j})}^{2}$ holds for every non-negative x₁, x₂, …, x_n, then $\begin{matrix} ({GAE}_{h_{E}} (r_{i} r_{j}))^{2} = {(\sum_{j = 1}^{n} | ϒ_{j} |)}^{2} \\ \geq \sum_{j = 1}^{n} | ϒ_{j} |^{2} = tr ({GA}_{h_{E}}^{2} (r_{i} r_{j}))) . \end{matrix}$ Therefore ${GAE}_{h_{E}} (r_{i} r_{j})^{2} \geq tr ({GA}_{h_{E}}^{2} (r_{i} r_{j})) .$ Analogously, we can show that ${GAE}_{g_{E}} (r_{i} r_{j})^{2} \geq tr ({GA}_{g_{E}}^{2} (r_{i} r_{j})) .$ Hence $GAE (G) \geq \sqrt{tr ({GA}^{2} (G))} .$ □

Theorem 3.3. Let $G = (D, E)$ be a DHq-ROFHG on n vertices. Then for any graph $G$ , $GAE (G) \geq \sqrt{tr ({GA}^{2} (G)) + n (n - 1) | \det (GA (G)) |^{\frac{2}{n}}} .$

Proof. We have $({GAE}_{h_{E}} (r_{i} r_{j}))^{2} = {(\sum_{j = 1}^{n} | ϒ_{j} |)}^{2} = \sum_{j = 1}^{n} ϒ_{j}^{2} + 2 \sum_{1 \leq i < j \leq n} | ϒ_{i} ϒ_{j} | = tr ({GA}_{h_{E}}^{2} (r_{i} r_{j})) + 2 \sum_{1 \leq i < j \leq n} | ϒ_{i} ϒ_{j} | .$ Since AM ≥ GM, we deduce $\frac{2}{n (n - 1)} \sum_{1 \leq i < j \leq n} | ϒ_{i} ϒ_{j} | \geq {(| ϒ_{1} |^{n - 1}, | ϒ_{2} |^{n - 1}, \dots, | ϒ_{n} |^{n - 1})}^{\frac{2}{n (n - 1)}} = | \det ({GA}_{h_{E}} (r_{i} r_{j})) |^{\frac{2}{n}},$ and

$\begin{matrix} ({GAE}_{h_{E}} (r_{i} r_{j}))^{2} \geq tr ({GA}_{h_{E}}^{2} (r_{i} r_{j})) \\ + 2 \sum_{1 \leq i < j \leq n} | ϒ_{i} ϒ_{j} | \geq tr ({GA}_{h_{E}}^{2} (r_{i} r_{j})) \\ + n (n - 1) | \det ({GA}_{h_{E}} (r_{i} r_{j}) |^{\frac{2}{n}} . \end{matrix}$

Therefore

$\begin{matrix} {GAE}_{h_{E}} (r_{i} r_{j}) \\ \geq \sqrt{tr ({GA}_{h_{E}}^{2} (r_{i} r_{j}) + n (n - 1) | \det ({GA}_{h_{E}} (r_{i} r_{j}) |^{\frac{2}{n}}} . \end{matrix}$

Similarly, we can show that $\begin{matrix} {GAE}_{g_{E}} (r_{i} r_{j}) \\ \geq \sqrt{tr ({GA}_{g_{E}}^{2} (r_{i} r_{j}) + n (n - 1) | \det ({GA}_{g_{E}} (r_{i} r_{j}) |^{\frac{2}{n}}} . \end{matrix}$

Hence $GAE (G) \geq \sqrt{tr ({GA}^{2} (G)) + n (n - 1) | \det (GA (G)) |^{\frac{2}{n}}} .$

□

3.2 Atom bond connectivity energy of a DHq-ROFHG

This subsection defines and investigates the atom bond connectivity energy of a DHq-ROFHG and provides some of its properties.

Definition 3.6. Let $G = (D, E)$ be a DHq-ROFHG on n vertices. The atom bond connectivity matrix, $ABC (G) = (ABC (h_{E} (r_{i} r_{j})), ABC (g_{E} (r_{i} r_{j}))) = [a_{ij}]$ of $G$ is a n × n matrix defined as:

$a_{ij} = {\begin{matrix} 0 & if i = j, \\ \sum_{r_{i} r_{j} \in E (G)} \sqrt{\frac{d_{G} (r_{i}) + d_{G} (r_{j}) - 2}{d_{G} (r_{i}) . d_{G} (r_{j})}} & if the vertices r_{i} and r_{j} of the DHq - ROFHG G are adjacent, \\ 0 & if the vertices r_{i} and r_{j} of the DHq - ROFHG G are non - adjacent . \end{matrix}$ It is worth noting that the sum of vertices’ degrees $d_{G} (r_{i})$ and $d_{G} (r_{j})$ must be greater than 2.

Definition 3.7. The atom bond connectivity energy of a DHq-ROFHG $G = (D, E)$ is defined as: $\begin{matrix} ABCE (G) = (ABCE (h_{E} (r_{i} r_{j})), ABCE (g_{E} (r_{i} r_{j})) \\ = (\sum_{j = 1 Γ_{j} \in Y_{ABC}}^{n} | Γ_{j} |, \sum_{j = 1 Ω_{j} \in Z_{ABC}}^{n} | Ω_{j} |) . \end{matrix}$ where Y_ABC and Z_ABC are the sets of atom bond connectivity eigenvalues of $ABC (h_{E} (r_{i} r_{j}))$ and $ABC (g_{E} (r_{i} r_{j}))$ , respectively.

Example 3.3. {Let $G = (D, E)$ be a DHq-ROFHG on V = {f₁, f₂, f₃, f₄, f₅, f₆} and E = {f₁f₂, f₂f₃, f₃f₄, f₄f₁, f₁f₅, f₅f₆, f₆f₃} as in Fig. 3, defined by:

Fig. 3

A dual hesitant 5-rung orthopair fuzzy Hamacher graph.

The adjacency matrix of a DH5-ROFHG given in Fig. 3 is:

Therefore, $Spec (ABC (S (A (G))))$ = {(-0.8814, -0.7674), (-0.6923, -0.6973), (-0.5797, -0.6938), (-0.4958, -0.6908), (-0.3829, -0.6897), (3.0321, 3.5390)}.

$ABCE (G) = (6.0642, 7.0781)$ .

Theorem 3.4. Let $G = (D, E)$ be a connected graph with m edges, maximum degree ▵_d = (▵ _h, ▵ _g), and minimum degree δ_d = (δ_h, δ_g). Then, $ABC (G) < \frac{m (▵_{d} + δ_{d})}{\sqrt{▵_{d} δ_{d}}} .$

Proof. We know that $\frac{▵_{h}}{δ_{h}} \leq \frac{d_{h} (r_{i})}{d_{h} (r_{j})} \leq \frac{δ_{h}}{▵_{h}}$ for all $r_{i}, r_{j} \in D (G)$ , so we have: $\begin{matrix} \sqrt{\frac{d_{h} (r_{i})}{d_{h} (r_{j})} + \frac{d_{h} (r_{j})}{d_{h} (r_{i})} - \frac{2}{d_{h} (r_{i}) . d_{h} (r_{j})}} \\ \leq \sqrt{\frac{d_{h} (r_{i})}{d_{h} (r_{j})} + \frac{d_{h} (r_{j})}{d_{h} (r_{i})}} \\ \leq \sqrt{\frac{d_{h}^{2} (r_{i}) + d_{h}^{2} (r_{j})}{d_{h} (r_{i}) . d_{h} (r_{j})} + 2} \end{matrix}$ and clearly, $\frac{d_{h}^{2} (r_{i}) + d_{h}^{2} (r_{j})}{d_{h} (r_{i}) . d_{h} (r_{j})} + 2 < {(\sqrt{\frac{δ_{h}}{▵_{h}}} + \sqrt{\frac{▵_{h}}{δ_{h}}})}^{2}$ so, $\sqrt{\frac{d_{h}^{2} (r_{i}) + d_{h}^{2} (r_{j})}{d_{h} (r_{i}) . d_{h} (r_{j})} + 2} < \sqrt{\frac{(δ_{h} + ▵_{h})^{2}}{▵_{h} δ_{h}}}$ Therefore, we have ${ABC}_{h_{E}} (r_{i} r_{j}) < m \frac{(▵_{h} + δ_{h})}{\sqrt{▵_{h} δ_{h}}} .$ Similarly, we can show that ${ABC}_{g_{E}} (r_{i} r_{j}) < m \frac{(▵_{g} + δ_{g})}{\sqrt{▵_{g} δ_{g}}} .$ Hence $ABC (G) < \frac{m (▵_{d} + δ_{d})}{\sqrt{▵_{d} δ_{d}}} .$

□

Definition 3.8. Let $G = (D, E)$ be a DHq-ROFHG on n vertices. The general Randić matrix, $R_{α} (G) = (R_{α} (h_{E} (r_{i} r_{j})), (R_{α} (g_{E} (r_{i} r_{j})))) = [r_{ij}],$ of $G$ is a n × n matrix defined as: $r_{ij} = {\begin{matrix} 0 & if i = j, \\ (d_{G} (r_{i}) d_{G} (r_{j}))^{α} & if the vertices r_{i} and r_{j} of the DHq - ROFHG G are adjacent, \\ 0 & if the vertices r_{i} and r_{j} of the DHq - ROFHG G are non - adjacent . \end{matrix}$ The matrix defined above is called the Randić matrix $R (G)$ if we take α = - 1/2.

Definition 3.9. The general Randić energy of a DHq-ROFHG $G = (D, E)$ is defined as: $\begin{matrix} {RE}_{α} (G) = ({RE}_{α} (h_{E} (r_{i} r_{j})), ({RE}_{α}) (g_{E} (r_{i} r_{j}))) \\ = (\sum_{j = 1 ϖ_{j} \in Y_{R}}^{n} | ϖ_{j} |, \sum_{j = 1 η_{j} \in Z_{R}}^{n} | η_{j} |) . \end{matrix}$ where $Y_{R}$ and $Z_{R}$ are the sets of general Randić eigenvalues of $R_{α} (h_{E} (r_{i} r_{j}))$ and $R_{α} (h_{E} (r_{i} r_{j}))$ , respectively.

Theorem 3.5. Let $G = (D, E)$ be a connected graph with m edges, maximum degree ▵_d = (▵ _h, ▵ _g), and minimum degree δ_d = (δ_h, δ_g). Then $R (G) \sqrt{2 δ_{d} - 2} \leq ABC (G) \leq R (G) \sqrt{2 ▵_{d} - 2} .$

Proof. We know that 2δ_h - 2 ≤ d_h (e_j) ≤ 2 ▵ _h - 2 for j = 1, 2, …, m . Thus we have

$\begin{matrix} \sqrt{2 δ_{h} - 2} \sum_{r_{i} r_{j} \in E (G)} \frac{1}{\sqrt{d_{h} (r_{i}) d_{h} (r_{j})}} \\ \leq {ABC}_{h_{E}} (r_{i} r_{j}) \leq \sqrt{2 ▵_{h} - 2} \\ \sum_{r_{i} r_{j} \in E (G)} \frac{1}{\sqrt{d_{h} (r_{i}) d_{h} (r_{j})}} \end{matrix}$ $\begin{matrix} R_{h_{E}} (r_{i} r_{j}) \sqrt{2 δ_{h} - 2} \\ \leq {ABC}_{h_{E}} (r_{i} r_{j}) \leq R_{h_{E}} (r_{i} r_{j}) \sqrt{2 ▵_{h} - 2} . \end{matrix}$ Similarly, we can show that $\begin{matrix} R_{g_{E}} (r_{i} r_{j}) \sqrt{2 δ_{g} - 2} \leq {ABC}_{g_{E}} (r_{i} r_{j}) \\ \leq R_{g_{E}} (r_{i} r_{j}) \sqrt{2 ▵_{g} - 2} . \end{matrix}$ Hence $R (G) \sqrt{2 δ_{d} - 2} \leq ABC (G) \leq R (G) \sqrt{2 ▵_{d} - 2} .$ □

4 A graph based MADM approach with dual hesitant q-rung orthopair fuzzy information

Let A = {A₁, A₂, …, A_n} be a discrete set of alternatives, Q = {Q₁, Q₂, …, Q_m} be the set of decision organizations, and ω = {ω₁, ω₂, …, ω_m} be the weight vector satisfying $\sum_{k = 1}^{m} ω_{k} = 1$ and ω_k ≥ 0, k = 1, 2, …, m . Each expert in an organization provides preference for each pair of alternatives, so for each organization its DHq-ROFPR is constructed with $H^{(k)} = (ς_{ij}^{(k)})_{n \times n} = (h_{ij}^{(k)}, g_{ij}^{(k)})_{n \times n}$ .

Step 1. For the i (i = 1, 2, …, n), we calculate the LCM of l (h_{ς
₁} (r_i)) , l (g_{ς
₁} (r_i)) , l (h_{ς
₂} (r_i)) , l (g_{ς
₂} (r_i)) , …, l (h_{ς
_n} (ri)) , l (g_{ς
_n} (r_i)), respectively, we denote it as $L (ς_{ji})_{j = 1}^{n}$ , then we construct the following multiple DHq-ROF matrix $D^{*} = (ς_{ji}^{*})_{n \times n}$ , which can be expressed as follows:

D^*= $[\begin{matrix} ς_{11}^{*} & ς_{12}^{*} & ς_{13}^{*} & \dots & ς_{1 n}^{*} \\ ς_{21}^{*} & ς_{22}^{*} & ς_{23}^{*} & \dots & ς_{2 n}^{*} \\ ς_{31}^{*} & ς_{32}^{*} & ς_{33}^{*} & \dots & ς_{3 n}^{*} \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ ς_{n 1}^{*} & ς_{n 2}^{*} & ς_{n 3}^{*} & \dots & ς_{nn}^{*} \end{matrix}],$

where $ς_{ji}^{*} (i, j = 1, 2, \dots, n)$ are described as:

$\begin{matrix} ς_{ji}^{*} = \\ 〈 {φ_{ς_{j}^{1}} (r_{i}), \dots, φ_{ς_{j}^{2}} (r_{i}), \dots, φ_{ς_{j}^{l (h_{ς_{ji}})}} (r_{i}), \dots}, \\ {{ϑ_{ς_{j}^{1}} (r_{i}), \dots, ϑ_{ς_{j}^{2}} (r_{i}), \dots, ϑ_{ς_{j}^{l (g_{ς_{ji}})}} (r_{i}), \dots} 〉}_{q}, \end{matrix}$

the numbers of $φ_{ς_{j}^{k_{1}}} (r_{i})$ and $ϑ_{ς_{j}^{k_{2}}} (r_{i}) (k_{1} = 1, 2, \dots, l (h_{ς_{j}}); k_{2} = 1, 2, \dots, l (g_{ς_{j}}))$ are determined by the values of $\frac{L (ς_{ji})_{j = 1}^{n}}{l (h_{ς_{ji}})}$ and $\frac{L (ς_{ji})_{j = 1}^{n}}{l (g_{ς_{ji}})}$ , respectively, l (h_{ς
_j}) and l (g_{ς
_j}) represent the numbers of elements in h_{ς
_ji} and g_{ς
_ji}, respectively.

Step 2. Draw DHq-ROFDGs $D_{i}$ corresponding to DHq-ROFPRs R_k (k = 1, 2, 3, 4) whose vertices and directed edges (arcs) represent the organizations considered and the preference of organizations, respectively.

Step 3. We utilize the dual hesitant q-rung orthopair fuzzy Hamacher weighting average (DHq-ROFHWA) operator [27] to aggregate all $ς_{ij}^{(k)} (j = 1, 2, \dots, n)$ that correspond to the alternative A_i. We get the DHq-ROFE $ς_{i}^{(k)}$ of the alternative A_i over all the other alternatives for the organization Q_k. $\begin{matrix} DHq - ROFHWA (ς_{1}, ς_{2}, \dots, ς_{n}) = \oplus_{i = 1}^{n} ω_{i} ς_{i} \\ = ⋃_{φ_{i} \in h_{i}, ϑ_{i} \in g_{i}} ({\sqrt[q]{\frac{\prod_{i = 1}^{n} (1 + (γ - 1) (φ_{i})^{q})^{ω_{i}} - \prod_{i = 1}^{n} (1 - (φ_{i})^{q})^{ω_{i}}}{\prod_{i = 1}^{n} (1 + (γ - 1) (φ_{i})^{q})^{ω_{i}} + (γ - 1) \prod_{i = 1}^{n} (1 - (φ_{i})^{q})^{ω_{i}}}}}, \\ {\frac{\sqrt[q]{γ} \prod_{i = 1}^{n} (ϑ_{i})^{ω_{i}}}{\sqrt[q]{\prod_{i = 1}^{n} (1 + (γ - 1) (1 - (ϑ_{i})^{q}))^{ω_{i}} + (γ - 1) \prod_{i = 1}^{n} (ϑ_{i})^{q ω_{i}}}}}) . \end{matrix}$ where ω_j = (ω₁, ω₂, …, ω_n) ^T, is the weighting vector, which satisfies ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1 .$

Step 4. Determine the weights of decision organizations as follows.

Compute the differences between any two organizations Q_l and Q_k utilizing DHq-ROFNE distance measure given by Definition (2.5), and obtain $\begin{matrix} D_{lk} = (d_{ij}^{(lk)})_{n \times n} \\ = {(d (ς_{ij}^{(l)}, ς_{ij}^{(k)}))}_{n \times n}, \\ l, k = 1, 2, \dots, m . \end{matrix}$ (4.1) where

(a) $d_{ij}^{(lk)} \geq 0$ , especially, if l = k, then $d_{ij}^{(lk)} = 0, i, j = 1, 2, \dots, n$ ;

(b) $d_{ij}^{(lk)} = 0, i = j$ ;

Determine the average value of the matrix $D_{lk}$ by ${\bar{d}}_{lk} = \frac{1}{n^{2}} \sum_{i = 1}^{n} \sum_{j = 1}^{n} d_{ij}^{(lk)}$ (4.2)

Afterwards, let ${\bar{d}}_{l} = \sum_{k = 1, k \neq l}^{m} {\bar{d}}_{lk}$ , then ${\bar{d}}_{l} = \frac{1}{n^{2}} \sum_{k = 1, k \neq l}^{m} \sum_{i = 1}^{n} \sum_{j = 1}^{n} d_{ij}^{(lk)},$ (4.3) which denotes the deviation of the organization Q_l from the rest organizations.

The weights ω_l of the organizations Q_l can be calculated as $w_{l} = \frac{({\bar{d}}_{l})^{- 1}}{\sum_{l = 1}^{m} ({\bar{d}}_{l})^{- 1}}, l = 1, 2, \dots, m$ (4.4)

Step 5. To aggregate all $ς_{i}^{(k)} (k = 1, 2, \dots, m)$ into a collective DHq-ROFE $ς_{i}^{(k)}$ for the alternative A_i, utilize the DHq-ROFHWA (or the DHq-ROFHWG) operator.

Step 6. Compute the score functions of ς_i (i = 1, 2, …, n) by Definition (2.2), and rank all the alternatives A_i (i = 1, 2, …, n) according to S (A_i) (i = 1, 2, …, n).

In the following, “selection of most significant defensive factor" is employed to demonstrate the validity of our approach:

5 Illustrated example and comparative analysis

Within this section, we apply the idea of DHq-ROFGs to a decision-making problem. A group decision-making problem relating to the “selection of most significant defensive factor" is solved to demonstrate the applicability of the proposed idea of DHq-ROFGs in reasonable situation dependent on DHq-ROFPRs.

5.1 Selection of most significant defensive factor

The fire system is extracted as a complex system by collocating and allocating various kinds of firearms in a timely way. The tank unit fire system plays a significant role in the execution of fire distribution in defensive fighting. In addition, the fire deployment would significantly affect the improvement of defensive stability, the combat strength of the enemies being annihilated, and the capacity to defend. The two companies that come from our tank unit are organizing a defensive battle in Xiaoshan area. Four critical factors influencing fire system proposals are given here: z₁: concealment by land-form use; z₂: reduction of enemy aircraft mobility; z₃: cooperation with mutual firepower; z₄: air-defense capability. Decision makers e_k (k = 1, 2, 3, 4) are invited to assess these z_i (i = 1, 2, 3, 4) factors and to provide their assessments with DHq-ROFEs, as shown below.

Step 1. Construct the DHq-ROFPRs $R_{k} = (ς_{ij}^{(k)})_{4 \times 4} (k = 1, 2, 3, 4)$ , as follows:

Step 2. Now, we calculate the LCM of the numbers of their corresponding DHq-ROFEs, the results are $L (ς_{j 1})_{j = 1}^{4} = L (ς_{j 2})_{j = 1}^{4} = L (ς_{j 3})_{j = 1}^{4} = L (ς_{j 4})_{j = 1}^{4} = 6$ . Then we construct the multiple DHq-ROFR matrix D^* = (ς_ji) _4×4, where $\begin{matrix} ς_{11}^{(1)} = 〈 {0.5, 0.5, 0.5, 0.5, 0.5, 0.5}, {0.5, 0.5, 0.5, 0.5, 0.5, 0.5} 〉_{q}; ς_{12}^{(1)} = 〈 {0.3, 0.3, 0.3, 0.5, 0.5, 0.5}, {0.4, 0.4, 0.4, 0.8, 0.8, 0.8} 〉_{q}; \\ ς_{13}^{(1)} = 〈 {0.1, 0.1, 0.2, 0.2, 0.5, 0.5}, {0.4, 0.4, 0.4, 0.6, 0.6, 0.6} 〉_{q}; ς_{14}^{(1)} = 〈 {0.4, 0.4, 0.4, 0.5, 0.5, 0.5}, {0.2, 0.2, 0.2, 0.3, 0.3, 0.3} 〉_{q}; \\ ς_{21}^{(1)} = 〈 {0.4, 0.4, 0.4, 0.8, 0.8, 0.8}, {0.3, 0.3, 0.3, 0.5, 0.5, 0.5} 〉_{q}; ς_{22}^{(1)} = 〈 {0.5, 0.5, 0.5, 0.5, 0.5, 0.5}, {0.5, 0.5, 0.5, 0.5, 0.5, 0.5} 〉_{q}; \\ ς_{23}^{(1)} = 〈 {0.3, 0.3, 0.3, 0.4, 0.4, 0.4}, {0.2, 0.2, 0.2, 0.4, 0.4, 0.4} 〉_{q}; ς_{24}^{(1)} = 〈 {0.4, 0.4, 0.5, 0.5, 0.6, 0.6}, {0.2, 0.2, 0.2, 0.3, 0.3, 0.3} 〉_{q}; \\ ς_{31}^{(1)} = 〈 {0.4, 0.4, 0.4, 0.6, 0.6, 0.6}, {0.1, 0.1, 0.2, 0.2, 0.5, 0.5} 〉_{q}; ς_{32}^{(1)} = 〈 {0.2, 0.2, 0.2, 0.4, 0.4, 0.4}, {0.3, 0.3, 0.3, 0.4, 0.4, 0.4} 〉_{q}; \\ ς_{33}^{(1)} = 〈 {0.5, 0.5, 0.5, 0.5, 0.5, 0.5}, {0.5, 0.5, 0.5, 0.5, 0.5, 0.5} 〉_{q}; ς_{34}^{(1)} = 〈 {0.3, 0.3, 0.3, 0.4, 0.4, 0.4}, {0.1, 0.1, 0.2, 0.2, 0.6, 0.6} 〉_{q}; \end{matrix}$ $\begin{matrix} ς_{41}^{(1)} = 〈 {0.2, 0.2, 0.2, 0.3, 0.3, 0.3}, {0.4, 0.4, 0.4, 0.5, 0.5, 0.5} 〉_{q}; ς_{42}^{(1)} = 〈 {0.2, 0.2, 0.2, 0.3, 0.3, 0.3}, {0.4, 0.4, 0.5, 0.5, 0.6, 0.6} 〉_{q}; \\ ς_{43}^{(1)} = 〈 {0.1, 0.1, 0.2, 0.2, 0.6, 0.6}, {0.3, 0.3, 0.3, 0.4, 0.4, 0.4} 〉_{q}; ς_{44}^{(1)} = 〈 {0.5, 0.5, 0.5, 0.5, 0.5, 0.5}, {0.5, 0.5, 0.5, 0.5, 0.5, 0.5} 〉_{q}; \\ ς_{11}^{(2)} = 〈 {0.5, 0.5, 0.5, 0.5, 0.5, 0.5}, {0.5, 0.5, 0.5, 0.5, 0.5, 0.5} 〉_{q}; ς_{12}^{(2)} = 〈 {0.3, 0.3, 0.5, 0.5, 0.7, 0.7}, {0.1, 0.1, 0.1, 0.2, 0.2, 0.2} 〉_{q}; \\ ς_{13}^{(2)} = 〈 {0.1, 0.1, 0.1, 0.4, 0.4, 0.4}, {0.7, 0.7, 0.7, 0.9, 0.9, 0.9} 〉_{q}; ς_{14}^{(2)} = 〈 {0.2, 0.2, 0.2, 0.3, 0.3, 0.3}, {0.2, 0.2, 0.4, 0.4, 0.5, 0.5} 〉_{q}; \\ ς_{21}^{(2)} = 〈 {0.1, 0.1, 0.1, 0.2, 0.2, 0.2}, {0.3, 0.3, 0.5, 0.5, 0.7, 0.7} 〉_{q}; ς_{22}^{(2)} = 〈 {0.5, 0.5, 0.5, 0.5, 0.5, 0.5}, {0.5, 0.5, 0.5, 0.5, 0.5, 0.5} 〉_{q}; \\ ς_{23}^{(2)} = 〈 {0.3, 0.3, 0.3, 0.4, 0.4, 0.4}, {0.1, 0.1, 0.1, 0.6, 0.6, 0.6} 〉_{q}; ς_{24}^{(2)} = 〈 {0.1, 0.1, 0.3, 0.3, 0.5, 0.5}, {0.2, 0.2, 0.2, 0.4, 0.4, 0.4} 〉_{q}; \\ ς_{31}^{(2)} = 〈 {0.7, 0.7, 0.7, 0.9, 0.9, 0.9}, {0.1, 0.1, 0.1, 0.4, 0.4, 0.4} 〉_{q}; ς_{32}^{(2)} = 〈 {0.1, 0.1, 0.1, 0.6, 0.6, 0.6}, {0.3, 0.3, 0.3, 0.4, 0.4, 0.4} 〉_{q}; \\ ς_{33}^{(2)} = 〈 {0.5, 0.5, 0.5, 0.5, 0.5, 0.5}, {0.5, 0.5, 0.5, 0.5, 0.5, 0.5} 〉_{q}; ς_{34}^{(2)} = 〈 {0.5, 0.5, 0.5, 0.6, 0.6, 0.6}, {0.1, 0.1, 0.1, 0.3, 0.3, 0.3} 〉_{q}; \\ ς_{41}^{(2)} = 〈 {0.2, 0.2, 0.4, 0.4, 0.5, 0.5}, {0.2, 0.2, 0.2, 0.3, 0.3, 0.3} 〉_{q}; ς_{42}^{(2)} = 〈 {0.2, 0.2, 0.2, 0.4, 0.4, 0.4}, {0.1, 0.1, 0.3, 0.3, 0.5, 0.5} 〉_{q}; \\ ς_{43}^{(2)} = 〈 {0.1, 0.1, 0.1, 0.3, 0.3, 0.3}, {0.5, 0.5, 0.5, 0.6, 0.6, 0.6} 〉_{q}; ς_{44}^{(2)} = 〈 {0.5, 0.5, 0.5, 0.5, 0.5, 0.5}, {0.5, 0.5, 0.5, 0.5, 0.5, 0.5} 〉_{q}; \end{matrix}$ $\begin{matrix} ς_{11}^{(3)} = 〈 {0.5, 0.5, 0.5, 0.5, 0.5, 0.5}, {0.5, 0.5, 0.5, 0.5, 0.5, 0.5} 〉_{q}; ς_{12}^{(3)} = 〈 {0.3, 0.3, 0.3, 0.4, 0.4, 0.4}, {0.2, 0.2, 0.2, 0.3, 0.3, 0.3} 〉_{q}; \\ ς_{13}^{(3)} = 〈 {0.2, 0.2, 0.2, 0.5, 0.5, 0.5}, {0.1, 0.1, 0.3, 0.3, 0.4, 0.4} 〉_{q}; ς_{14}^{(3)} = 〈 {0.4, 0.4, 0.4, 0.6, 0.6, 0.6}, {0.2, 0.2, 0.3, 0.3, 0.4, 0.4} 〉_{q}; \\ ς_{21}^{(3)} = 〈 {0.2, 0.2, 0.2, 0.3, 0.3, 0.3}, {0.3, 0.3, 0.3, 0.4, 0.4, 0.4} 〉_{q}; ς_{22}^{(3)} = 〈 {0.5, 0.5, 0.5, 0.5, 0.5, 0.5}, {0.5, 0.5, 0.5, 0.5, 0.5, 0.5} 〉_{q}; \\ ς_{23}^{(3)} = 〈 {0.4, 0.4, 0.4, 0.5, 0.5, 0.5}, {0.2, 0.2, 0.2, 0.3, 0.3, 0.3} 〉_{q}; ς_{24}^{(3)} = 〈 {0.5, 0.5, 0.5, 0.6, 0.6, 0.6}, {0.2, 0.2, 0.6, 0.6, 0.7, 0.7} 〉_{q}; \\ ς_{31}^{(3)} = 〈 {0.1, 0.1, 0.3, 0.3, 0.4, 0.4}, {0.2, 0.2, 0.2, 0.5, 0.5, 0.5} 〉_{q}; ς_{32}^{(3)} = 〈 {0.2, 0.2, 0.2, 0.3, 0.3, 0.3}, {0.4, 0.4, 0.4, 0.5, 0.5, 0.5} 〉_{q}; \\ ς_{33}^{(3)} = 〈 {0.5, 0.5, 0.5, 0.5, 0.5, 0.5}, {0.5, 0.5, 0.5, 0.5, 0.5, 0.5} 〉_{q}; ς_{34}^{(3)} = 〈 {0.2, 0.2, 0.2, 0.3, 0.3, 0.3}, {0.5, 0.5, 0.5, 0.7, 0.7, 0.7} 〉_{q}; \\ ς_{41}^{(3)} = 〈 {0.2, 0.2, 0.3, 0.3, 0.4, 0.4}, {0.4, 0.4, 0.4, 0.6, 0.6, 0.6} 〉_{q}; ς_{42}^{(3)} = 〈 {0.2, 0.2, 0.6, 0.6, 0.7, 0.7}, {0.5, 0.5, 0.5, 0.6, 0.6, 0.6} 〉_{q}; \\ ς_{43}^{(3)} = 〈 {0.5, 0.5, 0.5, 0.7, 0.7, 0.7}, {0.2, 0.2, 0.2, 0.3, 0.3, 0.3} 〉_{q}; ς_{44}^{(3)} = 〈 {0.5, 0.5, 0.5, 0.5, 0.5, 0.5}, {0.5, 0.5, 0.5, 0.5, 0.5, 0.5} 〉_{q}; \\ ς_{11}^{(4)} = 〈 {0.5, 0.5, 0.5, 0.5, 0.5, 0.5}, {0.5, 0.5, 0.5, 0.5, 0.5, 0.5} 〉_{q}; ς_{12}^{(4)} = 〈 {0.3, 0.3, 0.3, 0.5, 0.5, 0.5}, {0.4, 0.4, 0.4, 0.8, 0.8, 0.8} 〉_{q}; \\ ς_{13}^{(4)} = 〈 {0.2, 0.2, 0.2, 0.5, 0.5, 0.5}, {0.1, 0.1, 0.3, 0.3, 0.4, 0.4} 〉_{q}; ς_{14}^{(4)} = 〈 {0.1, 0.1, 0.2, 0.2, 0.3, 0.3}, {0.4, 0.4, 0.4, 0.5, 0.5, 0.5} 〉_{q}; \\ ς_{21}^{(4)} = 〈 {0.4, 0.4, 0.4, 0.8, 0.8, 0.8}, {0.3, 0.3, 0.3, 0.5, 0.5, 0.5} 〉_{q}; ς_{22}^{(4)} = 〈 {0.5, 0.5, 0.5, 0.5, 0.5, 0.5}, {0.5, 0.5, 0.5, 0.5, 0.5, 0.5} 〉_{q}; \\ ς_{23}^{(4)} = 〈 {0.3, 0.3, 0.3, 0.5, 0.5, 0.5}, {0.4, 0.4, 0.4, 0.6, 0.6, 0.6} 〉_{q}; ς_{24}^{(4)} = 〈 {0.1, 0.1, 0.2, 0.2, 0.3, 0.3}, {0.5, 0.5, 0.5, 0.6, 0.6, 0.6} 〉_{q}; \\ ς_{31}^{(4)} = 〈 {0.1, 0.1, 0.3, 0.3, 0.4, 0.4}, {0.2, 0.2, 0.2, 0.5, 0.5, 0.5} 〉_{q}; ς_{32}^{(4)} = 〈 {0.4, 0.4, 0.4, 0.6, 0.6, 0.6}, {0.3, 0.3, 0.3, 0.5, 0.5, 0.5} 〉_{q}; \\ ς_{33}^{(4)} = 〈 {0.5, 0.5, 0.5, 0.5, 0.5, 0.5}, {0.5, 0.5, 0.5, 0.5, 0.5, 0.5} 〉_{q}; ς_{34}^{(4)} = 〈 {0.1, 0.1, 0.1, 0.2, 0.2, 0.2}, {0.4, 0.4, 0.4, 0.6, 0.6, 0.6} 〉_{q}; \\ ς_{41}^{(4)} = 〈 {0.4, 0.4, 0.4, 0.5, 0.5, 0.5}, {0.1, 0.1, 0.2, 0.2, 0.3, 0.3} 〉_{q}; ς_{42}^{(4)} = 〈 {0.5, 0.5, 0.5, 0.6, 0.6, 0.6}, {0.1, 0.1, 0.2, 0.2, 0.3, 0.3} 〉_{q}; \\ ς_{43}^{(4)} = 〈 {0.4, 0.4, 0.4, 0.6, 0.6, 0.6}, {0.1, 0.1, 0.1, 0.2, 0.2, 0.2} 〉_{q}; ς_{44}^{(4)} = 〈 {0.5, 0.5, 0.5, 0.5, 0.5, 0.5}, {0.5, 0.5, 0.5, 0.5, 0.5, 0.5} 〉_{q} . \end{matrix}$

Step 3. The DHq-ROFDGs $D_{i}$ corresponding to DHq-ROFPRs $R_{k}$ (k = 1, 2, 3, 4) given in Tables 2 –5, are shown in Figure 4.

Table 2
DHq-ROFPR of the first decision maker

R ₁ z ₁ z ₂ z ₃ z ₄

z ₁ 〈{0.5} , {0.5} 〉_q 〈{0.3, 0.5} , {0.4, 0.8} 〉_q 〈{0.1, 0.2, 0.5} , {0.4, 0.6} 〉_q 〈{0.4, 0.5} , {0.2, 0.3} 〉_q

z ₂ 〈{0.4, 0.8} , {0.3, 0.5} 〉_q 〈{0.5} , {0.5} 〉_q 〈{0.3, 0.4} , {0.2, 0.4} 〉_q 〈{0.4, 0.5, 0.6} , {0.2, 0.3} 〉_q

z ₃ 〈{0.4, 0.6} , {0.1, 0.2, 0.5} 〉_q 〈{0.2, 0.4} , {0.3, 0.4} 〉_q 〈{0.5} , {0.5} 〉_q 〈{0.3, 0.4} , {0.1, 0.2, 0.6} 〉_q

z ₄ 〈{0.2, 0.3} , {0.4, 0.5} 〉_q 〈{0.2, 0.3} , {0.4, 0.5, 0.6} 〉_q 〈{0.1, 0.2, 0.6} , {0.3, 0.4} 〉_q 〈{0.5} , {0.5} 〉_q

R ₁	z ₁	z ₂	z ₃	z ₄
z ₁	〈{0.5} , {0.5} 〉_q	〈{0.3, 0.5} , {0.4, 0.8} 〉_q	〈{0.1, 0.2, 0.5} , {0.4, 0.6} 〉_q	〈{0.4, 0.5} , {0.2, 0.3} 〉_q
z ₂	〈{0.4, 0.8} , {0.3, 0.5} 〉_q	〈{0.5} , {0.5} 〉_q	〈{0.3, 0.4} , {0.2, 0.4} 〉_q	〈{0.4, 0.5, 0.6} , {0.2, 0.3} 〉_q
z ₃	〈{0.4, 0.6} , {0.1, 0.2, 0.5} 〉_q	〈{0.2, 0.4} , {0.3, 0.4} 〉_q	〈{0.5} , {0.5} 〉_q	〈{0.3, 0.4} , {0.1, 0.2, 0.6} 〉_q
z ₄	〈{0.2, 0.3} , {0.4, 0.5} 〉_q	〈{0.2, 0.3} , {0.4, 0.5, 0.6} 〉_q	〈{0.1, 0.2, 0.6} , {0.3, 0.4} 〉_q	〈{0.5} , {0.5} 〉_q

Table 3

DHq-ROFPR of the second decision maker

R ₂	z ₁	z ₂	z ₃	z ₄
z ₁	〈{0.5} , {0.5} 〉_q	〈{0.3, 0.5, 0.7} , {0.1, 0.2} 〉_q	〈{0.1, 0.4} , {0.7, 0.9} 〉_q	〈{0.2, 0.3} , {0.2, 0.4, 0.5} 〉_q
z ₂	〈{0.1, 0.2} , {0.3, 0.5, 0.7} 〉_q	〈{0.5} , {0.5} 〉_q	〈{0.3, 0.4} , {0.1, 0.6} 〉_q	〈{0.1, 0.3, 0.5} , {0.2, 0.4} 〉_q
z ₃	〈{0.7, 0.9} , {0.1, 0.4} 〉_q	〈{0.1, 0.6} , {0.3, 0.4} 〉_q	〈{0.5} , {0.5} 〉_q	〈{0.5, 0.6} , {0.1, 0.3} 〉_q
z ₄	〈{0.2, 0.4, 0.5} , {0.2, 0.3} 〉_q	〈{0.2, 0.4} , {0.1, 0.3, 0.5} 〉_q	〈{0.1, 0.3} , {0.5, 0.6} 〉_q	〈{0.5} , {0.5} 〉_q}

Table 4

DHq-ROFPR of the third decision maker

R ₃	z ₁	z ₂	z ₃	z ₄
z ₁	〈{0.5} , {0.5} 〉_q	〈{0.3, 0.4} , {0.2, 0.3} 〉_q	〈{0.2, 0.5} , {0.1, 0.3, 0.4} 〉_q	〈{0.4, 0.6} , {0.2, 0.3, 0.4} 〉_q
z ₂	〈{0.2, 0.3} , {0.3, 0.4} 〉_q	〈{0.5} , {0.5} 〉_q	〈{0.4, 0.5} , {0.2, 0.3} 〉_q	〈{0.5, 0.6} , {0.2, 0.6, 0.7} 〉_q
z ₃	〈{0.1, 0.3, 0.4} , {0.2, 0.5} 〉_q	〈{0.2, 0.3} , {0.4, 0.5} 〉_q	〈{0.5} , {0.5} 〉_q	〈{0.2, 0.3} , {0.5, 0.7} 〉_q
z ₄	〈{0.2, 0.3, 0.4} , {0.4, 0.6} 〉_q	〈{0.2, 0.6, 0.7} , {0.5, 0.6} 〉_q	〈{0.5, 0.7} , {0.2, 0.3} 〉_q	〈{0.5} , {0.5} 〉_q

Table 5

DHq-ROFPR of the fourth decision maker

R ₄	z ₁	z ₂	z ₃	z ₄
z ₁	〈{0.5} , {0.5} 〉_q	〈{0.3, 0.5} , {0.4, 0.8} 〉_q	〈{0.2, 0.5} , {0.1, 0.3, 0.4} 〉_q	〈{0.1, 0.2, 0.3} , {0.4, 0.5} 〉_q
z ₂	〈{0.4, 0.8} , {0.3, 0.5} 〉_q	〈{0.5} , {0.5} 〉_q	〈{0.3, 0.5} , {0.4, 0.6} 〉_q	〈{0.1, 0.2, 0.3} , {0.5, 0.6} 〉_q
z ₃	〈{0.1, 0.3, 0.4} , {0.2, 0.5} 〉_q	〈{0.4, 0.6} , {0.3, 0.5} 〉_q	〈{0.5} , {0.5} 〉_q	〈{0.1, 0.2} , {0.4, 0.6} 〉_q
z ₄	〈{0.4, 0.5} , {0.1, 0.2, 0.3} 〉_q	〈{0.5, 0.6} , {0.1, 0.2, 0.3} 〉_q	〈{0.4, 0.6} , {0.1, 0.2} 〉_q	〈{0.5} , {0.5} 〉_q

Fig. 4

Dual hesitant 4-rung orthopair fuzzy digraphs.

Step 4. Compute the averaged DHq-ROFHE $ς_{i}^{(k)}$ of the influencing factor z_i over all the other influencing factors for the experts e_k (k = 1, 2, 3, 4) by the DHq-ROFHWA operator with γ = 3, q = 4 and assume the weight vector ω_i = (ω₁, ω₂, . . . , ω_n)^T = ( $\frac{1}{n}$ , $\frac{1}{n}$ , . . . , $\frac{1}{n}$ )T = (0.25, 0.25, 0.25, 0.25), the aggregation results of the experts e_k (k = 1, 2, 3, 4) are as follows:

[e₁ :] $ς_{1}^{(1)} = 〈 {0.3929, 0.3929, 0.3945, 0.4656, 0.5000, 0.5000}, {0.3564, 0.3564, 0.3564, 0.5261, 0.5261, 0.5261} 〉$ ; $ς_{2}^{(1)} = 〈 {0.4172, 0.4172, 0.4457, 0.6092, 0.6274, 0.6274}, {0.2789, 0.2789, 0.2789, 0.4169, 0.4169, 0.4169} 〉$ ; $ς_{3}^{(1)} = 〈 {0.3945, 0.3945, 0.3945, 0.4954, 0.4954, 0.4954}, {0.1973, 0.1973, 0.2789, 0.2998, 0.4958, 0.4958} 〉$ ; $ς_{4}^{(1)} = 〈 {0.3565, 0.3565, 0.3585, 0.3752, 0.4756, 0.4756}, {0.3940, 0.3940, 0.4169, 0.4731, 0.4958, 0.4958} 〉$ ;

[e₂ :] $ς_{1}^{(2)} = 〈 {0.3652, 0.3652, 0.4205, 0.4457, 0.5350, 0.5350}, {0.2929, 0.2929, 0.3482, 0.4501, 0.4759, 0.4759} 〉$ ; $ς_{2}^{(2)} = 〈 {0.3632, 0.3632, 0.3734, 0.3945, 0.4408, 0.4408}, {0.2346, 0.2346, 0.2671, 0.4958, 0.5414, 0.5414} 〉$ ; $ς_{3}^{(2)} = 〈 {0.5467, 0.5467, 0.5467, 0.7107, 0.7107, 0.7107}, {0.1973, 0.1973, 0.1973, 0.3940, 0.3940, 0.3940} 〉$ ; $ς_{4}^{(2)} = 〈 {0.3565, 0.3565, 0.3859, 0.4172, 0.4457, 0.4457}, {0.2671, 0.2671, 0.3511, 0.4071, 0.4622, 0.4622} 〉$ ;

[e₃ :] $ς_{1}^{(3)} = 〈 {0.3945, 0.3945, 0.3945, 0.5136, 0.5136, 0.5136}, {0.2120, 0.2120, 0.3086, 0.3413, 0.3940, 0.3940} 〉$ ; $ς_{2}^{(3)} = 〈 {0.4408, 0.4408, 0.4408, 0.5050, 0.5050, 0.5050}, {0.2789, 0.2789, 0.3684, 0.4371, 0.4562, 0.4562} 〉$ ; $ς_{3}^{(3)} = 〈 {0.3565, 0.3565, 0.3671, 0.3828, 0.4010, 0.4010}, {0.3772, 0.3772, 0.3772, 0.5459, 0.5459, 0.5459} 〉$ ; $ς_{4}^{(3)} = 〈 {0.4218, 0.4218, 0.5050, 0.5736, 0.6118, 0.6118}, {0.3772, 0.3772, 0.3772, 0.4844, 0.4844, 0.4844} 〉$ ;

[e₄ :] $ς_{1}^{(4)} = 〈 {0.3652, 0.3652, 0.3671, 0.4656, 0.4697, 0.4697}, {0.3001, 0.3001, 0.3940, 0.5023, 0.5382, 0.5382} 〉$ ; $ς_{2}^{(4)} = 〈 {0.3929, 0.3929, 0.3945, 0.6019, 0.6039, 0.6039}, {0.4169, 0.4169, 0.4169, 0.5483, 0.5483, 0.5483} 〉$ ; $ς_{3}^{(4)} = 〈 {0.3842, 0.3842, 0.3929, 0.4717, 0.4821, 0.4821}, {0.3317, 0.3317, 0.3317, 0.5237, 0.5237, 0.5237} 〉$ ; $ς_{4}^{(4)} = 〈 {0.4578, 0.4578, 0.4578, 0.5562, 0.5562, 0.5562}, {0.1499, 0.1499, 0.2120, 0.2520, 0.3086, 0.3086} 〉$ .

Step 5.

To determine the weights of the experts, we first utilize Definition 2.5 of the DHq-ROFNE distance measure between two DHq-ROFEs, to compute $d (ς_{ij}^{(l)}, ς_{ij}^{(k)}), i, j = 1, 2, 3, 4; l, k = 1, 2, 3, 4$ and obtain the difference matrix $D_{lk} = (d_{ij}^{(lk)})_{n \times n}$ by utilizing Equation (4.1) as follows: $D_{12} = D_{21} = (\begin{matrix} 0 & 0.2175 & 0.2848 & 0.0387 \\ 0.2175 & 0 & 0.0520 & 0.0374 \\ 0.2848 & 0.0520 & 0 & 0.0768 \\ 0.0387 & 0.0374 & 0.0768 & 0 \end{matrix}),$ $D_{13} = D_{31} = (\begin{matrix} 0 & 0.2020 & 0.0592 & 0.0346 \\ 0.2020 & 0 & 0.3007 & 0.1105 \\ 0.0592 & 0.3007 & 0 & 0.0883 \\ 0.0346 & 0.1105 & 0.0883 & 0 \end{matrix}),$ $D_{14} = D_{41} = (\begin{matrix} 0 & 0 & 0.0592 & 0.0424 \\ 0 & 0 & 0.0566 & 0.0883 \\ 0.0592 & 0.0566 & 0 & 0.0412 \\ 0.0424 & 0.0883 & 0.0412 & 0 \end{matrix}),$ $D_{23} = D_{32} = (\begin{matrix} 0 & 0.0894 & 0.3401 & 0.0640 \\ 0.0894 & 0 & 0.0640 & 0.1131 \\ 0.3401 & 0.0640 & 0 & 0.1378 \\ 0.0640 & 0.1131 & 0.1378 & 0 \end{matrix}),$ $D_{24} = D_{42} = (\begin{matrix} 0 & 0.2175 & 0.3401 & 0.1977 \\ 0.2175 & 0 & 0.3035 & 0.0640 \\ 0.3401 & 0.3035 & 0 & 0.0943 \\ 0.1977 & 0.0640 & 0.0943 & 0 \end{matrix}),$ $D_{34} = D_{43} = (\begin{matrix} 0 & 0.2020 & 0 & 0.0678 \\ 0.2020 & 0 & 0.0624 & 0.0883 \\ 0 & 0.0624 & 0 & 0.0583 \\ 0.0678 & 0.0883 & 0.0583 & 0 \end{matrix}),$ $D_{11} = D_{22} = D_{33} = D_{44} = (\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) .$

Utilize Equation (4.2) to determine the average values of the difference matrix $\begin{matrix} {\bar{d}}_{12} = {\bar{d}}_{21} = \frac{1.4144}{16} = 0.0884, {\bar{d}}_{13} = {\bar{d}}_{31} = \frac{1.5905}{16} = 0.0994, {\bar{d}}_{14} = {\bar{d}}_{41} = \frac{0.5754}{16} = 0.0360, \\ {\bar{d}}_{23} = {\bar{d}}_{32} = \frac{1.6173}{16} = 0.1011, {\bar{d}}_{24} = {\bar{d}}_{42} = \frac{2.4344}{16} = 0.1522, {\bar{d}}_{34} = {\bar{d}}_{43} = \frac{0.9578}{16} = 0.0599 . \end{matrix}$

Using Equation (4.3), we find the deviation of the expert e₁ from the remaining experts as follows: $\begin{matrix} {\bar{d}}_{1} = 0.2238, {\bar{d}}_{2} = 0.3416, {\bar{d}}_{3} = 0.2603, \\ {\bar{d}}_{4} = 0.2480 . \end{matrix}$

Utilizing Equation (4.4), we compute the weights of the experts. $\begin{matrix} w_{1} = 0.2084, w_{2} = 0.3182, w_{3} = 0.2425, \\ w_{4} = 0.2310 . \end{matrix}$

Step 6. Determine a collective DHq-ROFE ς_i (i = 1, 2, …, n) of the influencing factor z_i over all the other influencing factors utilizing DHq-ROFHWA operator. That is, $\begin{matrix} ς_{1} = 〈 {0.3789, 0.3789, 0.3979, 0.4729, 0.5091, 0.5091}, {0.2837, 0.2837, 0.3496, 0.4465, 0.4780, 0.4780} 〉; \\ ς_{2} = 〈 {0.4034, 0.4034, 0.4131, 0.5347, 0.5479, 0.5479}, {0.2898, 0.2898, 0.3231, 0.4751, 0.4937, 0.4937} 〉; \\ ς_{3} = 〈 {0.4528, 0.4528, 0.4554, 0.5736, 0.5765, 0.5765}, {0.2604, 0.2604, 0.2799, 0.4312, 0.4783, 0.4783} 〉; \end{matrix}$

Step 7. Utilizing Definition 2.2, we calculate the score function of ς_i (i = 1, 2, 3, 4), and rank all the influencing factors z_i (i = 1, 2, 3, 4) according to the values of S (ς_i) (i = 1, 2, 3, 4). $\begin{matrix} S (ς_{1}) = 0.5065, S (ς_{2}) = 0.5124, \\ S (ς_{3}) = 0.5251, S (ς_{4}) = 0.5159 . \end{matrix}$ Then z₃ ≻ z₄ ≻ z₂ ≻ z₁. Thus, the optimal choice is z₃ .

5.2 Influence of the parameter on the ranking results

To explain the effects on the ranking results by adjusting parameters of γ in the DHq-ROFHWA (DHq-ROFHWG) operators, all results are described in Tables 6 and 7.

Table 6
Ranking results by adjusting γ in the DHq-ROFHWA operator with q = 4

γ S (d₁) S (d₂) S (d₃) S (d₄) Ranking orders

γ = 1 0.5081 0.5157 0.5324 0.5173 z₃ ≻ z₄ ≻ z₂ ≻ z₁

γ = 2 0.5071 0.5138 0.5278 0.5165 z₃ ≻ z₄ ≻ z₂ ≻ z₁

γ = 3 0.5065 0.5124 0.5251 0.5159 z₃ ≻ z₄ ≻ z₂ ≻ z₁

γ = 4 0.5061 0.5115 0.5233 0.5154 z₃ ≻ z₄ ≻ z₂ ≻ z₁

γ = 5 0.5058 0.5107 0.5219 0.5150 z₃ ≻ z₄ ≻ z₂ ≻ z₁

γ = 6 0.5055 0.5101 0.5207 0.5146 z₃ ≻ z₄ ≻ z₂ ≻ z₁

γ	S (d₁)	S (d₂)	S (d₃)	S (d₄)	Ranking orders
γ = 1	0.5081	0.5157	0.5324	0.5173	z₃ ≻ z₄ ≻ z₂ ≻ z₁
γ = 2	0.5071	0.5138	0.5278	0.5165	z₃ ≻ z₄ ≻ z₂ ≻ z₁
γ = 3	0.5065	0.5124	0.5251	0.5159	z₃ ≻ z₄ ≻ z₂ ≻ z₁
γ = 4	0.5061	0.5115	0.5233	0.5154	z₃ ≻ z₄ ≻ z₂ ≻ z₁
γ = 5	0.5058	0.5107	0.5219	0.5150	z₃ ≻ z₄ ≻ z₂ ≻ z₁
γ = 6	0.5055	0.5101	0.5207	0.5146	z₃ ≻ z₄ ≻ z₂ ≻ z₁

Table 7

Ranking results by adjusting γ in the DHq-ROFHWG operator with q = 4

γ	S (d₁)	S (d₂)	S (d₃)	S (d₄)	Ranking orders
γ = 1	0.4603	0.4845	0.4902	0.4909	z₄ ≻ z₃ ≻ z₂ ≻ z₁
γ = 2	0.4658	0.4854	0.4911	0.4915	z₄ ≻ z₃ ≻ z₂ ≻ z₁
γ = 3	0.4692	0.4860	0.4917	0.4919	z₄ ≻ z₃ ≻ z₂ ≻ z₁
γ = 4	0.4717	0.4865	0.4921	0.4922	z₄ ≻ z₃ ≻ z₂ ≻ z₁
γ = 5	0.4735	0.4869	0.4925	0.4924	z₃ ≻ z₄ ≻ z₂ ≻ z₁
γ = 6	0.4750	0.4872	0.4927	0.4926	z₃ ≻ z₄ ≻ z₂ ≻ z₁

Based on the calculated results mentioned in Tables 6 and 7, the rank of all alternatives is identical with distinct parameters γ in DHq-ROFHWA operator and rank of all alternatives (except γ = 5 and γ = 6) is same, with distinct parameters γ in DHq-ROFHWG operator. According to Fig. 5 and 6, when the parameter γ becomes larger, the fused results by DHq-ROFHWA and DHq-ROFHWG operators become smaller and larger, respectively, at the same time, the fused results become more and more steady.

Fig. 5

Scores of alternative z_i (i = 1, 2, 3, 4) when q = 4 and γ ∈ [1, 6] based on the DHq-ROFHWA operator.

Fig. 6

Scores of alternative z_i (i = 1, 2, 3, 4) when q = 4 and γ ∈ [1, 6] based on the DHq-ROFHWG operator.

5.3 Comparative analysis

In this subsection, to show the effectiveness and superiority of our developed approach. We utilize other existing techniques to handel with this example and compare the results with the developed approach.

Comparing with the DHq-ROF weighted averaging (DHq-ROFWA) operator (i.e., taking Algebraic t-norm and t-conorm) proposed by Wang et al. [27]: According to DHq-ROFWA operator, the ranking results are obtained as: z₃ ≻ z₄ ≻ z₂ ≻ z₁. Clearly, the approach in the literature has the same ranking result as our. However, our graph based approach adopted the Hamacher t-norm and t-conorm, and the literature utilized the Algebraic t-norm and t-conorm. As the Algebraic t-norm and t-conorm are just a special case of the Hamacher t-norm and t-conorm. So, the approach developed in this paper is more general than the approach proposed by [27].

Comparing with the DHq-ROF weighted geometric (DHq-ROFWG) operator proposed by Wang et al. [27]: According to DHq-ROFWG operator, the ranking results are obtained as: z₄ ≻ z₃ ≻ z₂ ≻ z₁.

Comparing with the DHq-ROF Einstein weighted (DHq-ROFEW) operator (i.e., taking Einstein t-norm and t-conorm) proposed by Wang et al. [27]: According to DHq-ROFEW operator, the ranking results are obtained as: z₃ ≻ z₄ ≻ z₂ ≻ z₁. Clearly, the approach in the literature has the same ranking result as our. However, our approach adopted the Hamacher t-norm and t-conorm, and the literature utilized the Einstein t-norm and t-conorm. As the Einstein t-norm and t-conorm are just a special case of the Hamacher t-norm and t-conorm. So, the approach developed in this paper is more general than the approach proposed in literature.

Comparing with the DHq-ROF Einstein weighted geometric (DHq-ROFEWG) operator proposed by Wang et al. [27]: According to DHq-ROFEWG operator, the ranking results are obtained as: z₄ ≻ z₃ ≻ z₂ ≻ z₁.

Comparing with the DHIF Hamacher weighted (DHIFHW) operator (i.e., takeing q=1): According to DHIFHW operator, the ranking results are obtained as: z₃ ≻ z₄ ≻ z₂ ≻ z₁. The DHIFHW operator just aggregates the DHIFNs, and the DHIFN must satisfy the conditional limit of 0 ≤ φ + ϑ ≤ 1, φ, ϑ ∈ [0, 1]. Therefore, the application range of the DHIFHW operator is limited. Obviously, most of the evaluation values do not satisfy the conditional limit of 0 ≤ φ + ϑ ≤ 1 in this example, so the DHIFHW operator is not suitable in this example.

Comparing with the DHPF Hamacher weighted (DHPFHW) operator (i.e., takeing q=2) in the literature proposed by Wang et al. [27]: According to DHPFHW operator, the ranking results are obtained as: z₃ ≻ z₄ ≻ z₂ ≻ z₁. The DHPFHW operator aggregates only the DHPFNs and the DHPFN must satisfy the conditional limit of 0 ≤ φ² + ϑ² ≤ 1, φ, ϑ ∈ [0, 1]. Therefore, the application range of the DHPFHW operator is wider than the DHIFHW operator but limited than the DHq-ROF weighted Hamacher operator in our proposed method. Most of the evaluation values in this decision-making problem meet the conditional limit of 0 ≤ φ² + ϑ² ≤ 1. So in this example, the DHPFHW operator presented by [27] cannot cope entirely with the decision-making problem.

From this analysis, it is easy to see that our developed method based on graph theory and DHq-ROFHWA operator is more versatile and reasonable to solve MADM problems.

The merits of our developed approach are summarized as follows:

(1) Apparently, proposed approach is clear and have less loss of data, and can be easily utilized to other MADM problems in DHq-ROF setting.

(2) One of the superiorities of the created approach is utilizing graph theory.

(3) The DHq-ROFSs of developed technique can accessibly depict more general decision-making circumstances.

(4) The Hamacher operator is more valid to deal MADM problems under DHq-ROF circumstances.

6 Conclusions

DHq-ROFS can flexibly denote decision-making information as well as effectively characterize the reliability of information. In this paper, the concept of DHq-ROFS has been applied to the theory of graphs and put forward the innovative concept of dual hesitant q-rung orthopair fuzzy graphs based on Hamacher operator called DHq-ROFHGs. The DHq-ROFHG is a good tool to express decision makers’ hesitancy in the process of MADM and can well describe the fuzziness of networks. Further, the theory of graph spectra has been discussed in the context of DHq-ROFSs and geometric-arithmetic energy, and atom bond connectivity energy of DHq-ROFHGs are determined. Moreover, the proposed concept of DHq-ROFHGs has been applied to solve the MADM problems with DHq-ROF information. Finally, we applied the newly proposed concept of DHq-ROFHGs to the selection of most significant defensive factor to demonstrate its validity and effectiveness. Comparative analysis has been conducted and the superiorities have been illustrated. In future, our research work will be extended to: (1) Linguistic interval-valued dual hesitant q-rung orthopair fuzzy graphs; (2) Linguistic complex hesitant fuzzy graphs; and (3) Linguistic complex dual hesitant q-rung orthopair fuzzy graphs.

Conflict of interest

The authors declare no conflicts of interest.

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