Some Properties of Double Domination in Vague Graphs with an Application

Abstract

This paper is devoted to the study of the double domination in vague graphs, and it is a contribution to the Special Issue “Advances in graph theory and Symmetry/Asymmetry” of Symmetry. Symmetry is one of the most important criteria that illustrate the structure and properties of fuzzy graphs. It has many applications in dominating sets and helps find a suitable place for construction. Vague graphs (VGs), which are a family of fuzzy graphs (FGs), are a well-organized and useful tool for capturing and resolving a range of real-world scenarios involving ambiguous data. In the graph theory, a dominating set (DS) for a graph $G^{*}=(X,E)$ is a subset $\mathfrak{D}$ of the vertices X so that every vertex which is not in $\mathfrak{D}$ is adjacent to at least one member of $\mathfrak{D}$. The subject of energy in graph theory is one of the most attractive topics serving a very important role in biological and chemical sciences. Hence, in this work, we express the notion of energy on a dominating vague graph (DVG) and also use the concept of energy in modeling problems related to DVGs. Moreover, we introduce a new notion of a double dominating vague graph (DDVG) and provide some examples to explain various concepts introduced. Finally, we present an application of energy on DVGs.

1. Introduction

Zadeh [1] introduced the subject of a fuzzy set (FS) in 1995. Rosenfeld [2] proposed the subject of FGs. The definition of FGs was presented by Kaufmann [3]. Akram et al. [4,5,6] explained several concepts in FGs. Gau and Buehrer [7] introduced the notion of a vague set (VS) in 1993. The concept of VGs was defined by Ramakrishna [8]. VGs belong to the FG family, such that have several applications in real-life systems. The system varies with time and has different accuracy levels. Rashmanlou et al. [9,10,11] investigated different subjects of VGs. Moreover, Akram et al. [12,13,14] developed several results on VGs. Kosari et al. [15] defined vague graph structure and studied its properties. Borzooei [16] proposed the degree of vertices in VGs. Haynes et al. [17] expressed the fundamentals of domination in graphs.

Symmetry is a kind of invariant or a feature that a mathematical object remains the same under some operations or transformations. However, symmetry is a significant feature in FG theory, especially in fuzzy DSs. Symmetric graphs have been the subject of much research. For instance, there are known and famous connections between symmetric configurations and regular bipartite graphs. The concept of DSs in VGs, both theoretically and practically, is very valuable. DSs in VGs are used for solving problems of different branches in applied sciences such as location problems. In this way, the study of new concepts such as the DS is essential in VGs. The domination in VGs has applications in several fields. The domination emerges in the facility location problems, where the number of facilities is fixed, and one endeavors to minimize the distance that a person needs to travel to get to the closest facility. Nagoor Gani and Prasanna Devi [18] suggested the reduction in the domination number of an FG and the notion of 2-domination in FGs [19] as the extension of 2-domination in crisp graphs. In another study, Somasundram [20] proposed the domination notion in FGs. The domination in product FGs and intuitionistic FGs were studied by Mahioub [21,22]. Karunambigai et al. and Rao et al. [23,24,25] expressed certain domination properties in vague incidence graphs. Kosari et al. [26,27,28] studied the domination of product VGs. The DS concept in FGs, both theoretically and practically, is very valuable. A DS in FGs is used for solving problems of different branches in applied sciences such as location problems. In this way, the study of new concepts, such as DSs, seems essential in FGs. The domination in VGs has numerous applications in several fields. Qiang et al. [29] introduced new domination concepts in VGs.

The DDS and double domination number (DDN) were first defined and introduced by Harary and Haynes in [30] as cited in [31]. Rashmanlou [32] expressed a new concept of Ring sum in product intuitionistic FGs. Gutman [33] introduced the graph energy concept. New results on energy are proposed in [34,35,36]. Anjali and Sunil Mathew [37] extended the energy of the graph to the energy of FGs. Gutman et al. [38] introduced that the Laplacian energy of a graph is the sum of the absolute deviations (i.e., the distance from the mean) of the eigenvalues of its Laplacian Matrix. Although VGs are better at expressing uncertain variables than FGs, they do not perform well in many real-world situations, such as IT management. Therefore, when the data come from several factors, it is necessary to use VGs. Zeng et al. presented new results in [39,40].

In this paper, we introduced a new notion of DSs in VGs. Finally, an application was proposed.

2. Preliminaries

In this section, we present some preliminary results which will be used throughout the paper. In Table 1, we show the essential notations.

Table 1. Some essential notations.

Notation	Meaning
FS	Fuzzy Set
FG	Fuzzy Graph
VS	Vague Set
VG	Vague Graph
DS	Dominating Set
DVG	Dominating Vague Graph
DDVG	Double Dominating Vague Graph
DDS	Double Dominating Set
AM	Adjacency Matrix
DN	Dominating Number
DDN	Double Dominating Number

Definition 1.

A graph $G^{*}$ is a pair $(X,E)$ where X is called the vertex set and $E\subseteq X\times X$ is called the edge set.

Definition 2.

A pair $\mathfrak{G}=(\zeta ,\eta )$ is an FG on a graph $G^{*}=(X,E)$ where ζ is an FS on X and η is an FS on E, such that

$$\eta (r,s)\le min\left\{\zeta \right(r),\zeta (s\left)\right\},$$

for all $rs\in E.$

Definition 3

([7]). A vague set (VS) $\mathfrak{M}$ is a pair $(t_{\mathfrak{M}},f_{\mathfrak{M}})$ on set X, where $t_{\mathfrak{M}}$ and $f_{\mathfrak{M}}$ are real-valued functions that can be defined on $X\to [0,1]$ so that, $t_{\mathfrak{M}}\left(r\right)+f_{\mathfrak{M}}\left(r\right)\le 1,\forall r\in X.$

Definition 4

([8]). A pair $\mathfrak{G}=(\mathfrak{M},\mathfrak{N})$ is called a VG on graph $G^{*}=(X,E)$, where $\mathfrak{M}=(t_{\mathfrak{M}},f_{\mathfrak{M}})$ is a VS on X and $\mathfrak{N}=(t_{\mathfrak{N}},f_{\mathfrak{N}})$ is a VS on E such that

$$t_{\mathfrak{N}}(r,s)\le min\{t_{\mathfrak{M}}\left(r\right),t_{\mathfrak{M}}\left(s\right)\},$$

$$f_{\mathfrak{N}}(r,s)\ge max\{f_{\mathfrak{M}}\left(r\right),f_{\mathfrak{M}}\left(s\right)\},$$

for all $r,s\in X$. $\mathfrak{M}$ is called VS on $\mathfrak{N}$. A VG $\mathfrak{G}$ is named strong if

$$t_{\mathfrak{N}}(r,s)=min\{t_{\mathfrak{M}}\left(r\right),t_{\mathfrak{M}}\left(s\right)\},$$

$$f_{\mathfrak{N}}(r,s)=max\{f_{\mathfrak{M}}\left(r\right),f_{\mathfrak{M}}\left(s\right)\},$$

for all $rs\in E.$

Definition 5.

A quadruple form $\mathfrak{G}=(t_{\mathfrak{N}},f_{\mathfrak{N}},t_{1_{\mathfrak{N}}},f_{1_{\mathfrak{N}}})$ is named DVG, where $t_{1_{\mathfrak{N}}}:X\to [0,1]$ denotes a degree of membership and $f_{1_{\mathfrak{N}}}:X\to [0,1]$ denotes a degree of non-membership, defined as

$$t_{1_{\mathfrak{N}}}\left(r\right)=\underset{s}{max}\left\{t_{\mathfrak{N}}(r,s)\right\},f_{1_{\mathfrak{N}}}\left(r\right)=\underset{s}{min}\left\{f_{\mathfrak{N}}(r,s)\right\},$$

for $r\in X.$

Note: The DS means a subset $\mathfrak{D}\subseteq X$ is named DS in $\mathfrak{G}$ if for each $s\in X-\mathfrak{D}$, there exists one vertex r in $\mathfrak{D}$ so that r dominates s, i.e.,

$$t_{\mathfrak{N}}(r,s)=t_{1_{\mathfrak{N}}}\left(r\right)\wedge t_{1_{\mathfrak{N}}}\left(s\right)$$

$$f_{\mathfrak{N}}(r,s)=f_{1_{\mathfrak{N}}}\left(r\right)\vee f_{1_{\mathfrak{N}}}\left(s\right)$$

Definition 6.

Suppose $\mathfrak{G}=(t_{\mathfrak{N}},f_{\mathfrak{N}},t_{1_{\mathfrak{N}}},f_{1_{\mathfrak{N}}})$ is a DVG. Suppose $r,s\in X$, we say that r dominates s in $\mathfrak{G}$ if there exists strong edge from r to s. A subset $\mathfrak{D}\subseteq X$ is named DS in $\mathfrak{G}$ if for each $s\in X-\mathfrak{D}$, there exists r in $\mathfrak{D}$ such that r dominates s.

A DS of $\mathfrak{D}$ is called to be a minimal DS if no proper subset of $\mathfrak{D}$ is a DS of $\mathfrak{G}$.

The minimum cardinality of a minimal DS in $\mathfrak{G}$ is named the domination number (DN) of $\mathfrak{G}$ and is denoted by $\alpha \left(\mathfrak{G}\right)$ and the corresponding minimal DS is named the minimum DS of $\mathfrak{G}$.

Definition 7.

Suppose $\mathfrak{G}=(t_{\mathfrak{N}},f_{\mathfrak{N}},t_{1_{\mathfrak{N}}},f_{1_{\mathfrak{N}}})$ is a DVG. The adjacency matrix (AM) of a DVG $\mathfrak{G}$ is defined as $A_{\mathfrak{D}}\left(\mathfrak{G}\right)=\left[d_{kl}\right]$, in which $d_{kl}=(t_{{\mathfrak{N}}_{kl}},f_{{\mathfrak{N}}_{kl}})$ where $t_{{\mathfrak{N}}_{kl}}=t_{\mathfrak{N}}(z_k,z_l)$ and $f_{{\mathfrak{N}}_{kl}}=f_{\mathfrak{N}}(z_k,z_l)$ represent the strength of relationship between $z_k$ and $z_l$, respectively.

$$d_{kl}=\left\{\begin{array}{c}(t_{{\mathfrak{N}}_{kl}},f_{{\mathfrak{N}}_{kl}}),(z_k,z_l)\in E,\hfill \\ (1,1),k=l and z_k\in \mathfrak{D},\hfill \\ 0,otherwise.\hfill \end{array}\right.$$

This AM of a DVG $\mathfrak{G}$ can be written in two different matrices as $A_{\mathfrak{D}}\left(\mathfrak{G}\right)=(t_{{\mathfrak{N}}_{\mathfrak{D}}}\left(\mathfrak{G}\right),f_{{\mathfrak{N}}_{\mathfrak{D}}}\left(\mathfrak{G}\right))$, where

$$t_{{\mathfrak{N}}_{\mathfrak{D}}}\left(\mathfrak{G}\right)=\left\{\begin{array}{c}{t}_{{\mathfrak{N}}_{kl}},(z_k,z_l)\in E,\hfill \\ 1,k=l and z_k\in \mathfrak{D},\hfill \\ 0,otherwise.\hfill \end{array}\right.$$

and

$$f_{{\mathfrak{N}}_{\mathfrak{D}}}\left(\mathfrak{G}\right)=\left\{\begin{array}{c}{f}_{{\mathfrak{N}}_{kl}},(z_k,z_l)\in E,\hfill \\ 1,k=l and z_k\in \mathfrak{D},\hfill \\ 0,otherwise.\hfill \end{array}\right.$$

Definition 8.

The spectrum of AM of a DVG $\mathfrak{G}$ is defined as $(W_{\mathfrak{D}},Z_{\mathfrak{D}})$, where $W_{\mathfrak{D}}$ and $Z_{\mathfrak{D}}$ are the sets of eigenvalues of $t_{{\mathfrak{N}}_{\mathfrak{D}}}\left(\mathfrak{G}\right)$ and $f_{{\mathfrak{N}}_{\mathfrak{D}}}\left(\mathfrak{G}\right)$, respectively.

Definition 9.

The energy of a DVG $\mathfrak{G}=(t_{\mathfrak{N}},f_{\mathfrak{N}},t_{1_{\mathfrak{N}}},f_{1_{\mathfrak{N}}})$ is defined as

$$E_{\mathfrak{D}}\left(\mathfrak{G}\right)=\left(E_{t_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right),E_{f_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right)\right)=\left(\sum _{k=1}^p|{\psi }_k|,\sum _{k=1}^p\left|{\pi }_k\right|\right)$$

where $W_{\mathfrak{D}}={\left\{{\psi }_k\right\}}_{k=1}^p$ and $Z_{\mathfrak{D}}={\left\{{\pi }_k\right\}}_{k=1}^p$.

Example 1.

Consider a DVG $\mathfrak{G}=(t_{\mathfrak{N}},f_{\mathfrak{N}},t_{1_{\mathfrak{N}}},f_{1_{\mathfrak{N}}})$ on $X=\{a_1,a_2,a_3,a_4,a_5,a_6\}$ and $t_{1_{\mathfrak{N}}},f_{1_{\mathfrak{N}}}$ are defined by $t_{1_{\mathfrak{N}}}:X\to [0,1]$ and $f_{1_{\mathfrak{N}}}:X\to [0,1]$, as shown in Figure 1.

$$t_{1_{\mathfrak{N}}}\left(a_1\right)=\underset{a_i}{max}\left\{t_{\mathfrak{N}}(a_1,a_i)\right\}$$

$$=max\{t_{\mathfrak{N}}(a_1,a_2),t_{\mathfrak{N}}(a_1,a_3)\}$$

$$=max\{0.1,0.2\}=0.2$$

Similarly,

$$t_{1_{\mathfrak{N}}}\left(a_2\right)=0.2,t_{1_{\mathfrak{N}}}\left(a_3\right)=0.2,t_{1_{\mathfrak{N}}}\left(a_4\right)=0.2,t_{1_{\mathfrak{N}}}\left(a_5\right)=0.2,t_{1_{\mathfrak{N}}}\left(a_6\right)=0.2.$$

Moreover,

$$f_{1_{\mathfrak{N}}}\left(a_1\right)=\underset{a_i}{min}\left\{f_{\mathfrak{N}}(a_1,a_i)\right\}$$

$$=min\{f_{\mathfrak{N}}(a_1,a_2),f_{\mathfrak{N}}(a_1,a_3)\}$$

$$=min\{0.5,0.6\}=0.5$$

Similarly,

$$f_{1_{\mathfrak{N}}}\left(a_2\right)=0.5,f_{1_{\mathfrak{N}}}\left(a_3\right)=0.5,f_{1_{\mathfrak{N}}}\left(a_4\right)=0.7,f_{1_{\mathfrak{N}}}\left(a_5\right)=0.6,f_{1_{\mathfrak{N}}}\left(a_6\right)=0.6.$$

Here, $a_3$ dominates $a_1,a_2$ and $a_5$ dominates $a_4,a_6$ because

$$t_{\mathfrak{N}}(a_3,a_2)=t_{1_{\mathfrak{N}}}\left(a_3\right)\wedge t_{1_{\mathfrak{N}}}\left(a_2\right),f_{\mathfrak{N}}(a_3,a_2)=f_{1_{\mathfrak{N}}}\left(a_3\right)\vee f_{1_{\mathfrak{N}}}\left(a_2\right)$$

$$t_{\mathfrak{N}}(a_3,a_1)=t_{1_{\mathfrak{N}}}\left(a_3\right)\wedge t_{1_{\mathfrak{N}}}\left(a_1\right),f_{\mathfrak{N}}(a_3,a_1)=f_{1_{\mathfrak{N}}}\left(a_3\right)\vee f_{1_{\mathfrak{N}}}\left(a_1\right)$$

$$t_{\mathfrak{N}}(a_5,a_6)=t_{1_{\mathfrak{N}}}\left(a_5\right)\wedge t_{1_{\mathfrak{N}}}\left(a_6\right),f_{\mathfrak{N}}(a_5,a_6)=f_{1_{\mathfrak{N}}}\left(a_5\right)\vee f_{1_{\mathfrak{N}}}\left(a_6\right)$$

$$t_{\mathfrak{N}}(a_5,a_4)=t_{1_{\mathfrak{N}}}\left(a_5\right)\wedge t_{1_{\mathfrak{N}}}\left(a_4\right),f_{\mathfrak{N}}(a_5,a_4)=f_{1_{\mathfrak{N}}}\left(a_5\right)\vee f_{1_{\mathfrak{N}}}\left(a_4\right)$$

Therefore, $\mathfrak{D}=\{a_3,a_5\}$ is a DS because every vertex in $X-\mathfrak{D}$ is dominated by at least one vertex in $\mathfrak{D}.$ The AM of DVG $\mathfrak{G}$ is given below

Figure 1. Dominating VG.

$A_{\mathfrak{D}}(\mathfrak{G})$ = $\left[\begin{array}{cccccc}(0,0)& (0.1,0.5)& (0.2,0.6)& (0,0)& (0,0)& (0,0)\\ (0.1,0.5)& (0,0)& (0.2,0.5)& (0,0)& (0.2,0.7)& (0,0)\\ (0.2,0.6)& (0.2,0.5)& (1,1)& (0.2,0.8)& (0,0)& (0,0)\\ (0,0)& (0,0)& (0.2,0.8)& (0,0)& (0.2,0.7)& (0,0)\\ (0,0)& (0.2,0.7)& (0,0)& (0.2,0.7)& (1,1)& (0.2,0.7)\\ (0,0)& (0,0)& (0,0)& (0,0)& (0.2,0.6)& (0,0)\end{array}\right]$

We can write in two different matrices as

$A(t_{{\mathfrak{N}}_{\mathfrak{D}}}(\mathfrak{G}))$ = $\left[\begin{array}{cccccc}0& 0.1& 0.2& 0& 0& 0\\ 0.1& 0& 0.2& 0& 0.2& 0\\ 0.2& 0.2& 1& 0.2& 0& 0\\ 0& 0& 0.2& 0& 0.2& 0\\ 0& 0.2& 0& 0.2& 1& 0.2\\ 0& 0& 0& 0& 0.2& 0\end{array}\right]$

and

$A(f_{{\mathfrak{N}}_{\mathfrak{D}}}(\mathfrak{G}))$ = $\left[\begin{array}{cccccc}0& 0.5& 0.6& 0& 0& 0\\ 0.5& 0& 0.5& 0& 0.7& 0\\ 0.6& 0.5& 1& 0.8& 0& 0\\ 0& 0& 0.8& 0& 0.7& 0\\ 0& 0.7& 0& 0.7& 1& 0.7\\ 0& 0& 0& 0& 0.6& 0\end{array}\right]$

We obtain

$Spec\left(A\left(t_{{\mathfrak{N}}_{\mathfrak{D}}}\left(\mathfrak{G}\right)\right)\right)=(-0.161,-0.089,-0.017,0.053,1.039,1.177)$

$Spec\left(A\left(f_{{\mathfrak{N}}_{\mathfrak{D}}}\left(\mathfrak{G}\right)\right)\right)=(-1.031,-0.57,-0.14,0.292,1.302,2.147)$

Therefore,

$Spec\left(A_{\mathfrak{D}}\left(\mathfrak{G}\right)\right)=\left((-0.161,-1.031)(-0.089,-0.57)(-0.017,-0.14)(0.053,0.292)(1.039,1.302)\right)(1.177,2.147)$.

The energy of DVG $\mathfrak{G}$ is

$$E_{\mathfrak{D}}\left(\mathfrak{G}\right)=\left(E_{t_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right),E_{f_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right)\right)$$

$$=\left(\sum _{k=1}^p|{\psi }_k|,\sum _{k=1}^p|{\pi }_k|\right)=(2.536,5.482).$$

Theorem 1.

Suppose $\mathfrak{G}=(t_{\mathfrak{N}},f_{\mathfrak{N}},t_{1_{\mathfrak{N}}},f_{1_{\mathfrak{N}}})$ is a DVG with p vertices and m edges. Suppose $\mathfrak{D}=\{z_1,z_2,\dots ,z_q\}$ is a DS. If ${\tau }_1,{\tau }_2,\dots ,{\tau }_p$ are the eigenvalues of AM $t_{{\mathfrak{N}}_{\mathfrak{D}}}$, then

$$\left(i\right)\sum _{k=1}^p{\tau }_k=\alpha ,$$

$$\left(ii\right)\sum _{k=1}^p{\tau }_k^2=\sum _{k=1}^p{\left(t_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{t}_{{\mathfrak{N}}_{kl}}{t}_{{\mathfrak{N}}_{lk}},$$

and if ${\eta }_1,{\eta }_2,\dots ,{\eta }_p$ are the eigenvalues of AM $f_{{\mathfrak{N}}_{\mathfrak{D}}}$, then

$$\left(iii\right)\sum _{k=1}^p{\eta }_k=\alpha ,$$

$$\left(iv\right)\sum _{k=1}^p{\eta }_k^2=\sum _{k=1}^p{\left(f_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{f}_{{\mathfrak{N}}_{kl}}{f}_{{\mathfrak{N}}_{lk}},$$

where $\alpha =\mathfrak{D}.$

Proof. 

(i) By the trace property of matrices, we have

$$\sum _{k=1}^p{\tau }_k=\sum t_{{\mathfrak{N}}_{kk}}=\alpha .$$

(ii) Equivalently, the sum of square of eigenvalues of ${\left(t_{{\mathfrak{N}}_{\mathfrak{D}}}\left(\mathfrak{G}\right)\right)}^2$

$$\sum _{k=1}^p{\tau }_k^2=traceof{\left(t_{{\mathfrak{N}}_{\mathfrak{D}}}\left(\mathfrak{G}\right)\right)}^2$$

$$=t_{{\mathfrak{N}}_{11}}{t}_{{\mathfrak{N}}_{11}}+t_{{\mathfrak{N}}_{12}}{t}_{{\mathfrak{N}}_{21}}+\dots +t_{{\mathfrak{N}}_{1p}}{t}_{{\mathfrak{N}}_{p1}}$$

$$+t_{{\mathfrak{N}}_{21}}{t}_{{\mathfrak{N}}_{12}}+t_{{\mathfrak{N}}_{22}}{t}_{{\mathfrak{N}}_{22}}+\dots +t_{{\mathfrak{N}}_{2p}}{t}_{{\mathfrak{N}}_{p2}}$$

$$+\dots +t_{{\mathfrak{N}}_{p1}}{t}_{{\mathfrak{N}}_{1p}}+t_{{\mathfrak{N}}_{p2}}{t}_{{\mathfrak{N}}_{2p}}+\dots +t_{{\mathfrak{N}}_{pp}}{t}_{{\mathfrak{N}}_{pp}}$$

$$=\sum _{k=1}^p{\left(t_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{t}_{{\mathfrak{N}}_{kl}}{t}_{{\mathfrak{N}}_{lk}}.$$

Similarity, we can show that

$$\left(iii\right)\sum _{k=1}^p{\eta }_k=\alpha ,$$

$$\left(iv\right)\sum _{k=1}^p{\eta }_k^2=\sum _{k=1}^p{\left(f_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{f}_{{\mathfrak{N}}_{kl}}{f}_{{\mathfrak{N}}_{lk}}.$$

□

Theorem 2.

Suppose $\mathfrak{G}=(t_{\mathfrak{N}},f_{\mathfrak{N}},t_{1_{\mathfrak{N}}},f_{1_{\mathfrak{N}}})$ is a DVG with p vertices and m edges. If $\mathfrak{D}$ is the DS, then

$$\left(i\right)\sqrt{\sum _{k=1}^p{\left(t_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{t}_{{\mathfrak{N}}_{kl}}{t}_{{\mathfrak{N}}_{lk}}+p(p-1){\left(\mathfrak{a}\right)}^{\frac{2}{p}}}$$

$$\le E_{t_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right)\le \sqrt{p\left(\sum _{k=1}^p{\left(t_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{t}_{{\mathfrak{N}}_{kl}}{t}_{{\mathfrak{N}}_{lk}}\right)},$$

where $\mathfrak{b}=det\left(t_{{\mathfrak{N}}_{\mathfrak{D}}}\left(\mathfrak{G}\right)\right)$ and $\mathfrak{a}=\left|\mathfrak{b}\right|.$

$$\left(ii\right)\sqrt{\sum _{k=1}^p{\left(f_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{f}_{{\mathfrak{N}}_{kl}}{f}_{{\mathfrak{N}}_{lk}}+p(p-1){\left(\mathfrak{c}\right)}^{\frac{2}{p}}}$$

$$\le E_{f_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right)\le \sqrt{p\left(\sum _{k=1}^p{\left(f_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{f}_{{\mathfrak{N}}_{kl}}{f}_{{\mathfrak{N}}_{lk}}\right)},$$

where $\mathfrak{f}=det\left(f_{{\mathfrak{N}}_{\mathfrak{D}}}\left(\mathfrak{G}\right)\right)$ and $\mathfrak{c}=\left|\mathfrak{f}\right|.$

Proof. 

(i) By Cauchy Schwarz inequality, we have

$${\left(\sum _{k=1}^p{r}_k{s}_k\right)}^2\le \left(\sum _{k=1}^p{r}_k^2\right)\left(\sum _{k=1}^p{s}_k^2\right).$$

Upper bound

If $r_k=1$ and $s_k=|{\pi }_k|$ then

$${\left(\sum _{k=1}^p|{\pi }_k|\right)}^2\le \left(\sum _{k=1}^{p}1\right)\left(\sum _{k=1}^p{\pi }_k^2\right).$$

$${\left(E_{t_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right)\right)}^2\le p\left(\sum _{k=1}^p{\left(t_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{t}_{{\mathfrak{N}}_{kl}}{t}_{{\mathfrak{N}}_{lk}}\right).$$

$$E_{t_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right)\le \sqrt{p\left(\sum _{k=1}^p{\left(t_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{t}_{{\mathfrak{N}}_{kl}}{t}_{{\mathfrak{N}}_{lk}}\right)}.$$

Lower bound

$${\left(E_{t_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right)\right)}^2={\left(\sum _{k=1}^p|{\pi }_k|\right)}^2$$

$$=\left(\sum _{k=1}^p|t_{{\mathfrak{N}}_{kk}}{|}^2+2\sum _{1\le k\le l\le p}|t_{{\mathfrak{N}}_{kl}}||t_{{\mathfrak{N}}_{lk}}|\right)$$

$$=\left(\sum _{k=1}^p{\left(t_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{t}_{{\mathfrak{N}}_{kl}}{t}_{{\mathfrak{N}}_{lk}}\right)+2\frac{p(p-1)}{2}A{M}_{1\le k\le l\le p}\left\{|{\pi }_k||{\pi }_l|\right\}.$$

However,

$$A{M}_{1\le k\le l\le p}\left\{|{\pi }_k||{\pi }_l|\right\}\ge G{M}_{1\le k\le l\le p}\left\{|{\pi }_k||{\pi }_l|\right\}.$$

Therefore,

$${\left(E_{t_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right)\right)}^2\ge \sqrt{\sum _{k=1}^p{\left(t_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{t}_{{\mathfrak{N}}_{kl}}{t}_{{\mathfrak{N}}_{lk}}+p(p-1)G{M}_{1\le k\le l\le p}\left\{|{\pi }_k||{\pi }_l|\right\}}$$

$$G{M}_{1\le k\le l\le p}\left\{|{\pi }_k||{\pi }_l|\right\}={\left(\prod _{1\le k\le l\le p}|{\pi }_k||{\pi }_l|\right)}^{\frac{2}{p(p-1)}}$$

$$={\left(\prod _{k=1}^p{|{\pi }_k|}^{p-1}\right)}^{\frac{2}{p(p-1)}}={\left(\prod _{k=1}^p|{\pi }_k|\right)}^{\frac{2}{p}}={\mathfrak{a}}^{\frac{2}{p}}$$

$${\left(E_{t_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right)\right)}^2\ge \sqrt{\sum _{k=1}^p{\left(t_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{t}_{{\mathfrak{N}}_{kl}}{t}_{{\mathfrak{N}}_{lk}}+p(p-1){\mathfrak{a}}^{\frac{2}{p}}}.$$

Combining upper bound and lower bound, we have

$$\left(i\right)\sqrt{\sum _{k=1}^p{\left(t_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{t}_{{\mathfrak{N}}_{kl}}{t}_{{\mathfrak{N}}_{lk}}+p(p-1){\left(\mathfrak{a}\right)}^{\frac{2}{p}}}$$

$$\le E_{t_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right)\le \sqrt{p\left(\sum _{k=1}^p{\left(t_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{t}_{{\mathfrak{N}}_{kl}}{t}_{{\mathfrak{N}}_{lk}}\right)}.$$

Similarity, we can show that

$$\sqrt{\sum _{k=1}^p{\left(f_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{f}_{{\mathfrak{N}}_{kl}}{f}_{{\mathfrak{N}}_{lk}}+p(p-1){\left(\mathfrak{c}\right)}^{\frac{2}{p}}}$$

$$\le E_{t_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right)\le \sqrt{p\left(\sum _{k=1}^p{\left(f_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{f}_{{\mathfrak{N}}_{kl}}{f}_{{\mathfrak{N}}_{lk}}\right)}.$$

For example, the equality is satisfied for the family of graphs in the form of union $K{}_2$. □

Theorem 3.

Suppose $\mathfrak{G}=(t_{\mathfrak{N}},f_{\mathfrak{N}}$ is a VG and $A\left(\mathfrak{G}\right)=(t_{\mathfrak{N}}\left(\mathfrak{G}\right),f_{\mathfrak{N}}\left(\mathfrak{G}\right))$ is the AM of $\mathfrak{G}.$ Suppose ${\mathfrak{G}}_{\mathfrak{1}}=(t_{\mathfrak{N}},f_{\mathfrak{N}},t_{1_{\mathfrak{N}}},f_{1_{\mathfrak{N}}})$ is the DVG of $\mathfrak{G}$ and $A_{\mathfrak{D}}\left(\mathfrak{G}\right)=(t_{{\mathfrak{N}}_{\mathfrak{D}}}\left(\mathfrak{G}\right),f_{{\mathfrak{N}}_{\mathfrak{D}}}\left(\mathfrak{G}\right))$ is the DVG AM of ${\mathfrak{G}}_{\mathfrak{1}}$. Then

$$\left(i\right)E{{(}_{t_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right))}^2\le p\left(\sum _{k=1}^p{\left(t_{{\mathfrak{N}}_{kk}}\right)}^2+E{{(}_{t_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right))}^2\right),$$

$$\left(ii\right)E{{(}_{f_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right))}^2\le p\left(\sum _{k=1}^p{\left(f_{{\mathfrak{N}}_{kk}}\right)}^2+E{{(}_{f_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right))}^2\right).$$

Proof. 

$$\left(i\right)E{{(}_{t_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right))}^2\ge 2\sum _{1\le k\le l\le p}{t}_{{\mathfrak{N}}_{kl}}{t}_{{\mathfrak{N}}_{lk}}+p(p-1){\left(\mathfrak{a}\right)}^{\frac{2}{p}}$$

$$\ge 2\sum _{1\le k\le l\le p}{t}_{{\mathfrak{N}}_{kl}}{t}_{{\mathfrak{N}}_{lk}}$$

$$\mathit{\text{i.e.}},2\sum _{1\le k\le l\le p}{t}_{{\mathfrak{N}}_{kl}}{t}_{{\mathfrak{N}}_{lk}}\le E{{(}_{t_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right))}^2$$

(1)

Now

$$E{{(}_{t_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right))}^2\le p\sum _{k=1}^p{\left(t_{{\mathfrak{N}}_{kk}}\right)}^2+2p\sum _{1\le k\le l\le p}{t}_{{\mathfrak{N}}_{kl}}{t}_{{\mathfrak{N}}_{lk}}$$

$$E{{(}_{t_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right))}^2\le p\left(\sum _{k=1}^p{\left(t_{{\mathfrak{N}}_{kk}}\right)}^2+E{{(}_{t_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right))}^2\right)\left(byEquation\left(1\right)\right)$$

Similarity, we can show that

$$E{{(}_{f_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right))}^2\le p\left(\sum _{k=1}^p{\left(f_{{\mathfrak{N}}_{kk}}\right)}^2+E{{(}_{f_{{\mathfrak{N}}_{\mathfrak{D}}}}\left(\mathfrak{G}\right))}^2\right).$$

□

3. Energy of Double Dominating Vague Graphs

In this section, we defined the notion of energy of double dominating vague graphs.

Definition 10.

A graph $\tilde{\mathfrak{G}}=(t_{\mathfrak{N}},f_{\mathfrak{N}},t_{1_{\mathfrak{N}}},f_{1_{\mathfrak{N}}})$ is named DDVG, where $t_{1_{\mathfrak{N}}}:X\to [0,1]$ denotes a degree of membership and $f_{1_{\mathfrak{N}}}:X\to [0,1]$ denotes a degree of non-membership, defined as $t_{1_{\mathfrak{N}}}\left(r\right)=max\left\{t_{\mathfrak{N}}(r,s)\right\}$ and $f_{1_{\mathfrak{N}}}\left(r\right)=min\left\{f_{\mathfrak{N}}(r,s)\right\},r\in X.$

Note: The double dominating set (DDS) means a subset $\tilde{\mathfrak{D}}\subseteq X$ is named DDS in $\tilde{\mathfrak{G}}$ if for each $s\in X-\tilde{\mathfrak{D}}$, there exists two vertices r in $\tilde{\mathfrak{D}}$ such that r dominates s, i.e.,

$$t_{\mathfrak{N}}(r,s)=t_{1_{\mathfrak{N}}}\left(r\right)\wedge t_{1_{\mathfrak{N}}}\left(s\right)$$

$$f_{\mathfrak{N}}(r,s)=f_{1_{\mathfrak{N}}}\left(r\right)\vee f_{1_{\mathfrak{N}}}\left(s\right)$$

Definition 11.

Suppose $\tilde{\mathfrak{G}}=(t_{\mathfrak{N}},f_{\mathfrak{N}},t_{1_{\mathfrak{N}}},f_{1_{\mathfrak{N}}})$ is a DDVG. Suppose $r,s\in X$, we say that r dominates s in $\tilde{\mathfrak{G}}$ if there exists strong edge from r to s. A subset $\tilde{\mathfrak{D}}\subseteq X$ is named DDS in $\tilde{\mathfrak{G}}$ if for each $s\in X-\tilde{\mathfrak{D}}$, there exists r in $\tilde{\mathfrak{D}}$ such that r dominates s.

A DDS $\tilde{\mathfrak{D}}$ of X is said to be a minimal DDS if no proper subset of $\tilde{\mathfrak{D}}$ is a DDS of $\tilde{\mathfrak{G}}$. The minimum cardinality of a minimal DDS in $\tilde{\mathfrak{G}}$ is called the DDN of $\tilde{\mathfrak{G}}$ and is denoted by ${\alpha }_{\tilde{\mathfrak{D}}}\left(\tilde{\mathfrak{G}}\right)$ and the corresponding minimal DDS is called the minimum DDS of $\tilde{\mathfrak{G}}$.

Example 2.

From the VG in Figure 2, we have

The DS of $\mathfrak{G}$ is $\mathfrak{D}=\{d_2,d_4\}$.

The DN of $\mathfrak{G}$ is $\alpha \left(\mathfrak{G}\right)=1.25$

The DDS of $\mathfrak{G}=\{d_1,d_4,d_5\}$.

The DDN of $\mathfrak{G}$ is ${\alpha }_{\tilde{\mathfrak{D}}}=2$

Figure 2. DVG.

Figure 2. DVG.

Theorem 4.

For any VG, then $\alpha \left(\mathfrak{G}\right)\le {\alpha }_{\tilde{\mathfrak{D}}}\left(\tilde{\mathfrak{G}}\right)$.

Proof. 

Suppose $\mathfrak{G}=(t_{\mathfrak{N}},f_{\mathfrak{N}})$ is a VG. Suppose $\mathfrak{D}\subseteq X$ is a DS and $\tilde{\mathfrak{D}}\subseteq X$ is a DDS of $\tilde{\mathfrak{G}}$. If $\mathfrak{D}=\tilde{\mathfrak{D}}$, then $\alpha \left(\mathfrak{G}\right)={\alpha }_{\tilde{\mathfrak{D}}}\left(\tilde{\mathfrak{G}}\right)$. If $\mathfrak{D}\ne \tilde{\mathfrak{D}}$, then $\tilde{\mathfrak{D}}$ has at least one vertices more than $\mathfrak{D}$ and hence, $\alpha \left(\mathfrak{G}\right)<{\alpha }_{\tilde{\mathfrak{D}}}\left(\tilde{\mathfrak{G}}\right)$. Therefore, $\alpha \left(\mathfrak{G}\right)\le {\alpha }_{\tilde{\mathfrak{D}}}\left(\tilde{\mathfrak{G}}\right)$. □

Theorem 5.

Suppose $\mathfrak{G}=(t_{\mathfrak{N}},f_{\mathfrak{N}})$ is a VG with DDS. Then $\alpha \left(\mathfrak{G}\right)+{\alpha }_{\tilde{\mathfrak{D}}}\left(\tilde{\mathfrak{G}}\right)\le P$.

Proof. 

Suppose $\mathfrak{G}=(t_{\mathfrak{N}},f_{\mathfrak{N}})$ is a VG. Suppose $\tilde{\mathfrak{D}}$ is the DDS. Then, $\alpha \left(\mathfrak{G}\right)\le P-{\alpha }_{\tilde{\mathfrak{D}}}\left(\tilde{\mathfrak{G}}\right)$. Therefore, $\alpha \left(\mathfrak{G}\right)+{\alpha }_{\tilde{\mathfrak{D}}}\left(\tilde{\mathfrak{G}}\right)\le P$. □

Theorem 6.

Suppose $\mathfrak{G}=(t_{\mathfrak{N}},f_{\mathfrak{N}})$ is a VG, then ${\alpha }_{\tilde{\mathfrak{D}}}\left(\tilde{\mathfrak{G}}\right)<P$.

Proof. 

Suppose $\mathfrak{G}=(t_{\mathfrak{N}},f_{\mathfrak{N}})$ is a VG. Then, by Theorem 5, $\alpha \left(\mathfrak{G}\right)+{\alpha }_{\tilde{\mathfrak{D}}}\left(\tilde{\mathfrak{G}}\right)\le P$. Therefore, ${\alpha }_{\tilde{\mathfrak{D}}}\left(\tilde{\mathfrak{G}}\right)\le P$. □

Definition 12.

Suppose $\tilde{\mathfrak{G}}=(t_{\mathfrak{N}},f_{\mathfrak{N}},t_{1_{\mathfrak{N}}},f_{1_{\mathfrak{N}}})$ is a DDVG. The adjacency matrix (AM) of $\tilde{\mathfrak{G}}$ is defined as $A_{\tilde{\mathfrak{D}}}(\tilde{\mathfrak{G}})=\left[d_{kl}\right]$, where

$$d_{kl}=\left\{\begin{array}{c}(t_{{\mathfrak{N}}_{kl}},f_{{\mathfrak{N}}_{kl}}),(k,l)\in E,\hfill \\ (1,1),k=l and r\in \tilde{\mathfrak{D}},\hfill \\ 0,otherwise.\hfill \end{array}\right.$$

This AM of $\tilde{\mathfrak{G}}$ can be written in two different matrices as $A_{\tilde{\mathfrak{D}}}(\tilde{\mathfrak{G}})=(t_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}(\tilde{\mathfrak{G}}),f_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}(\tilde{\mathfrak{G}}))$, where

$$t_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}(\tilde{\mathfrak{G}})=\left\{\begin{array}{c}{t}_{{\mathfrak{N}}_{kl}},(k,l)\in E,\hfill \\ 1,k=l and r\in \tilde{\mathfrak{D}},\hfill \\ 0,otherwise.\hfill \end{array}\right.$$

and

$$f_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}(\tilde{\mathfrak{G}})=\left\{\begin{array}{c}{f}_{{\mathfrak{N}}_{kl}},(k,l)\in E,\hfill \\ 1,k=l and r\in \tilde{\mathfrak{D}},\hfill \\ 0,otherwise.\hfill \end{array}\right.$$

Definition 13.

The spectrum of AM of a DDVG $\tilde{\mathfrak{G}}$ is defined as $(W_{\tilde{\mathfrak{D}}},Z_{\tilde{\mathfrak{D}}})$, where $W_{\tilde{\mathfrak{D}}}$ and $Z_{\tilde{\mathfrak{D}}}$ are the sets of eigenvalues of $t_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}(\tilde{\mathfrak{G}})$ and $f_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}(\tilde{\mathfrak{G}})$, respectively.

Definition 14.

The energy of a DDVG $\tilde{\mathfrak{G}}=(t_{\mathfrak{N}},f_{\mathfrak{N}},t_{1_{\mathfrak{N}}},f_{1_{\mathfrak{N}}})$ is defined as

$$E_{\tilde{\mathfrak{D}}}(\tilde{\mathfrak{G}})=\left(E_{t_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}}(\tilde{\mathfrak{G}}),E_{f_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}}(\tilde{\mathfrak{G}})\right)=\left(\sum _{k=1}^p|{\psi }_k|,\sum _{k=1}^p|{\pi }_k|\right)$$

where $W_{\tilde{\mathfrak{D}}}={\left\{{\psi }_k\right\}}_{k=1}^p$ and $Z_{\tilde{\mathfrak{D}}}={\left\{{\pi }_k\right\}}_{k=1}^p$.

Example 3.

Consider a DVG $\tilde{\mathfrak{G}}=(t_{\mathfrak{N}},f_{\mathfrak{N}},t_{1_{\mathfrak{N}}},f_{1_{\mathfrak{N}}})$ on $X=\{p_1,p_2,p_3,p_4,p_5,p_6\}$ and $t_{1_{\mathfrak{N}}},f_{1_{\mathfrak{N}}}$ are defined by $t_{1_{\mathfrak{N}}}:X\to [0,1]$ and $f_{1_{\mathfrak{N}}}:X\to [0,1]$, as shown in Figure 3.

$$t_{1_{\mathfrak{N}}}\left(p_1\right)=\underset{p_i}{max}\left\{t_{\mathfrak{N}}(p_1,p_i)\right\}$$

$$=max\{t_{\mathfrak{N}}(p_1,p_2),t_{\mathfrak{N}}(p_1,p_4),t_{\mathfrak{N}}(p_1,p_6)\}$$

$$=max\{0.2,0.2,0.2\}=0.2$$

Similarly,

$$t_{1_{\mathfrak{N}}}\left(p_2\right)=0.3,t_{1_{\mathfrak{N}}}\left(p_3\right)=0.2,t_{1_{\mathfrak{N}}}\left(p_4\right)=0.4,t_{1_{\mathfrak{N}}}\left(p_5\right)=0.4,t_{1_{\mathfrak{N}}}\left(p_6\right)=0.4.$$

Moreover,

$$f_{1_{\mathfrak{N}}}\left(p_1\right)=\underset{p_i}{min}\left\{f_{\mathfrak{N}}(p_1,p_i)\right\}$$

$$=min\{f_{\mathfrak{N}}(p_1,p_2),f_{\mathfrak{N}}(p_1,p_4),f_{\mathfrak{N}}(p_1,p_6)\}$$

$$=min\{0.6,0.6,0.6\}=0.6$$

Similarly,

$$f_{1_{\mathfrak{N}}}\left(p_2\right)=0.5,f_{1_{\mathfrak{N}}}\left(p_3\right)=0.5,f_{1_{\mathfrak{N}}}\left(p_4\right)=0.6,f_{1_{\mathfrak{N}}}\left(p_5\right)=0.6,f_{1_{\mathfrak{N}}}\left(p_6\right)=0.7.$$

Here, $p_1$ dominates $p_2,p_4,p_6$ and $p_3$ dominates $a_2,a_4$ and $p_5$ dominates $p_2,p_4,p_6$ because,

$$t_{\mathfrak{N}}(p_1,p_2)=t_{1_{\mathfrak{N}}}\left(p_1\right)\wedge t_{1_{\mathfrak{N}}}\left(p_2\right),f_{\mathfrak{N}}(p_1,p_2)=f_{1_{\mathfrak{N}}}\left(p_1\right)\vee f_{1_{\mathfrak{N}}}\left(p_2\right)$$

$$t_{\mathfrak{N}}(p_2,p_3)=t_{1_{\mathfrak{N}}}\left(p_2\right)\wedge t_{1_{\mathfrak{N}}}\left(p_3\right),f_{\mathfrak{N}}(p_2,p_3)=f_{1_{\mathfrak{N}}}\left(p_2\right)\vee f_{1_{\mathfrak{N}}}\left(p_3\right)$$

$$t_{\mathfrak{N}}(p_3,p_4)=t_{1_{\mathfrak{N}}}\left(p_3\right)\wedge t_{1_{\mathfrak{N}}}\left(p_4\right),f_{\mathfrak{N}}(p_3,p_4)=f_{1_{\mathfrak{N}}}\left(p_3\right)\vee f_{1_{\mathfrak{N}}}\left(p_4\right)$$

$$t_{\mathfrak{N}}(p_4,p_5)=t_{1_{\mathfrak{N}}}\left(p_4\right)\wedge t_{1_{\mathfrak{N}}}\left(p_5\right),f_{\mathfrak{N}}(p_4,p_5)=f_{1_{\mathfrak{N}}}\left(p_4\right)\vee f_{1_{\mathfrak{N}}}\left(p_5\right)$$

$$t_{\mathfrak{N}}(p_5,p_6)=t_{1_{\mathfrak{N}}}\left(p_5\right)\wedge t_{1_{\mathfrak{N}}}\left(p_6\right),f_{\mathfrak{N}}(p_5,p_6)=f_{1_{\mathfrak{N}}}\left(p_5\right)\vee f_{1_{\mathfrak{N}}}\left(p_6\right)$$

$$t_{\mathfrak{N}}(p_6,p_1)=t_{1_{\mathfrak{N}}}\left(p_6\right)\wedge t_{1_{\mathfrak{N}}}\left(p_1\right),f_{\mathfrak{N}}(p_6,p_1)=f_{1_{\mathfrak{N}}}\left(p_6\right)\vee f_{1_{\mathfrak{N}}}\left(p_1\right)$$

$$t_{\mathfrak{N}}(p_1,p_4)=t_{1_{\mathfrak{N}}}\left(p_1\right)\wedge t_{1_{\mathfrak{N}}}\left(p_4\right),f_{\mathfrak{N}}(p_1,p_4)=f_{1_{\mathfrak{N}}}\left(p_1\right)\vee f_{1_{\mathfrak{N}}}\left(p_4\right)$$

$$t_{\mathfrak{N}}(p_2,p_5)=t_{1_{\mathfrak{N}}}\left(p_2\right)\wedge t_{1_{\mathfrak{N}}}\left(p_5\right),f_{\mathfrak{N}}(p_2,p_5)=f_{1_{\mathfrak{N}}}\left(p_2\right)\vee f_{1_{\mathfrak{N}}}\left(p_5\right)$$

Figure 3. Double dominating VG.

Therefore, $\tilde{\mathfrak{D}}=\{p_1,p_3,p_5\}$ is a DDS because every vertex in $X-\tilde{\mathfrak{D}}$, is dominated by atleast two vertices in $\tilde{\mathfrak{D}}.$ The AM of DVG $\mathfrak{G}$ is given below

$A_{\tilde{\mathfrak{D}}}(\tilde{\mathfrak{G}})$ = $\left[\begin{array}{cccccc}(1,1)& (0.2,0.6)& (0,0)& (0.2,0.6)& (0,0)& (0.2,0.7)\\ (0.2,0.6)& (0,0)& (0.2,0.5)& (0,0)& (0.3,0.6)& (0,0)\\ (0,0)& (0.2,0.5)& (1,1)& (0.2,0.6)& (0,0)& (0,0)\\ (0.2,0.6)& (0,0)& (0.2,0.6)& (0,0)& (0.4,0.6)& (0,0)\\ (0,0)& (0.3,0.6)& (0,0)& (0.4,0.6)& (1,1)& (0.4,0.7)\\ (0.2,0.7)& (0,0)& (0,0)& (0,0)& (0.4,0.7)& (0,0)\end{array}\right]$

We can write in two different matrices as

$A(t_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}(\tilde{\mathfrak{G}}))$ = $\left[\begin{array}{cccccc}1& 0.2& 0& 0.2& 0& 0.2\\ 0.2& 0& 0.2& 0& 0.3& 0\\ 0& 0.2& 1& 0.2& 0& 0\\ 0.2& 0& 0.2& 0& 0.4& 0\\ 0& 0.3& 0& 0.4& 1& 0.4\\ 0.2& 0& 0& 0& 0.4& 0\end{array}\right]$

and

$A(f_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}(\tilde{\mathfrak{G}}))$ = $\left[\begin{array}{cccccc}1& 0.6& 0& 0.6& 0& 0.7\\ 0.6& 0& 0.5& 0& 0.6& 0\\ 0& 0.5& 1& 0.6& 0& 0\\ 0.6& 0& 0.6& 0& 0.6& 0\\ 0& 0.6& 0& 0.6& 1& 0.7\\ 0.7& 0& 0& 0& 0.7& 0\end{array}\right]$

We obtain

$Spec(A(t_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}(\tilde{\mathfrak{G}})))=(-0.411,-0.028,-0.001,1.001,1.028,1.411)$

$Spec(A(f_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}(\tilde{\mathfrak{G}})))=(0,1,1.182,-0.182,2.251,-1.251)$

Therefore,

$Spec(A_{\tilde{\mathfrak{D}}}(\tilde{\mathfrak{G}}))=((-0.411,0)(-0.028,1)(-0.001,1.182)(1.001,-0.182)(1.028,2.251))(1.411,-1.251)$.

The energy of DVG $\tilde{\mathfrak{G}}$ is

$$E_{\tilde{\mathfrak{D}}}(\tilde{\mathfrak{G}})=\left(E_{t_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}}(\tilde{\mathfrak{G}}),E_{f_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}}(\tilde{\mathfrak{G}})\right)$$

$$=\left(\sum _{k=1}^p|{\psi }_k|,\sum _{k=1}^p|{\pi }_k|\right)=(3.88,5.866).$$

Theorem 7.

Suppose $\tilde{\mathfrak{G}}=(t_{\mathfrak{N}},f_{\mathfrak{N}},t_{1_{\mathfrak{N}}},f_{1_{\mathfrak{N}}})$ is a DDVG with p vertices and m edges. Suppose $\tilde{\mathfrak{D}}=\{z_1,z_2,\dots ,z_q\}$ is a DDS. If ${\tau }_1,{\tau }_2,\dots ,{\tau }_p$ are the eigenvalues of AM $t_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}$, then

$$\left(i\right)\sum _{k=1}^p{\tau }_k=\alpha ,$$

$$\left(ii\right)\sum _{k=1}^p{\tau }_k^2=\sum _{k=1}^p{\left(t_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{t}_{{\mathfrak{N}}_{kl}}{t}_{{\mathfrak{N}}_{lk}},$$

and if ${\eta }_1,{\eta }_2,\dots ,{\eta }_p$ are the eigenvalues of AM $f_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}$, then

$$\left(iii\right)\sum _{k=1}^p{\eta }_k=\alpha ,$$

$$\left(iv\right)\sum _{k=1}^p{\eta }_k^2=\sum _{k=1}^p{\left(f_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{f}_{{\mathfrak{N}}_{kl}}{f}_{{\mathfrak{N}}_{lk}},$$

where $\alpha =\tilde{\mathfrak{D}}.$

Proof. 

By using similar information as used in Theorem 1. □

Theorem 8.

Suppose $\tilde{\mathfrak{G}}=(t_{\mathfrak{N}},f_{\mathfrak{N}},t_{1_{\mathfrak{N}}},f_{1_{\mathfrak{N}}})$ is a DDVG with p vertices and m edges. If $\tilde{\mathfrak{D}}$ is the DDS, then

$$\left(i\right)\sqrt{\sum _{k=1}^p{\left(t_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{t}_{{\mathfrak{N}}_{kl}}{t}_{{\mathfrak{N}}_{lk}}+p(p-1){\left(\mathfrak{a}\right)}^{\frac{2}{p}}}$$

$$\le E_{t_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}}\left(\tilde{\mathfrak{G}}\right)\le \sqrt{p\left(\sum _{k=1}^p{\left(t_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{t}_{{\mathfrak{N}}_{kl}}{t}_{{\mathfrak{N}}_{lk}}\right)},$$

where $\mathfrak{b}=det\left(t_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}\left(\tilde{\mathfrak{G}}\right)\right)$ and $\mathfrak{a}=|\mathfrak{b}|.$

$$\left(ii\right)\sqrt{\sum _{k=1}^p{\left(f_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{f}_{{\mathfrak{N}}_{kl}}{f}_{{\mathfrak{N}}_{lk}}+p(p-1){\left(\mathfrak{c}\right)}^{\frac{2}{p}}}$$

$$\le E_{f_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}}\left(\tilde{\mathfrak{G}}\right)\le \sqrt{p\left(\sum _{k=1}^p{\left(f_{{\mathfrak{N}}_{kk}}\right)}^2+2\sum _{1\le k\le l\le p}{f}_{{\mathfrak{N}}_{kl}}{f}_{{\mathfrak{N}}_{lk}}\right)},$$

where $\mathfrak{f}=det(f_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}(\tilde{\mathfrak{G}}))$ and $\mathfrak{c}=|\mathfrak{f}|.$

Proof. 

By using similar information as used in Theorem 2. □

Theorem 9.

Suppose $\tilde{\mathfrak{G}}=(t_{\mathfrak{N}},f_{\mathfrak{N}})$ is a VFG and $A(\tilde{\mathfrak{G}})=(t_{\mathfrak{N}}(\tilde{\mathfrak{G}}),f_{\mathfrak{N}}(\tilde{\mathfrak{G}}))$ is the AM of $\tilde{\mathfrak{G}}.$ Suppose $\widetilde{{\mathfrak{G}}_{\mathfrak{1}}}=(t_{\mathfrak{N}},f_{\mathfrak{N}},t_{1_{\mathfrak{N}}},f_{1_{\mathfrak{N}}})$ is the DDVG of $\tilde{\mathfrak{G}}$ and $A_{\tilde{\mathfrak{D}}}(\tilde{\mathfrak{G}})=(t_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}(\tilde{\mathfrak{G}}),f_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}(\tilde{\mathfrak{G}}))$ is the AM of $\widetilde{{\mathfrak{G}}_{\mathfrak{1}}}$. Then

$$\left(i\right)E{{(}_{t_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}}(\tilde{\mathfrak{G}}))}^2\le p\left(\sum _{k=1}^p{\left(t_{{\mathfrak{N}}_{kk}}\right)}^2+E{{(}_{t_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}}(\tilde{\mathfrak{G}}))}^2\right),$$

$$\left(ii\right)E{{(}_{f_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}}(\tilde{\mathfrak{G}}))}^2\le p\left(\sum _{k=1}^p{\left(f_{{\mathfrak{N}}_{kk}}\right)}^2+E{{(}_{f_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}}(\tilde{\mathfrak{G}}))}^2\right).$$

Proof. 

By using similar information as used in Theorem 3. □

4. Application to Select the Best Medical Laboratories

Assume that there are six different medical labs working in a city for conducting tests. The vertices show the laboratories and the edges show the contract conditions among the laboratories to share the facilities or test kits. To make a domination collection among these labs, a collection of labs that have higher quality and have good communication with other laboratories is assumed. It is necessary to find the minimal dominant set to obtain a dominance number. Furthermore, to construct a minimal dominant set with a lab that has higher features (fuzzy vertex value), the maximum neighborhood with the value of the effective edge should be preferred.

Consider a city with 6 labs framed as a VG $\mathfrak{G}$ with 6 vertices $X=\{l_1,l_2,l_3,l_4,l_5,l_6\}$ as shown in Figure 4. The labs are considered vertex sets of VG, as shown in Table 2. The relationship between the labs is considered an edge set of VG, as shown in Table 3. Here, we have to choose the labs that have the quality level and available facilities from the rest of the labs, which are the DSs.

Figure 4. Dominating VG $\mathfrak{G}$.

Table 2. The membership and non-membership of vertices of DVFG $\mathfrak{G}$.

${\mathit{l}}_1$	${\mathit{l}}_2$	${\mathit{l}}_3$	${\mathit{l}}_4$	${\mathit{l}}_5$	${\mathit{l}}_6$
$(0.3,0.5)$	$(0.2,0.5)$	$(0.2,0.8)$	$(0.6,0.7)$	$(0.3,0.4)$	$(0.4,0.6)$

Table 3. The membership and non-membership of edges of DVFG $\mathfrak{G}$.

$({\mathit{l}}_1,{\mathit{l}}_2)$	$({\mathit{l}}_1,{\mathit{l}}_4)$	$({\mathit{l}}_4,{\mathit{l}}_6)$
$(0.2,0.5)$	$(0.2,0.7)$	$(0.4,0.7)$
$({\mathit{l}}_{\mathbf{2}},{\mathit{l}}_{\mathbf{3}})$	$({\mathit{l}}_{\mathbf{3}},{\mathit{l}}_{\mathbf{6}})$	$({\mathit{l}}_{\mathbf{1}},{\mathit{l}}_{\mathbf{5}})$
$(0.2,0.8)$	$(0.1,0.8)$	$(0.3,0.5)$
$({\mathit{l}}_{\mathbf{2}},{\mathit{l}}_{\mathbf{5}})$	$({\mathit{l}}_{\mathbf{3}},{\mathit{l}}_{\mathbf{4}})$	$({\mathit{l}}_{\mathbf{5}},{\mathit{l}}_{\mathbf{6}})$
$(0.2,0.6)$	$(0.2,0.8)$	$(0.3,0.6)$

Consider DVG $\mathfrak{G}=(t_{\mathfrak{N}},f_{\mathfrak{N}},t_{1_{\mathfrak{N}}},f_{1_{\mathfrak{N}}})$ on X and $t_{1_{\mathfrak{N}}},f_{1_{\mathfrak{N}}}$ are, therefore, defined by $t_{1_{\mathfrak{N}}}:X\to [0,1]$ and $f_{1_{\mathfrak{N}}}:X\to [0,1]$. The quality level of labs is shown in Table 4.

Table 4. The quality level of labs.

Labs	${\mathit{t}}_{1_{\mathfrak{N}}}$	${\mathit{f}}_{1_{\mathfrak{N}}}$
$l_1$	0.3	0.5
$l_2$	0.2	0.5
$l_3$	0.2	0.8
$l_4$	0.4	0.7
$l_5$	0.3	0.5
$l_6$	0.4	0.6

Here, lab $l_2$ dominates labs $l_1,l_3$ and lab $l_5$ dominates labs $l_1,l_6$ and lab $l_4$ dominates labs $l_3,l_7$, because

$$t_{\mathfrak{N}}(l_1,l_2)=t_{1_{\mathfrak{N}}}\left(l_1\right)\wedge t_{1_{\mathfrak{N}}}\left(l_2\right),f_{\mathfrak{N}}(l_1,l_2)=f_{1_{\mathfrak{N}}}\left(l_1\right)\vee f_{1_{\mathfrak{N}}}\left(l_2\right)$$

$$t_{\mathfrak{N}}(l_1,l_5)=t_{1_{\mathfrak{N}}}\left(l_1\right)\wedge t_{1_{\mathfrak{N}}}\left(l_5\right),f_{\mathfrak{N}}(l_1,l_5)=f_{1_{\mathfrak{N}}}\left(l_1\right)\vee f_{1_{\mathfrak{N}}}\left(l_5\right)$$

$$t_{\mathfrak{N}}(l_4,l_6)=t_{1_{\mathfrak{N}}}\left(l_4\right)\wedge t_{1_{\mathfrak{N}}}\left(l_6\right),f_{\mathfrak{N}}(l_4,l_6)=f_{1_{\mathfrak{N}}}\left(l_4\right)\vee f_{1_{\mathfrak{N}}}\left(l_6\right)$$

$$t_{\mathfrak{N}}(l_5,l_6)=t_{1_{\mathfrak{N}}}\left(l_5\right)\wedge t_{1_{\mathfrak{N}}}\left(l_6\right),f_{\mathfrak{N}}(l_5,l_6)=f_{1_{\mathfrak{N}}}\left(l_5\right)\vee f_{1_{\mathfrak{N}}}\left(l_6\right)$$

$$t_{\mathfrak{N}}(l_2,l_3)=t_{1_{\mathfrak{N}}}\left(l_2\right)\wedge t_{1_{\mathfrak{N}}}\left(l_3\right),f_{\mathfrak{N}}(l_2,l_3)=f_{1_{\mathfrak{N}}}\left(l_2\right)\vee f_{1_{\mathfrak{N}}}\left(l_3\right)$$

$$t_{\mathfrak{N}}(l_3,l_4)=t_{1_{\mathfrak{N}}}\left(l_3\right)\wedge t_{1_{\mathfrak{N}}}\left(l_4\right),f_{\mathfrak{N}}(l_3,l_4)=f_{1_{\mathfrak{N}}}\left(l_3\right)\vee f_{1_{\mathfrak{N}}}\left(l_4\right)$$

It means that labs $\tilde{\mathfrak{D}}=\{l_2,l_4,l_5\}$ have a quality level and available facilities from the rest of the labs, thus, $\tilde{\mathfrak{D}}=\{l_2,l_4,l_5\}$ is a DS because every lab is dominated by at least two labs. After selecting the minimum labs that have access to the rest of the labs and are at a high level in terms of facilities and diagnosis of diseases, we will examine the performance of the labs of this city, so by calculating a new concept of energy on the DVG, we presented the quality of the labs. First, we obtain the AM of DVG $\mathfrak{G}$, which is given below

$A_{\tilde{\mathfrak{D}}}\left(\mathfrak{G}\right)$ = $\left[\begin{array}{cccccc}(0,0)& (0.2,0.5)& (0,0)& (0.2,0.7)& (0.3,0.5)& (0,0)\\ (0.2,0.5)& (1,1)& (0.2,0.8)& (0,0)& (0.2,0.6)& (0,0)\\ (0,0)& (0.2,0.8)& (0,0)& (0.2,0.8)& (0,0)& (0.1,0.8)\\ (0.2,0.7)& (0,0)& (0.2,0.8)& (1,1)& (0,0)& (0.4,0.7)\\ (0.2,0.5)& (0.2,0.6)& (0,0)& (0,0)& (1,1)& (0.3,0.6)\\ (0,0)& (0,0)& (0.1,0.8)& (0.4,0.7)& (0.3,0.6)& (0,0)\end{array}\right]$

Then we can write the two different matrices as

$A(t_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}(\mathfrak{G}))$ = $\left[\begin{array}{cccccc}0& 0.2& 0& 0.2& 0.3& 0\\ 0.2& 1& 0.2& 0& 0.2& 0\\ 0& 0.2& 0& 0.2& 0& 0.1\\ 0.2& 0& 0.2& 1& 0& 0.4\\ 0.2& 0.2& 0& 0& 1& 0.3\\ 0& 0& 0.1& 0.4& 0.3& 0\end{array}\right]$

and

$A(f_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}(\mathfrak{G}))$ = $\left[\begin{array}{cccccc}0& 0.5& 0& 0.7& 0.5& 0\\ 0.5& 1& 0.8& 0& 0.6& 0\\ 0& 0.8& 0& 0.8& 0& 0.8\\ 0.7& 0& 0.8& 1& 0& 0.7\\ 0.5& 0.6& 0& 0& 1& 0.6\\ 0& 0& 0.8& 0.7& 0.6& 0\end{array}\right]$

We obtain

$Spec(A(t_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}(\mathfrak{G})))=(-0.274,-0.128,0.026,0.857,1.116,1.403)$

$Spec(A(f_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}(\mathfrak{G})))=(-1.148,-0.849,0.302,0.724,1.394,2.577)$

The energy of DVG $\mathfrak{G}$ is

$$E_{\tilde{\mathfrak{D}}}(\mathfrak{G})=\left(E_{t_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}}(\mathfrak{G}),E_{f_{{\mathfrak{N}}_{\tilde{\mathfrak{D}}}}}(\mathfrak{G})\right)$$

$$=\left(\sum _{k=1}^p|{\psi }_k|,\sum _{k=1}^p|{\pi }_k|\right)=(3.804,6.994).$$

In this part, we conclude that by choosing three laboratories $\{l_2,l_4,l_5\}$ and equipping them and raising their quality level, the level of diagnosis performance in the labs will increase.

5. Conclusions

A VG is suitable for modeling uncertainty-related problems that necessitate human knowledge and evaluation. Moreover, DSs have a wide range of applications in VGs for the analysis of vague information, and also, serve as one of the most widely used topics in VGs in various sciences. In this research, we described a new concept of the DS in VGs. We also defined DDVG. We have presented the notion of the energy of the DVG and we discussed some properties and bounds for the energy of DVG and DDVG. Finally, an application of DVG was presented. In future work, we will define a DVG structure and study the DS energy of VG structure.