1 Introduction

Graphs have long been used to describe objects and the relationships between them. Many of the issues and phenomena around us are associated with complexities and ambiguities that make it difficult to express certainty. These difficulties were alleviated by the introduction of fuzzy sets by Zadeh [48]. This concept established well-grounded allocation membership degree to elements of a set. The existence of a single degree for a membership could not resolve the ambiguity on uncertain issues, so the need for a degree of membership was felt. Afterward, to overcome the existing ambiguities, Atanassov [7] introduced non-membership degrees and defined an intuitionistic fuzzy set as the sum of degrees not greater than 1. This set is used in image processing [11], robotic system [17], decision making [23], medical diagnosis [43] and etc. Fuzzy graphs were presented in different types, and researchers conducted many studies about their properties and characteristics [3132, 34, 35, 37, 38, 40, 44,45,46]. Liu et al. [24] presented t-spherical fuzzy 2-tuple linguistic muirhead mean aggregation operators. Naz et al. [26] explained decision analysis under the hesitant dual fuzzy environment q-rung orthopair. An extension of social network group decision-making based on trust rank and personas was proposed by Cai et al. [10]. Pal et al. [25] investigated new concepts in neutrosophic graphs. Certain properties of single-valued neutrosophic graph, presented by Zeng et al. [49]. Rao et al. [33], studied intuitionistic fuzzy tree.

Intuitionistic fuzzy graphs have only two modes of membership and non-membership, but in some issues such as voting, medical diagnosis, and etc., we also face a state of neutrality. When it comes to the climate impact on the human environment in different climatic conditions, ineffectiveness, in addition to being effective or not, can be proposed. To resolve this case, Cuong and Kreinevich [14] presented the concept of picture fuzzy set (PFS) as a combination of the fuzzy set (FS) and intuitionistic fuzzy set (IFS). Any element in this set is inclusive of three degrees of being a member, neutral, and non-member, where the sum of degrees is not greater than 1. Some PFS properties were studied by Cuong [13]. Phong et al. [30] investigated some picture fuzzy relations combinations. A fuzzy inference system on PFSs was proposed by Son et al. [42]. Akram et al. [3] examined the q-Rung PFG. Zuo et al. [51] gave some operations on PFG, i.e., union, joint, and Cartesian products. Meanwhile, Xiao et al. [47] studied regular PFGs and Garg [19] studied some picture fuzzy aggregation operations. Khan et al. [22] introduced bipolar PFGs. Amanathulla et al. [5] initiated the concept of balanced PFGs. An approach to decision-making via picture fuzzy soft graphs was introduced by Chellamani et al. [12]. Certain operations on picture fuzzy graph were studied by Shoaib et al. [41]. Picture fuzzy incidence graphs were introduced by Nazeer and Rashid [28].

On the advent of graph theory over algebraic graph theory, algebraic methods were used to study graphs. The main branch of algebraic graph theory is the spectral graph theory, which studies the characteristics of polynomial properties, eigenvalues, and eigenvectors related to graph matrices. Conceptual energy is related to the spectrum of a graph that plays an important role in recognizing patterns, modeling virus spread in computer networks, and securing personal data in databases. The energy of a graph is a useful tool in deciding which option to choose. When searching for the best option in decisions, the application of energy, because of the expression of the strength of the graph obtained from the selected vertices, is a great help in determining the desired option. This was a good incentive for researchers to study graph energies. Gutman [20] introduced the concept of a graph energy. He found this concept in the study of the electrons energy of specific molecules. In fact, the energy of a given molecular graph compared to the total energy of the electrons of a molecule is of interest to chemists. Obviously, the energy of a graph with all isolated vertices is zero, but for a complete graph with n vertices, it is \(2(n - 1)\). The energy of a graph is used for entropy [16], properties of proteins [18], Alzheimer’s disease [15], etc. The Laplacian energy (LE) of a graph is defined by Gutman and Zhou [21]. This was obtained from the sum of the absolute values of the differences of the mean degree vertices of a graph with its Laplacian eigenvalues. The LE is applied in image analysis [50], brain activity [9], etc. Pena and Rada analyzed the digraph energy [29]. The digraph skew energy was introduced by Adiga and Balakrishnan [2]. The skew Laplacian energy (SLE) of a digraph was defined by Adiga and Smitha [1]. Other types of energy were later introduced by researchers. The energy of pythagorean fuzzy graphs was proposed by Akram and Naz [4]. Certain notions of energy in single-valued neutrosophic graphs were introduced by Naz et al. [27].

The fuzzy graph (FG) energy was introduced by Anjali and Mathew [6]. The LE of an FG was defined by Sharbaf and Fayazi [39]. Basha and Kartheek [8] generalized the LE concept of an FG to the Laplacian energy (LE) of an IFG. Simultaneously, with the variety of FGs, different energies of fuzzy graphs were introduced. The limitation of previous definitions of energy in FGs and IFGs led us to define energy in a PFG. Since PFG is a good tool in fuzzy modeling of uncertain problems, for its degree of neutrality, it is a good incentive to examine the energy in it, especially in decisions based on the degree of neutrality. This shows that neutrality, as part of total energy, is effective in energy-based decisions.

In this research, with the aim of developing energy on a PFG and examining its properties, we sought to use the energy applications of this graph to solve real problems. We studied the PFG energy, PFG Laplacian energy and picture fuzzy digraph (PFDG) Laplacian energy. We also introduced the PFDG skew Laplacian energy. We compared three types of energy in varying degrees with examples. In addition, considering the decision making, a method was proposed to rank the available options using the PFG and its Laplacian energy/energy. Some of the energy boundaries were examined, and finally, the applications of energy were presented.

2 Preliminaries

In this section, we have an overview of the concepts we need in this article.

A graph \(G=(V,E)\) is a mathematical model consisting of a set of vertices V and a set of edges E, where each is an unordered pair of distinct vertices. If G is a graph with n vertices and m edges, Its adjacency matrix (AM) M is the \(n\times n\) matrix whose ij-th entry is the number of edges joining vertices i and j. The eigenvalues \(\lambda _i\), \(i=1,2,\ldots ,n\), of the AM of G are the eigenvalues of G. The spectrum \(\lbrace \lambda _1,\lambda _2,\ldots ,\lambda _n\rbrace\) of the AM of G is the spec(G). The eigenvalues of a graph satisfy the following relations:

$$\begin{aligned} \sum \limits _{i=1}^n\lambda _i=0,\quad \sum \limits _{i=1}^n \lambda _i^2=2m. \end{aligned}$$

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of the eigenvalues of G, i.e., \(E(G)=\sum _{i=1}^n \mid \lambda _i \mid\). The energy of a digraph D, denoted by E(D), is defined as the sum of the absolute values of the real part of eigenvalues of D, i.e., \(E(D)=\sum _{i=1}^n \mid Re(z_i)\mid\). A graph with all isolated vertices \(K_n^c\) has zero energy while the complete graph \(K_n\) with n vertices has \(2(n-1)\) energy. The LE of a graph G with n vertex and m edge is \(\mathrm{LE}(G)=\sum _{i=1}^n \mid \lambda _i-\frac{2m}{n} \mid\) where \(\lambda _i\) relates to the eigenvalues of LM of G. The skew energy of G is some of the absolute values [8] of eigenvalues. The skew adjacency matrix (SAM) of the digraph G is a matrix so that the (i, j)-entry of the SAM is \(+1\) if an edge is directed from the i-th vertex to the j-th vertex, vice versa the (j, i)-entry is \(-1\). If there is no directed edge between the nodes i and j, then, the respective matrix element is zero.

The SLE of the digraph G is defined as:

$$\begin{aligned} \mathrm{SLE}(G)=\sum _{i=1}^n\lambda _i^2, \end{aligned}$$

where \(\lambda _i\) refers to the eigenvalues of the LM of the digraph G.

Definition 1

[48] An FS \(\nu\) on a set V is defined through its membership function \(\nu :V\rightarrow [0,1]\), where \(\nu (x)\) represents the degree to which point \(x\in V\) belongs to the FS. The smallest and largest elements are the functions constantly equal to 0 and 1, respectively.

A fuzzy relation on a set V is a mapping \(\eta :V\times V\rightarrow [0,1]\) so that \(\eta (x,y)\le \min \lbrace \nu (x),\nu (y)\rbrace\) for all \(x,y\in V\). A fuzzy relation \(\eta\) is symmetric if \(\eta (x,y)=\eta (y,x)\), for all \(x,y\in V\).

Definition 2

[36] An FG \(Z=(V,\nu ,\eta )\) is a non-empty set V together with a pair of functions \(\nu :V\rightarrow [0,1]\) and \(\eta :V\times V\rightarrow [0,1]\) so that \(\eta (xy)\le \min \lbrace \nu (x),\nu (y)\rbrace\), for all \(x,y\in V\). Here \(\eta\) is a symmetric fuzzy relation on \(V\times V\).

Definition 3

[14] A PFS A on X is specified as follow

$$\begin{aligned} A=\lbrace (x,\mu _A(x),\eta _A(x),\nu _A(x))\mid x\in X\rbrace , \end{aligned}$$

so that

$$\begin{aligned} \mu _A,\eta _A,\nu _A:X\rightarrow [0,1], \end{aligned}$$

and

$$\begin{aligned} 0\le \mu _A(x)+\eta _A(x)+\nu _A(x)\le 1, \end{aligned}$$

where \(\mu _A(x)\), \(\eta _A(x)\) and \(\nu _A(x)\) are called the degrees of positive, neutral, and negative membership of x in A.

Definition 4

[51] A picture fuzzy relation B is a PFS of \(X\times Y\) as shown by

$$\begin{aligned} B=\lbrace (xy,\mu _A(xy),\eta _A(xy),\nu _A(xy)) \mid xy\in X\times Y\rbrace , \end{aligned}$$

so that

$$\begin{aligned} \mu _B,\eta _B,\nu _B:X\times Y\rightarrow [0,1], \end{aligned}$$

and

$$\begin{aligned} 0\le \mu _B(xy)+\eta _B(xy)+\nu _B(xy)\le 1\quad xy\in X\times Y. \end{aligned}$$

Definition 5

[51] A pair \(G=(A,B)\) is called a PFG on \(G^*=(V, {\mathcal {E}})\) where A is a PFS on V and B is a picture fuzzy relation on \({\mathcal {E}}\subseteq V\times V\) so that for each \(xy\in {\mathcal {E}}\)

$$\begin{aligned}&\mu _B(xy)\le \mu _A(x)\wedge \mu _A(y),\\&\eta _B(xy)\le \eta _A(x)\wedge \eta _A(y),\\&\nu _B(xy)\ge \nu _A(x)\vee \nu _A(y). \end{aligned}$$

Definition 6

[51] The degree of a vertex x on PFG \(G=(A,B)\) is specified as follow:

$$\begin{aligned} d(x)=(d_{\mu }(x),d_{\eta }(x),d_{\nu }(x)), \end{aligned}$$

where for every \(xy\in {\mathcal {E}}\)

$$\begin{aligned} d_{\mu }(x)=\sum _{x\ne y}\mu _B(xy),\quad d_{\eta }(x)=\sum _{x\ne y}\eta _B(xy),\quad d_{\nu }(x)=\sum _{x\ne y}\nu _B(xy). \end{aligned}$$

Example 1

Consider a PFG as shown in Fig. 1. The vertex degree z is (1, 3, 0.2, 0.8) while the vertex degree y is (0.8, 0.3, 0.4) .

Fig. 1
figure 1

A PFG

Full size image

Some notations are listed in the table of abbreviations.

3 Energy of Picture Fuzzy Graph

In this section, some of the most important PFG energies will be studied and their properties will be examined.

Definition 7

The AM M(G) of a PFG \(G=(A,B)\) is specified as a square matrix \(M(G)=[m_{ij}]\), \(m_{ij}=(\mu _B(x_ix_j),\eta _B(x_ix_j),\nu _B(x_ix_j))\), and this can be written as three matrices \(M(\mu _B(x_ix_j))\), \(M(\eta _B(x_ix_j))\), and \(M(\nu _B(x_ix_j))\), thus,

$$\begin{aligned} M(G)=(M(\mu _B(x_ix_j)),M(\eta _B(x_ix_j)),M(\nu _B(x_ix_j)) ). \end{aligned}$$

Definition 8

The energy of a PFG \(G=(A,B)\) is specified as the follow:

$$\begin{aligned} E(G)=(E(\mu _B(x_ix_j)),E(\eta _B(x_ix_j)),E(\nu _B(x_ix_j)) ), \end{aligned}$$

in other words,

$$\begin{aligned} E(G)=\Bigg (\sum _{i=1}^n\mid \alpha _i\mid ,\sum _{i=1}^n\mid \beta _i\mid ,\sum _{i=1}^n\mid \gamma _i\mid \Bigg ), \end{aligned}$$

where \(\alpha _i\), \(\beta _i\), and \(\gamma _i\) are eigenvalues of \(M(\mu _B(x_ix_j))\), \(M(\eta _B(x_ix_j))\), and \(M(\nu _B(x_ix_j))\), respectively.

Example 2

Consider a PFG \(G=(A,B)\) given in Fig. 2. The AM M is as follows:

$$\begin{aligned} M(G)=\left[ \begin{array}{ccccc} (0,0,0)&{}(0.2,0.2,0.5)&{}(0,0,0)&{}(0.2,0.4,0.3)&{}(0.3,0.2,0.3)\\ (0.2,0.2,0.5)&{}(0,0,0)&{}(0.3,0.2,0.4)&{}(0.3,0.2,0.4)&{}(0,0,0)\\ (0,0,0)&{}(0.3,0.2,0.4)&{}(0,0,0)&{}(0.2,0.3,0.3)&{}(0,0,0)\\ (0.2,0.4,0.3)&{}(0.3,0.2,0.4)&{}(0.2,0.3,0.3)&{}(0,0,0)&{}(0.2,0.3,0.4)\\ (0.3,0.2,0.3)&{}(0,0,0)&{}(0,0,0)&{}(0.2,0.3,0.4)&{}(0,0,0) \end{array}\right] . \end{aligned}$$
Fig. 2
figure 2

PFG \(G=(A,B)\)

Full size image

The AMs and eigenvalues of each degree of G are obtained as follows:

$$\begin{aligned}&M(\mu _B(x_ix_j))\\&\quad =\left[ \begin{array}{ccccc} 0&{}0.2&{}0&{}0.2&{}0.3\\ 0.2&{}0&{}0.3&{}0.3&{}0\\ 0&{}0.3&{}0&{}0.2&{}0\\ 0.2&{}0.3&{}0.2&{}0&{}0.2\\ 0.3&{}0&{}0&{}0.3&{}0 \end{array}\right] . \\&\mathrm{Spec}(\mu _B(x_ix_j))\\&\quad =\lbrace -0.434,-0.281,-0.211,0.218,0.708\rbrace , \\&M(\eta _B(x_ix_j))=\left[ \begin{array}{ccccc} 0&{}0.2&{}0&{}0.4&{}0.2\\ 0.2&{}0&{}0.2&{}0.2&{}0\\ 0&{}02&{}0&{}0.3&{}0\\ 0.4&{}0.2&{}0.3&{}0&{}0.3\\ 0.2&{}0&{}0&{}0.3&{}0 \end{array}\right] . \\&\mathrm{Spec}(\eta _B(x_ix_j))\\&\quad =\lbrace -0.527,-0.287,-0.096,0.127,0.784\rbrace , \\&M(\nu _B(x_ix_j))\\&\quad =\left[ \begin{array}{ccccc} 0&{}0.5&{}0&{}0.2&{}0.3\\ 0.5&{}0&{}0.4&{}0.4&{}0\\ 0&{}0.4&{}0&{}0.3&{}0\\ 0.3&{}0.4&{}0.3&{}0&{}0.4\\ 0.3&{}0&{}0&{}0.4&{}0 \end{array}\right] . \\&\mathrm{Spec}(\nu _B(x_ix_j))\\&\quad =\lbrace -0.716,-0.485,-0.056,0.182,1.075\rbrace . \end{aligned}$$

So, the energy of a PFG \(G=(A,B)\) is equal to \(E(G)=(1.852, 1.821, 2.514)\).

Theorem 1

Let \(G=(A,B)\) be a PFG and M(G) be its AM. If \(\alpha _1 \ge \alpha _2 \ge \dots \ge \alpha _n\), \(\beta _1 \ge \beta _2 \ge \dots \ge \beta _n\), and \(\gamma _1 \ge \gamma _2 \ge \dots \ge \gamma _n\), are the eigenvalues of \(M(\mu _B(x_ix_j))\), \(M(\eta _B(x_ix_j))\), and \(M(\nu _B(x_ix_j))\), respectively, then,

$$\begin{aligned} & \mathrm{(i)} \, \sum _{i=1}^n\alpha _i=0, \quad \sum _{i=1}^n\beta _i=0, \quad \sum _{i=1}^n\gamma _i=0 \\ & \mathrm{(ii)}\,\sum _{i=1}^n\alpha _i^2=2\sum _{1\le i<j\le n}(\mu _B(x_ix_j))^2,\quad \sum _{i=1}^n\beta _i^2=2\sum _{1\le i<j\le n}(\eta _B(x_ix_j))^2 , \\ & \sum _{i=1}^n\gamma _i^2=2\sum _{1\le i<j\le n}(\nu _B(x_ix_j))^2. \end{aligned}$$

Proof

(i) is held because M(G) is a symmetric matrix with zero trace.

(ii) According to the trace properties of a matrix, we have

$$\begin{aligned} \mathrm{tr}((M(\mu _\beta (x_i x_j)))^2) = \sum _{i=1}^n \alpha _i^2. \end{aligned}$$

On the other hand,

$$\begin{aligned}&\mathrm{tr}((M(\mu _\beta (x_i x_j)))^2) = \left( 0 + (\mu _\beta (x_1x_2))^2 + \dots + (\mu _\beta (x_1 x_n))^2 \right) \\&\qquad + \left( (\mu _\beta (x_2x_1))^2 + 0 + \dots + (\mu _\beta (x_2 x_n))^2 \right) \\&\qquad \vdots \\&\qquad + \left( (\mu _\beta (x_nx_1))^2 + (\mu _\beta (x_n x_2))^2 + \dots + 0 \right) \\&\quad = 2 \sum _{1\le i< j\le n } (\mu _\beta (x_i x_j))^2. \end{aligned}$$

Therefore,

$$\begin{aligned} \sum _{i=1}^n \alpha _i^2 = 2 \sum _{1\le i< j\le n } (\mu _\beta (x_i x_j))^2. \end{aligned}$$

Similarly, \(\sum _{i=1}^n\beta _i^2=2\sum _{1\le i<j\le n}(\eta _B(x_ix_j))^2\), and \(\sum _{i=1}^n\gamma _i^2=2\sum _{1\le i<j\le n}(\nu _B(x_ix_j))^2\). \(\square\)

Theorem 2

If \(G=(A,B)\) is a PFG on n vertices with the AM M(G), then,

$$\begin{aligned} \mathrm{(i)} \, \sqrt{2\sum _{1\le i<j\le n} (\mu _B(x_ix_j))^2+n(n-1)\mid \det (M(\mu _B(x_ix_j)))\mid ^{\frac{2}{n}}} &\le E(\mu _B(x_ix_j)) \\ & \le \sqrt{2n \sum _{1\le i<j \le n} (\mu _B(x_ix_j))^2},\\ \mathrm{(ii)} \, \sqrt{2\sum _{1\le i<j\le n} (\eta _B(x_ix_j))^2+n(n-1)\mid \det (M(\eta _B(x_ix_j)))\mid ^{\frac{2}{n}}} &\le E(\eta _B(x_ix_j)) \\ &\\le sqrt{2n \sum _{1\le i<j\le n}(\eta _B(x_ix_j))^2},\\ \mathrm{(iii)}\, \sqrt{2\sum _{1\le i<j\le n}(\nu _B(x_ix_j))^2+n(n-1)\mid \det (M(\nu _B(x_ix_j)))\mid ^{\frac{2}{n}}} &\le E(\nu _B(x_ix_j)) \\ & \le \sqrt{2n \sum _{1\le i<j\le n}(\nu _B(x_ix_j))^2}. \end{aligned}$$

Proof

According to the Cauchy–Schwarz inequality to the vectors \((1,1,\dots , 1)\) and \((\mid \alpha _1\mid , \mid \alpha _2\mid , \dots , \mid \alpha _n\mid )\) with n entries, we have

$$\begin{aligned}&\sum _{i=1}^n \mid \alpha _i\mid \le \sqrt{n} \sqrt{\sum _{i=1}^n \mid \alpha _i\mid ^2}, \end{aligned}$$
(1)
$$\begin{aligned}&\left( \sum _{i=1}^n \alpha _i \right) ^2 = \sum _{i=1}^n \mid \alpha _i\mid ^2 + 2 \sum _{1\le i< j\le n } \alpha _i \alpha _j. \end{aligned}$$
(2)

By comparing the coefficients of \(\alpha ^{n-2}\) in the characteristic

$$\begin{aligned}&\prod _{i=1}^{n} (\alpha - \alpha _i) = \mid M(G) - \alpha I \mid , \nonumber \\&\sum _{1\le i< j\le n } \alpha _i \alpha _j = - \sum _{1\le i< j\le n } (\mu _\beta (x_i x_j))^2. \end{aligned}$$
(3)

By placing (3) in (2), we have

$$\begin{aligned} \sum _{i=1}^n \mid \alpha _i\mid ^2 = 2 \sum _{1\le i< j\le n } (\mu _\beta (x_i x_j))^2. \end{aligned}$$
(4)

By placing (4) in (1), we have

$$\begin{aligned} \sum _{i=1}^n \mid \alpha _i\mid\le \sqrt{n} \sqrt{2 \sum _{1\le i< j\le n } (\mu _\beta (x_i x_j))^2} = \sqrt{2n \sum _{1\le i< j\le n } (\mu _\beta (x_i x_j))^2}. \end{aligned}$$

Hence,

$$\begin{aligned} E(\mu _\beta (x_i, x_j)) \le \sqrt{2n \sum _{1\le i< j\le n} (\mu _\beta (x_i x_j))^2}. \end{aligned}$$

For lower bound

$$\begin{aligned} \left( E(\mu _\beta (x_i x_j)) \right) ^2&= \left( \sum _{i=1}^n \mid \alpha _i\mid \right) ^2 = \sum _{i=1}^n \mid \alpha _i\mid ^2 + 2 \sum _{1\le i< j\le n } \mid \alpha _i \alpha _j\mid \\&= 2 \sum _{1\le i< j\le n } \left( \mu _\beta (x_i x_j) \right) ^2 + \dfrac{2n(n-1)}{2} \mathrm{AM} \{\mid \alpha _i \alpha _j\mid \}. \end{aligned}$$

Since \(AM\{\mid \alpha _i \alpha _j \mid \} \ge \mathrm{GM} \{\mid \alpha _i \alpha _j\mid \}\), \(1\le i< j \le n\), therefore,

$$\begin{aligned}&E(\mu _\beta (x_i x_j)) \\&\quad \ge \sqrt{2\sum _{1\le i< j\le n } \left( \mu _\beta (x_i x_j) \right) ^2 + n(n-1) \mathrm{GM} \{\mid \alpha _i\alpha _j\mid \} }. \end{aligned}$$

Also,

$$\begin{aligned}&\mathrm{GM} \{\mid \alpha _i\alpha _j\mid \} = \left( \prod _{1\le i< j \le n} \mid \alpha _i \alpha _j\mid \right) ^{\frac{2}{n(n-1)}} = \left( \prod _{i=1} ^n \mid \alpha _i \mid ^{n-1} \right) ^{\frac{2}{n(n-1)}} \\&\quad = \left( \prod _{i=1} ^n \mid \alpha _i \mid \right) ^{\frac{2}{n}} = \mid \det (M(\mu _\beta (x_i x_j)))\mid ^{\frac{2}{n}}. \end{aligned}$$

Therefore,

$$\begin{aligned} E(\mu _\beta (x_i x_j)) \ge \sqrt{2 \sum (\mu _\beta (x_i x_j))^2 + n(n-1) \mid \det (M(\mu _\beta (x_i x_j)))\mid ^{\frac{2}{n}}}. \end{aligned}$$

Hence,

$$\begin{aligned} & \sqrt{2 \sum _{1\le i< j\le n } (\mu _\beta (x_i x_j))^2 + n(n-1) \mid \det (M(\mu _\beta (x_i x_j)))\mid ^{\frac{2}{n}}} \\ & \quad \le E(\mu _\beta (x_i x_j)) \le \sqrt{2n \sum _{1\le i< j\le n } (\mu _\beta (x_i x_j))^2}. \end{aligned}$$

Similarly (ii) and (iii) are also established, of course, the upper bound is subjected to the following conditions \(n \le 2 \sum _{1\le i< j\le n } (\mu _\beta (x_i x_j))^2\), \(n \le 2 \sum _{1\le i< j\le n } (\eta _\beta (x_i x_j))^2\), and \(n \le 2 \sum _{1\le i< j\le n } (\gamma _\beta (x_i x_j))^2\). \(\square\)

Theorem 3

Let \(G=(A,B)\) be a PFG on n vertices. If \(n\le 2\sum _{1\le i<j\le n}(\mu _B(x_ix_j))^2\), \(n\le 2\sum _{1\le i<j\le n}(\eta _B(x_ix_j))^2\), and \(n\le 2\sum _{1\le i<j\le n}(\nu _B(x_ix_j))^2\), then

$$\begin{aligned}&\mathrm{(i)}\, E(\mu _B(x_ix_j)) \le \dfrac{2\sum _{1\le i<j\le n} (\mu _B(x_ix_j))^2}{n}\\&\quad + \sqrt{(n-1)\left\{ 2\sum _{1\le i<j\le n} (\mu _B(x_ix_j))^2 -\bigg ( \dfrac{2\sum _{1\le i<j\le n}(\mu _B(x_ix_j))^2}{n}\bigg )^2 \right\} }, \\&\mathrm{(ii)} \, E(\eta _B(x_ix_j)) \le \dfrac{2\sum _{1\le i<j\le n} (\eta _B(x_ix_j))^2}{n} \\&\quad + \sqrt{(n-1)\left\{ 2\sum _{1\le i<j\le n} (\eta _B(x_ix_j))^2- \bigg ( \dfrac{2\sum _{1\le i<j\le n} (\eta _B(x_ix_j))^2}{n} \bigg )^2\right\} }, \\&\mathrm{(iii)} \, E(\nu _B(x_ix_j)) \le \dfrac{2\sum _{1\le i<j\le n} (\nu _B(x_ix_j))^2}{n} \\&\quad + \sqrt{(n-1)\left\{ 2\sum _{1\le i<j\le n}(\nu _B(x_ix_j))^2-\bigg ( \dfrac{2\sum _{1\le i<j\le n}(\nu _B(x_ix_j))^2}{n}\bigg )^2\right\} }. \end{aligned}$$

Proof

If \(M=[m_{ij}]_{n \times n}\) is a symmetric matrix with zero trace, then, \(\alpha _{\max } \ge \dfrac{2 \sum _{1\le i< j\le n } m_{ij}}{n}\), where \(\alpha _{\max }\) is the maximum eigenvalue of M(G) and is the AM of a PFG G, then, \(\alpha _1 \ge \dfrac{2 \sum _{1\le i< j\le n }\mu _\beta (x_ix_j)}{n}\), where \(\alpha _1 \ge \alpha _2 \ge \dots \ge \alpha _n\). On the other hand, since

$$\begin{aligned}&\sum _{i=1}^n \alpha _i^2 = 2 \sum _{1\le i< j\le n } (\mu _\beta (x_ix_j))^2 \nonumber \\&\sum _{i=2}^n \alpha _i^2 = 2 \sum _{1\le i< j\le n } (\mu _\beta (x_ix_j))^2 - \alpha _1^2 . \end{aligned}$$
(5)

According to the Cauchy–Schwarz inequality to the vectors \((1,1,\dots , 1)\) and \((\mid \alpha _1\mid , \mid \alpha _2\mid , \dots , \mid \alpha _n\mid )\) with \(n-1\) entries, we have

$$\begin{aligned} E(\mu _\beta (x_ix_j)) - \alpha _1 = \sum _{i=2}^n \mid \alpha _i\mid \le \sqrt{(n-1) \sum _{i=2}^n \mid \alpha _i\mid ^2}. \end{aligned}$$
(6)

By placing (5) in (6), we get

$$\begin{aligned} E(\mu _\beta (x_ix_j))- \alpha _1 \le \sqrt{(n-1) \left( 2 \sum _{1\le i< j\le n } (\mu _\beta (x_ix_j))^2 - \alpha _1^2 \right) }, \nonumber \\ E(\mu _\beta (x_ix_j))\le \alpha _1 + \sqrt{(n-1) \left( 2 \sum _{1\le i< j\le n } (\mu _\beta (x_ix_j))^2 - \alpha _1^2 \right) }. \end{aligned}$$
(7)

Since the least value of the function \(F(x) = x+ \sqrt{(n-1) \left( 2 \sum _{1\le i< j\le n } (\mu _\beta (x_ix_j))^2 - x^2 \right) }\) is in the interval \(\left( \sqrt{\dfrac{2 \sum _{1\le i< j\le n } (\mu _\beta (x_ix_j))^2}{n}} , \sqrt{2 \sum _{1\le i< j\le n } (\mu _\beta (x_ix_j))^2} \right)\),

\(n \le 2 \sum _{1\le i< j\le n } (\mu _\beta (x_ix_j))^2\), \(1\le \dfrac{2 \sum _{1\le i< j\le n } (\mu _\beta (x_ix_j))^2}{n}\). So,

$$\begin{aligned} \sqrt{\dfrac{2 \sum _{1\le i< j\le n } (\mu _\beta (x_ix_j))^2}{n}} &\le \dfrac{2 \sum _{1\le i< j\le n } (\mu _\beta (x_ix_j))^2}{n} \\ &\le \dfrac{2 \sum _{1\le i< j\le n } \mu _\beta (x_ix_j)}{n}\\ &\le\alpha _1 \le \sqrt{2 \sum _{1\le i< j\le n } (\mu _\beta (x_ix_j))^2}. \end{aligned}$$

Therefore, the inequality (7) holds

$$\begin{aligned} E(\mu _\beta (x_ix_j)) \le \dfrac{2 \sum _{1\le i< j\le n } (\mu _\beta (x_ix_j))^2}{n}+ \sqrt{(n-1) \left\{ 2 \sum _{1\le i< j\le n } (\mu _\beta (x_ix_j))^2 - \left( \dfrac{2 \sum _{1\le i< j\le n } (\mu _\beta (x_ix_j))^2}{n} \right) ^2 \right\} }. \end{aligned}$$

Similarly, we can show (ii) and (iii). \(\square\)

Theorem 4

If \(G=(A,B)\) is a PFG on n vertices, then \(E(G)\le \dfrac{n}{2}(1+\sqrt{n})\).

Proof

Suppose that \(G=(A,B)\) is a PFG on n nodes and \(n \le 2 \sum _{1\le i< j\le n } (\mu _\beta (x_ix_j))^2 = 2 z\). It can be easily shown that \(f(z) = \dfrac{2z}{n} + \sqrt{(n-1) \left( 2 z - \left( \dfrac{2z}{n} \right) ^2 \right) }\) is maximized when \(z= \dfrac{n^2 + n \sqrt{n}}{4}\). By placing this value of z in Theorem 3; we have

$$\begin{aligned} E(\mu _\beta (x_ix_j)) \le \dfrac{n}{2} (1+\sqrt{n}). \end{aligned}$$

Similarly,

$$\begin{aligned}&E(\eta _\beta (x_ix_j)) \le \dfrac{n}{2}(1+\sqrt{n}), \\&E(\nu _\beta (x_ix_j)) \le \dfrac{n}{2}(1+\sqrt{n}). \end{aligned}$$

Therefore, \(E(G) \le \dfrac{n}{2} (1+\sqrt{n})\). \(\square\)

Definition 9

Let \(G=(A,B)\) be a PFG on n nodes. The degree matrix \(K(G)=[k_{ij}]\) of G is a \(n\times n\) diagonal matrix which is defined as:

$$\begin{aligned} k_{ij}=\left\{ \begin{array}{ll} d_G(x_i)&{}i=j,\\ 0&{}i\ne j. \end{array}\right. \end{aligned}$$

Definition 10

The LM of a PFG \(G=(A, B)\) is defined as \(L(G)=K(G)-M(G)\), where K(G) and M(G) are the degrees matrix and AM of a PFG, respectively.

Definition 11

The LE of PFG \(G=(A,B)\) is specified as the following:

$$\begin{aligned}&\mathrm{LE}(G)=\big (\mathrm{LE}(\mu _B(x_ix_j)),\mathrm{LE}(\eta _B(x_ix_j)),\mathrm{LE}(\nu _B(x_ix_j))\big ),\\&\mathrm{LE}(G)=\big (\sum _{i=1}^n \mid \varphi _i\mid ,\sum _{i=1}^n \mid \psi _i\mid ,\sum _{i=1}^n \mid \omega _i\mid \big ), \end{aligned}$$

where

$$\begin{aligned}&\varphi _i=\varphi _i^*-\dfrac{2\sum _{1\le i<j\le n}\mu _B(x_ix_j)}{n},\\&\psi _i=\psi _i^*-\dfrac{2\sum _{1\le i<j\le n}\eta _B(x_ix_j)}{n},\\&\omega _i=\omega _i^*-\dfrac{2\sum _{1\le i<j\le n}\nu _B(x_ix_j)}{n}. \end{aligned}$$

\(\varphi _i^*\), \(\psi _i^*\) and \(\omega _i^*\), \(i=1,2,\ldots ,n\) are the eigenvalues of \(L(\mu _B(x_ix_j))\), \(L(\eta _B(x_ix_j))\) and \(L(\nu _B(x_ix_j))\), respectively.

Example 3

Consider a PFG G given in Fig. 3.

Fig. 3
figure 3

PFG for LE

Full size image

The AM, degree matrix, and LM are as follows, respectively.

$$\begin{aligned}&M(G)=\left[ \begin{array}{ccccc} (0,0,0)&{}(0.3,0.1,0.4)&{}(0,0,0)&{}(0.2,0.2,0.6)&{}(0.3,0.2,0.3)\\ (0.3,0.1,0.4)&{}(0,0,0)&{}(0.3,0.1,0.5)&{}(0,0,0)&{}(0.3,0.3,0.4)\\ (0,0,0)&{}(0.3,0.1,0.5)&{}(0,0,0)&{}(0.2,0.3,0.5)&{}(0.2,0.2,0.5)\\ (0.2,0.2,0.6)&{}(0,0,0)&{}(0.2,0.3,0.5)&{}(0,0,0)&{}(0.2,0.1,0.6)\\ (0.3,0.2,0.3)&{}(0.3,0.3,0.4)&{}(0.2,0.2,0.5)&{}(0.2,0.1,0.6)&{}(0,0,0) \end{array}\right] . \\&K(G)=\left[ \begin{array}{ccccc} (1.2,0.5,0.9)&{} (0,0,0)&{} (0,0,0)&{} (0,0,0)&{} (0,0,0)\\ (0,0,0)&{} (0.9,0.5,1.3)&{} (0,0,0)&{} (0,0,0)&{} (0,0,0)\\ (0,0,0)&{} (0,0,0)&{} (0.7,0.6,1.5)&{} (0,0,0)&{} (0,0,0)\\ (0,0,0)&{} (0,0,0)&{} (0,0,0)&{} (0.6,0.6,1.7)&{} (0,0,0)\\ (0,0,0)&{} (0,0,0)&{} (0,0,0)&{} (0,0,0)&{} (1,0.8,1.8) \end{array} \right] . \\&L(G)=\left[ \begin{array}{ccccc} (1.2,0.5,0.9)&{}(-0.3,-0.1,-0.4)&{}(0,0,0)&{}(-0.2,-0.2,-0.6)&{}(-0.3,-0.2,-0.3)\\ (-0.3,-0.1,-0.4)&{}(0.9,0.5,1.3)&{}(-0.3,-0.1,-0.5)&{}(0,0,0)&{}(-0.3,-0.3,-0.4)\\ (0,0,0)&{}(-0.3,-0.1,-0.5)&{}(0,7,0,6,1.5)&{}(-0.2,-0.3,-0.5)&{}(-0.2,-0.2,-0.5)\\ (-0.2,-0.2,-0.6)&{}(0,0,0)&{}(-0.2,-0.3,-0.5)&{}(0.6,0.6,1.7)&{}(-0.2,-0.1,-0.6)\\ (-0.3,-0.2,-0.3)&{}(-0.3,-0.3,-0.4)&{}(-0.2,-0.2,-0.5)&{}(-0.2,-0.1,-0.6)&{}(1,0.8,1.8) \end{array}\right] . \end{aligned}$$

After computing we have \(\mathrm{LE}(\mu _B(x_ix_j))=1.681\), \(\mathrm{LE}(\eta _B(x_ix_j))=1.516\) and \(\mathrm{LE}(\nu _B(x_ix_j))=3.806\).

So, \(\mathrm{LE}(G)=(1.681,1.516.3.806)\).

Theorem 5

Let \(G=(A,B)\) be a PFG and L(G) be the LM of G. If \(\varphi _1^*\ge \varphi _2^*\ge \cdots \ge \varphi _n^*\), \(\psi _1^*\ge \psi _2^*\ge \cdots \ge \psi _n^*\) and \(\omega _1^*\ge \omega _2^*\ge \cdots \ge \omega _n^*\) are the eigenvalues of \(L(\mu _B(x_ix_j))\), \(L(\eta _B(x_ix_j))\) and \(L(\nu _B(x_ix_j))\), respectively, then,

$$\begin{aligned}&(i) \, \sum _{i=1}^n\varphi _i^*=2\sum _{1\le i<j\le n}\mu _B(x_ix_j), \\& \quad \sum _{i=1}^n\psi _i^*=2\sum _{1\le i<j\le n}\eta _B(x_ix_j), \\& \quad \sum _{i=1}^n\omega _i^*=2\sum _{1\le i<j\le n}\nu _B(x_ix_j). \\&\mathrm{(ii)}\,\sum _{i=1}^n{\varphi _i^*}^2 =2\sum _{1\le i<j \le n} (\mu _B(x_ix_j))^2+ \sum _{i=1}^n d_{\mu }^2(x_i), \\& \quad \sum _{i=1}^n {\psi _i^*}^2= 2\sum _{1\le i<j\le n} (\eta _B(x_ix_j))^2+ \sum _{i=1}^n d_{\eta }^2(x_i), \\& \quad \sum _{i=1}^n {\omega _i^*}^2= 2\sum _{1\le i<j\le n} (\nu _B(x_ix_j))^2+ \sum _{i=1}^n d_{\nu }^2(x_i). \end{aligned}$$

Proof

(i) Since L(G) is a symmetric matrix with non-negative Laplacian eigenvalues, therefore,

$$\begin{aligned} \sum _{i=1}^n \varphi _i^* = tr (L(G)) = \sum _{i=1}^n d_\mu (x_i) = 2 \sum _{1\le i< j\le n } \mu _\beta (x_i x_j). \end{aligned}$$

Then, \(\sum _{i=1}^n \varphi _i^* =2 \sum _{1\le i< j\le n } \mu _\beta (x_i x_j)\). Similarly, \(\sum _{i=1}^n \psi _i^* = 2 \sum _{1\le i< j\le n } \eta _\beta (x_i x_j)\) and \(\sum _{i=1}^n \omega _i^* = 2 \sum _{1\le i< j\le n } \nu _\beta (x_i x_j)\).

(ii) According to the trace properties of a matrix, we have

$$\begin{aligned} \mathrm{tr}((L(\mu _\beta (x_i x_j)))^2) = \sum _{i=1}^n \varphi _i^*, \end{aligned}$$

where

$$\begin{aligned}&\mathrm{tr} ((L(\mu _\beta (x_ix_j)))^2) = \left( d^2\mu (x_1) + \mu _\beta ^2(x_1x_2) + \dots + \mu _\beta ^2(x_1x_n) \right) \\&\qquad + \left( \mu _\beta ^2(x_2x_1) + d^2\mu (x_2) +\dots + \mu _\beta ^2(x_2x_n) \right) \\&\qquad + \dots + \left( \mu _\beta ^2(x_nx_1) + \mu _\beta ^2(x_nx_2) +\dots + d^2\mu (x_n) \right) \\&\quad = 2 \sum _{1\le i< j\le n } (\mu _\beta (x_i x_j))^2 + \sum _{i=1}^n d_\mu ^2(x_i). \end{aligned}$$

Therefore,

$$\begin{aligned} \sum _{i=1}^n {\varphi _i^*}^2 = 2 \sum _{1\le i< j\le n } (\mu _\beta (x_ix_j))^2 + \sum _{i=1}^n d^2_\mu (x_i). \end{aligned}$$

Similarly, the other equation are fixed. \(\square\)

Proposition 6

Let \(G=(A,B)\) be a PFG and L(G) be the LM of G. If \(\varphi _1^*\ge \varphi _2^*\ge \cdots \ge \varphi _n^*\), \(\psi _1^*\ge \psi _2^*\ge \cdots \ge \psi _n^*\), and \(\omega _1^*\ge \omega _2^*\ge \cdots \ge \omega _n^*\) are the eigenvalues of \(L(\mu _B(x_ix_j))\), \(L(\eta _B(x_ix_j))\), and \(L(\nu _B(x_ix_j))\), respectively, and

$$\begin{aligned}&\varphi _i=\varphi _i^*-\dfrac{2\sum _{1\le i< j\le n }\mu _B(x_ix_j)}{n},\\&\psi _i=\psi _i^*-\dfrac{2\sum _{1\le i< j\le n} \mu _B(x_ix_j)}{n},\\&\omega _i=\omega _i^*-\dfrac{2\sum _{1\le i< j\le n} \mu _B(x_ix_j)}{n}, \end{aligned}$$

then,

$$\begin{aligned}&\mathrm{(i)} \, \sum _{i=1}^n\varphi _i=0,\quad \sum _{i=1}^n\psi _i=0,~~~ \sum _{i=1}^n\omega _i=0,\\&\mathrm{(ii)} \, \sum _{i=1}^n \varphi _i^2=2M_{\mu },\quad \sum _{i=1}^n \psi _i^2=2M_{\eta },\quad \sum _{i=1}^n \omega _i^2=2M_{\nu }, \end{aligned}$$

where

$$\begin{aligned}&M_{\mu }=\sum _{1\le i<j\le n}\big (\mu _B(x_ix_j)\big )^2 +\dfrac{1}{2}\sum _{i=1}^n\Bigg (d_{\mu }(x_i)-\dfrac{2\sum _{1\le i<j\le n}\mu _B(x_ix_j)}{n}\Bigg )^2,\\&M_{\eta }=\sum _{1\le i<j\le n}\big (\eta _B(x_ix_j)\big )^2 +\dfrac{1}{2}\sum _{i=1}^n\Bigg (d_{\eta }(x_i)-\dfrac{2\sum _{1\le i<j\le n}\eta _B(x_ix_j)}{n}\Bigg )^2,\\&M_{\nu }=\sum _{1\le i<j\le n}\big (\nu _B(x_ix_j)\big )^2 +\dfrac{1}{2}\sum _{i=1}^n\Bigg (d_{\nu }(x_i)-\dfrac{2\sum _{1\le i<j\le n}\nu _B(x_ix_j)}{n}\Bigg )^2. \end{aligned}$$

Theorem 7

Let G be a PFG on n vertices and L(G) be the LM of G, then

$$\begin{aligned}&\mathrm{(i)}~\mathrm{LE}(\mu _B(x_ix_j))\\&\quad \le \sqrt{2n \sum _{1\le i<j\le n}\big (\mu _B(x_ix_j)\big )^2+n\sum _{i=1}^n\Bigg (d_{\mu }(x_i)-\dfrac{2\sum _{1\le i<j\le n}\mu _B(x_ix_j)}{n}\Bigg )^2},\\&\mathrm{(ii)}~\mathrm{LE}(\eta _B(x_ix_j))\\&\quad \le \sqrt{2n \sum _{1\le i<j\le n}\big (\eta _B(x_ix_j)\big )^2+n\sum _{i=1}^n\Bigg (d_{\eta }(x_i)-\dfrac{2\sum _{1\le i<j\le n}\eta _B(x_ix_j)}{n}\Bigg )^2},\\&\mathrm{(iii)}~\mathrm{LE}(\nu _B(x_ix_j))\\&\quad \le \sqrt{2n \sum _{1\le i<j\le n}\big (\nu _B(x_ix_j)\big )^2+n\sum _{i=1}^n\Bigg (d_{\nu }(x_i)-\dfrac{2\sum _{1\le i<j\le n}\nu _B(x_ix_j)}{n}\Bigg )^2}. \end{aligned}$$

Proof

According to the Cauchy–Schwarz inequality to the n numbers \(1,1,\ldots ,1\) and \(\mid \varphi _1\mid ,\mid \varphi _2\mid ,\ldots ,\mid \varphi _n\mid\), we have

$$\begin{aligned}&\sum _{i=1}^n\mid \varphi _i\mid \le \sqrt{n}\sqrt{\sum _{i=1}^n\mid \varphi _i\mid ^2},\\&\mathrm{LE}(\mu _B(x_ix_j))\le \sqrt{n}\sqrt{2M_{\mu }}=\sqrt{2nM_{\mu }}, \end{aligned}$$

because

$$\begin{aligned} M_{\mu }= 2n \sum _{1\le i<j\le n}\big (\mu _B(x_ix_j)\big )^2 +\dfrac{1}{2}\sum _{i=1}^n\bigg (d_{\mu }(x_i)-\dfrac{2\sum _{1\le i<j\le n}\mu _B(x_ix_j)}{n}\bigg )^2. \end{aligned}$$

So, we have

$$\begin{aligned}&\mathrm{LE}(\mu _B(x_ix_j))\\&\quad \le \sqrt{2n \sum _{1\le i<j\le n}\big (\mu _B(x_ix_j)\big )^2+n\sum _{i=1}^n\bigg (d_{\mu }(x_i)-\dfrac{2\sum _{1\le i<j\le n}\mu _B(x_ix_j)}{n}\bigg )^2}. \end{aligned}$$

Similarly, we can show other inequalities. \(\square\)

Theorem 8

Let G be a PFG on n vertices and L(G) be the LM of G. Then,

$$\begin{aligned}&\mathrm{(i)}~\mathrm{LE}(\mu _B(x_ix_j))\\&\quad \ge 2\sqrt{ \sum _{1\le i<j\le n}\big (\mu _B(x_ix_j)\big )^2+\dfrac{1}{2}\sum _{i=1}^n\Bigg (d_{\mu }(x_i)-\dfrac{2\sum _{1\le i<j\le n}\mu _B(x_ix_j)}{n}\Bigg )^2},\\&\mathrm{(ii)}~\mathrm{LE}(\eta _B(x_ix_j))\\&\quad \ge 2\sqrt{ \sum _{1\le i<j\le n}\big (\eta _B(x_ix_j)\big )^2+\dfrac{1}{2}\sum _{i=1}^n\Bigg (d_{\eta }(x_i)-\dfrac{2\sum _{1\le i<j\le n}\eta _B(x_ix_j)}{n}\Bigg )^2},\\&\mathrm{(iii)}~\mathrm{LE}(\nu _B(x_ix_j))\\&\quad \ge 2\sqrt{\sum _{1\le i<j\le n}\big (\nu _B(x_ix_j)\big )^2+\dfrac{1}{2}\sum _{i=1}^n\Bigg (d_{\nu }(x_i)-\dfrac{2\sum _{1\le i<j\le n}\nu _B(x_ix_j)}{n}\Bigg )^2}. \end{aligned}$$

Proof

According to the assumptions we have

$$\begin{aligned}&\big (\sum _{i=1}^n\mid \varphi _i\mid \big )^2=\sum _{i=1}^n\mid \varphi _i\mid ^2+2\sum _{1\le i<j\le j}\mid \varphi _i\varphi _j\mid \ge 4M_{\mu },\\&\mathrm{LE}(\mu _B(x_ix_j))\ge 2\sqrt{M_{\mu }}, \end{aligned}$$

which will be demonstrated with situation (i). Similarly, we can prove another inequality. \(\square\)

Theorem 9

Let G be a PFG on n vertices and L(G) be the LM of G. Then,

$$\begin{aligned}&\mathrm{(i)}\, \mathrm{LE}(\mu _\beta (x_i x_j)) \le \mid \varphi _i\mid \\&\quad + \sqrt{(n-1) \left( 2 \sum _{1\le i< j\le n } (\mu _\beta (x_ix_j))^2 + \sum _{i=1}^n \left( d_\mu (x_i) - \dfrac{2 \sum _{1\le i< j\le n } \mu _\beta (x_ix_j)}{n} \right) ^2 - \varphi _i^2 \right) }, \\&\mathrm{(ii)}\, \mathrm{LE}(\eta _\beta (x_i x_j)) \le \mid \psi _i\mid \\&\quad + \sqrt{(n-1) \left( 2 \sum _{1\le i< j\le n } (\eta _\beta (x_ix_j))^2 + \sum _{i=1}^n \left( d_\eta (x_i) - \dfrac{2 \sum _{1\le i< j\le n } \eta _\beta (x_ix_j)}{n} \right) ^2 - \psi _i^2 \right) },\\&\mathrm{(iii)}\, \mathrm{LE}(\nu _\beta (x_i x_j)) \le \mid \omega _i\mid \\&\quad + \sqrt{(n-1) \left( 2 \sum _{1\le i< j\le n } (\nu _\beta (x_ix_j))^2 + \sum _{i=1}^n \left( d_\nu (x_i) - \dfrac{2 \sum _{1\le i< j\le n } \nu _\beta (x_ix_j)}{n} \right) ^2 - \omega _i^2 \right) }. \end{aligned}$$

Proof

Using the Caushy–Schwarz inequality, we get

$$\begin{aligned}&\sum _{i=1}^n \mid \varphi _i\mid \le \sqrt{n \sum _{i=1}^n \mid \varphi _i\mid ^2}, \\&\sum _{i=2}^n \mid \varphi _i\mid \le \sqrt{(n-1) \sum _{i=2}^n \mid \varphi _i\mid ^2}, \\&\mathrm{LE}(\mu _\beta (x_ix_j)) - \mid \varphi _1\mid \le \sqrt{(n-1) (2 M_\mu -\varphi _1^2)},\\&\mathrm{LE}(\mu _\beta (x_ix_j)) \le \mid \varphi _1\mid + \sqrt{(n-1) (2 M_\mu -\varphi _1^2)}. \end{aligned}$$

Since \(M_\mu = \sum _{1\le i< j\le n } (\mu _\beta (x_ix_j))^2 + \dfrac{1}{2} \sum _{i=1}^n \left( d_\mu (x_i) - \dfrac{2 \sum _{1\le i< j\le n } \mu _\beta (x_ix_j)}{n} \right) ^2\), so

$$\begin{aligned}&\mathrm{LE}(\mu _\beta (x_ix_j)) \le \mid \varphi _1\mid \\&\quad + \sqrt{(n-1) \left( 2 \sum _{1\le i< j\le n } (\mu _\beta (x_ix_j))^2 + \sum _{i=1}^n \left( d_\mu (x_i) - \dfrac{2 \sum _{1\le i< j\le n } \mu _\beta (x_ix_j)}{n} \right) ^2 - \varphi _1^2 \right) }. \end{aligned}$$

Similarly, other inequalities are proved. \(\square\)

Remark

All results from the energy of a PFG are confirmed for a PFDG while eigenvalues are complex numbers we spot real part therefrom. Herein, suffice it to say the following definition.

Definition 12

Let \(D=(A,\overrightarrow{B})\) be a PFDG. The energy of D is defined as follow:

$$\begin{aligned} E(D)=\bigg (E(\mu _{\overrightarrow{B}}(x_ix_j)),E(\eta _{\overrightarrow{B}}(x_ix_j)),E(\nu _{\overrightarrow{B}}(x_ix_j))\bigg ) \end{aligned}$$

Here, we survey the LE of the PFDGs.

Definition 13

The out-degree matrix of a PFG is shown with symbol \(K^+(D)=[k_{ij}]\) and it is a \(n\times n\) diagonal matrix defined as

$$\begin{aligned} k_{ij}=\left\{ \begin{array}{ll} d^+_D(x_i)&{}i=j,\\ 0&{}i\ne j. \end{array}\right. \end{aligned}$$

Definition 14

The LM of PFDG \(D=(A,\overrightarrow{B})\) is defined as follow:

$$\begin{aligned}&L(D)=\Bigg (L(\mu _{\overrightarrow{B}}(x_ix_j)),L(\eta _{\overrightarrow{B}}(x_ix_j)),L(\nu _{\overrightarrow{B}}(x_ix_j))\Bigg )\\&L(D)=K^+(D)-M(D) \end{aligned}$$

where \(K^+(D)\) and M(D) are the out-degree matrix and AM of D, respectively.

Definition 15

The LE of a PFDG \(D=( A,\overrightarrow{B})\) is specified as follow:

$$\begin{aligned}&\mathrm{LE}(D)=\bigg (\mathrm{LE}(\mu _{\overrightarrow{B}}(x_ix_j)),\mathrm{LE}(\eta _{\overrightarrow{B}}(x_ix_j)),\mathrm{LE}(\nu _{\overrightarrow{B}}(x_ix_j))\bigg ),\\&\mathrm{LE}(D)=\bigg (\sum _{i=1}^n\mid t_i\mid ,\sum _{i=1}^n\mid w_i\mid ,\sum _{i=1}^n\mid z_i\mid \bigg ), \end{aligned}$$

where

$$\begin{aligned}&t_i=t_i^*-\dfrac{2\sum _{1\le i<j\le n}\mu _{\overrightarrow{B}}(x_ix_j)}{n},\\&w_i=w_i^*-\dfrac{2\sum _{1\le i<j\le n}\eta _{\overrightarrow{B}}(x_ix_j)}{n},\\&z_i=z_i^*-\dfrac{2\sum _{1\le i<j\le n}\nu _{\overrightarrow{B}}(x_ix_j)}{n}, \end{aligned}$$

\(t_i^*\), \(w_i^*\), and \(z_i^*\), are real part the eigenvalues of \(L\big (\mu _{\overrightarrow{B}}(x_ix_j)\big )\), \(L\big (\eta _{\overrightarrow{B}}(x_ix_j)\big )\) and \(L\big (\nu _{\overrightarrow{B}}(x_ix_j)\big )\), respectively.

Example 4

Consider a PFDG \(D=( A,\overrightarrow{B})\) which is given in Fig. 4.

Fig. 4
figure 4

A PFDG

Full size image
$$\begin{aligned}&M(D)=\left[ \begin{array}{ccccc} (0,0,0)&{}(0.6,0.1,0.3)&{}(0.5,0.2,0.3)&{}(0,0,0)&{}(0.3,0.2,0.4)\\ (0.5,0.1,0.4)&{}(0,0,0)&{}(0.4,0.1,0.3)&{}(0.6,0.1,0.3)&{}(0,0,0)\\ (0,0,0)&{}(0.5,0.1,0.4)&{}(0,0,0)&{}(0.4,0.1,0.4)&{}(0.3,0.2,0.4)\\ (0.5,0.1,0.4)&{}(0,0,0)&{}(0.5,0.1,0.3)&{}(0,0,0)&{}(0.4,0.1,0.5)\\ (0.4,0.1,0.3)&{}(0.4,0.1,0.4)&{}(0,0,0)&{}(0.3,0.1,0.4)&{}(0,0,0) \end{array}\right] . \\&K^+(D)=\left[ \begin{array}{ccccc} (1.4,0.5,1)&{}(0,0,0)&{}(0,0,0)&{}(0,0,0)&{}(0,0,0)\\ (0,0,0)&{}(1.5,0.3,1)&{}(0,0,0)&{}(0,0,0)&{}(0,0,0)\\ (0,0,0)&{}(0,0,0)&{}(1.2,0.4,1.2)&{}(0,0,0)&{}(0,0,0)\\ (0,0,0)&{}(0,0,0)&{}(0,0,0)&{}(1.4,0.3,1.2)&{}(0,0,0)\\ (0,0,0)&{}(0,0,0)&{}(0,0,0)&{}(0,0,0)&{}(1.1,0.3,1.1) \end{array}\right] . \\&L(D)=\left[ \begin{array}{ccccc} (1.4,0.5,1)&{}(-0.6,-0.1,-0.3)&{}(-0.5,-0.2,-0.3)&{}(0,0,0)&{}(-0.3,-0.2,-0.4)\\ (-0.5,-0.1,-0.4)&{}(1.5,0.3,1)&{}(-0.4,-0.1,-0.3)&{}(-0.6,-0.1,-0.3)&{}(0,0,0)\\ (0,0,0)&{}(-0.5,-0.1,-0.4)&{}(1.2,0.4,1.2)&{}(-0.4,-0.1,-0.4)&{}(-0.3,-0.2,-0.4)\\ (-0.5,-0.1,-0.4)&{}(0,0,0)&{}(-0.5,-0.1,-0.3)&{}(1.4,0.3,1.2)&{}(-0.4,-0.1,-0.5)\\ (-0.4,-0.1,-0.3)&{}(-0.4,-0.1,-0.4)&{}(0,0,0)&{}(-0.3,-0.1,-0.4)&{}(1.1,0.3,1.1) \end{array}\right] .\\&\mathrm{spec}(\mu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 1.332,-0.565+0.449i,-0.565-0.449i,-0.101\\&\qquad +0.274i,-0.101-0.274i\rbrace ,\\&\mathrm{spec}(\eta _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 0.355,-0.138+0.142i,-0.138-0.1421i,-0.039\\&\qquad +0.059i,-0.039-0.059i\rbrace ,\\&\mathrm{spec}(\nu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 1.095,-0.487+0.354i,-0.487-0.354i,\\&\qquad -0.610+0.188i,-0.610-0.188i\rbrace . \end{aligned}$$

So, the energy of PFDG is as follow:

$$\begin{aligned}&E(D)=\bigg ( E(\mu _{\overrightarrow{B}}(x_ix_j)),E(\eta _{\overrightarrow{B}}(x_ix_j)),E(\nu _{\overrightarrow{B}}(x_ix_j))\bigg )\\&E(D)=(2.664,0.709,4.144). \end{aligned}$$

On the other hand,

$$\begin{aligned}&\mathrm{Laplacian}\,\mathrm{spec}(\mu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 1.933+0.446i,1.933-0.446i,1.367\\&\qquad +0.237i,1.367-0.237i\rbrace ,\\&\mathrm{Laplacian}\,\mathrm{spec}(\eta _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 0.528+0.123i,0.528-0123i,0.371\\&\qquad +0.048i,0.371-0.048i\rbrace ,\\&\mathrm{Laplacian}\,\mathrm{spec}(\nu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 1.581+0.333i,1.581-0.333i,1.168\\&\qquad +0.175i,1.168-0.175i\rbrace . \end{aligned}$$

We have

$$\begin{aligned}&\mathrm{LE}(\mu _{\overrightarrow{B}}(x_ix_j))=2.532, \\&\mathrm{LE}(\eta _{\overrightarrow{B}}(x_ix_j))=0.754, \\ &\mathrm{LE}(\nu _{\overrightarrow{B}}(x_ix_j))=2.018. \end{aligned}$$

So, \(\mathrm{LE}(D)=(2.532,0.754,2.018)\).

One of the energies of a digraph is skew energy. Here, we define the SLE of a PFDG.

Definition 16

Let \(D^*=(V,\overrightarrow{{\mathcal {E}}})\) be a simple digraph without any loop and multiple arcs and \(D=(A,\overrightarrow{B})\) be a PFDG on \(D^*\).The SAM of D is the matrix in the form of

$$\begin{aligned} S(D)=\bigg (S(\mu _{\overrightarrow{B}}(x_ix_j)),S(\eta _{\overrightarrow{B}}(x_ix_j)),S(\nu _{\overrightarrow{B}}(x_ix_j)\bigg ) \end{aligned}$$

where

$$\begin{aligned} S(D)=[s_{ij}]= \left\{ \begin{array}{ll} \big (\mu _{\overrightarrow{B}}(x_ix_j),\eta _{\overrightarrow{B}}(x_ix_j),\nu _{\overrightarrow{B}}(x_ix_j)\big )&{}x_ix_j\in \overrightarrow{{\mathcal {E}}},\\ \big (-\mu _{\overrightarrow{B}}(x_ix_j),-\eta _{\overrightarrow{B}}(x_ix_j),-\nu _{\overrightarrow{B}}(x_ix_j)\big )&{}x_jx_i\in \overrightarrow{{\mathcal {E}}},\\ (0,0,0)&{} otherwise. \end{array}\right. \end{aligned}$$

Definition 17

The SLE of the PFDG \(D=(A,\overrightarrow{B})\) is specified as

$$\begin{aligned} \mathrm{SLE}(D)=\Bigg (\sum _{i=1}^np_i^2,\sum _{i=1}^nq_i^2,\sum _{i=1}^nr_i^2\Bigg ) \end{aligned}$$

where \(p_i\), \(q_i\) and \(r_i\) are eigenvalues of the LM \(L(D)=K^+(D)-S(D)\) of D.

Example 5

Consider a PFDG which is given in Fig. 5.

Fig. 5
figure 5

A PFDG \(G=(A, \mathbf {B})\)

Full size image

The SAM of D is as the follow:

$$\begin{aligned}&S(D)=\left[ \begin{array}{ccccc} (0,0,0)&{}(-0.3,-0.3,-0.2)&{}(-0.4,-0.1,-0.4)&{}(0,0,0)&{}(0.3,0.2,0.4)\\ (0.3,0.3,0.2)&{}(0,0,0)&{}(-0.3,-0.2,-0.4)&{}(0.4,0.2,0.3)&{}(0.3,0.2,0.5)\\ (0.4,0.1,0.4)&{}(0.3,0.2,0.4)&{}(0,0,0)&{}(-0.4,-0.1,-0.5)&{}(-0.3,-0.2,-0.4)\\ (0,0,0)&{}(-0.4,-0.2,-0.3)&{}(0.4,0.1,0.5)&{}(0,0,0)&{}(0,0,0)\\ (0,0,0)&{}(-0.3,-0.2,-0.5)&{}(0.3,0.2,0.4)&{}(0,0,0)&{}(0,0,0) \end{array}\right] . \\&K^+(D)=\left[ \begin{array}{ccccc} (0,0,0)&{}(0,0,0)&{}(0,0,0)&{}(0,0,0)&{}(0,0,0)\\ (0,0,0)&{}(1,0.7,1)&{}(0,0,0)&{}(0,0,0)&{}(0,0,0)\\ (0,0,0)&{}(0,0,0)&{}(0.7,0.3,0.8)&{}(0,0,0)&{}(0,0,0)\\ (0,0,0)&{}(0,0,0)&{}(0,0,0)&{}(0.4,0.1,0.5)&{}(0,0,0)\\ (0,0,0)&{}(0,0,0)&{}(0,0,0)&{}(0,0,0)&{}(0.3,0.2,0.4) \end{array}\right] . \\&L(D)=\left[ \begin{array}{ccccc} (0,0,0)&{}(0.3,0.3,0.2)&{}(0.4,0.1,0.4)&{}(0,0,0)&{}(0,0,0)\\ (-0.3,-0.3,-0.2)&{}(1,0.7,1)&{}(0.3,0.2,0.4)&{}(-0.4,-0.2,-0.3)&{}(-0.3,-0.2,-0.5)\\ (-0.4,-0.1,-0.4)&{}(-0.3,-0.2,-0.4)&{}(0.7,0.3,0.8)&{}(0.4,0.1,0.5)&{}(0.3,0.2,0.4)\\ (0,0,0)&{}(0.4,0.2,0.3)&{}(-0.4,-0.1,-0.5)&{}(0.4,0.1,0.5)&{}(0,0,0)\\ (0,0,0)&{}(0.3,0.2,0.5)&{}(-0.3,-0.2,-0.4)&{}(0,0,0)&{}(0.3,0.2,0.4) \end{array}\right] . \end{aligned}$$

The skew Laplacian spectrum of a PFDG D, given in Fig. 5, is

$$\begin{aligned}&\mathrm{skew}\,\mathrm{Laplacian}\,\mathrm{spec}(\mu _{\overrightarrow{B}}(x_ix_j)) \\&=\lbrace 0.625+0.726i,0.625-0.726i, 0.408\\&\quad +0.234i,0.408-0.234i,0.334\rbrace ,\\&\mathrm{skew}\,\mathrm{Laplacian}\,\mathrm{spec} (\eta _{\overrightarrow{B}}(x_ix_j))\\&=\lbrace 0.396+0.377i,0.396-0.377i, 0.197\\&\quad +0.123i,0.197-0123i,0.112\rbrace ,\\&\mathrm{skew}\,\mathrm{Laplacian}\,\mathrm{spec} (\nu _{\overrightarrow{B}}(x_ix_j))\\&=\lbrace 0.694+0.929i,0.694-0929i,0.541\\&\quad +0.147i,0.541-0.147i,0.229\rbrace . \end{aligned}$$

So, the SLE of D is

$$\begin{aligned} \mathrm{SLE}(D)=(1.225,0.405,1.602). \end{aligned}$$

Theorem 10

If D is a simple PFDG with vertex degrees of \(d(x_1),d(x_2),\ldots ,d(x_n)\) where \(d(x_i)=(d_{\mu }(x_i),d_{\eta }(x_i),d_{\nu }(x_i))\), then,

$$\begin{aligned} \mathrm{SLE}(D)= & {} \bigg (\sum _{i=1}^n d_{\mu }(x_i)(d_{\mu }(x_i)-1),\sum _{i=1}^n d_{\eta }(x_i)(d_{\eta }(x_i)-1),\\&\times \sum _{i=1}^n d_{\nu }(x_i)(d_{\nu }(x_i)-1)\bigg ). \end{aligned}$$

Proof

Let \(p_1,p_2,\ldots ,p_n\) be the eigenvalues of the LM \(L(\mu _{\overrightarrow{B}}(x_ix_j))=K^+(\mu _{\overrightarrow{B}}(x_ix_j))-S(\mu _{\overrightarrow{B}}(x_ix_j))\) where \(K^+(\mu _{\overrightarrow{B}}(x_ix_j))\) and \(S(\mu _{\overrightarrow{B}}(x_ix_j))\) are out-degree matrix and SAM of membership element of D. We have

$$\begin{aligned} \sum _{i=1}^np_i&= {\text {sum of determinants of all 1}}\\&\quad \times 1~{\text {principal submatrices of }} L(\mu _{\overrightarrow{B}}(x_ix_j)) ~\\&= {\text {trace of }}L(\mu _{\overrightarrow{B}}(x_ix_j)) \\&= \sum _{i=1}^n d_{\mu }(x_i) \\ \sum _{i<j}p_ip_j =&{\text {sum of determinants of all 2}}\\&\quad \times 2 {\text { principal submtrices of }} L(\mu _{\overrightarrow{B}}(x_ix_j))\\&= \sum _{i<j}\mid \begin{array}{ll} d_{\mu }(x_i)&{}-\mu _{ij}\\ -\mu _{ji}&{}d_{\mu }(x_j) \end{array}\mid =\sum _{i<j}d_{\mu }(x_i)d_{\mu }(x_j)-\mu _{ij}\mu _{ji} \\&= \sum _{i<j}\bigg (d_{\mu }(x_i)d_{\mu }(x_j)+\mu _{ij}^2\bigg ). \end{aligned}$$

Since \(\mu _{ij}^2=\mid \mu _{ij}\mid\) for \(i<j\), so,

$$\begin{aligned} \sum _{i\ne j}p_ip_j =&2\sum _{i<j}p_ip_j = \sum _{i\ne j}d_{\mu }(x_i)d_{\mu }(x_j)+\sum _{i\ne j}\mid a_{ij}\mid \\ =&\sum _{i\ne j}d_{\mu }(x_i)d_{\mu }(x_j)+\sum _{i=1}^nd_{\mu }(x_i). \end{aligned}$$

Thus,

$$\begin{aligned} \mathrm{SLE}(\mu _{\overrightarrow{B}}(x_ix_j)) =&\sum _{i=1}^np_i^2=\bigg (\sum _{i=1}^np_i\bigg )^2-\sum _{i\ne j}p_ip_j\\ =&\bigg (\sum _{i=1}^nd_{\mu }(x_i)\bigg )^2\\&\quad -\bigg [\sum _{i\ne j} d_{\mu }(x_i)d_{\mu }(x_j)+\sum _{i=1}^nd_{\mu }(x_i)\bigg ]\\ =&\sum _{i=1}^n\big ( d_{\mu }(x_i)\big )^2-\sum _{i=1}^nd_{\mu }(x_i)\\ =&\sum _{i=1}^nd_{\mu }(x_i)\big ( d_{\mu }(x_i)-1\big ). \end{aligned}$$

Similarly, we can show that

$$\begin{aligned}&\mathrm{SLE}(\eta _{\overrightarrow{B}}(x_ix_j))=\sum _{i=1}^nd_{\eta }(x_i)\big ( d_{\eta }(x_i)-1\big ),\\&\mathrm{SLE}(\nu _{\overrightarrow{B}}(x_ix_j))=\sum _{i=1}^nd_{\nu }(x_i)\big ( d_{\nu }(x_i)-1\big ). \end{aligned}$$

This completes the proof. \(\square\)

4 Applications

4.1 Selecting a Suitable Construction Plan in Different Climatic Zone

In studying the characteristics of housing and architecture, not only historical and economic factors should be carefully studied, but also the impact of natural geographical factors on housing architecture in different climatic regions should be considered. Architects have traditionally paid attention to nature and the surrounding environment and have maintained a peaceful relationship with nature in the traditional way, and in architecture, they have always paid attention to the environment and nature around them. The house in the warm and dry climate had dense, introverted plans facing the central courtyard. These houses showed all their beauties to the family. Traditional houses in the cold climate were also built as a central courtyard and had dense and compact plans. In temperate and humid climates, due to the high humidity in traditional houses, the roofs are sloping and the porches around the building have been considered as an important space in the house. The shape of the building is also unlike the hot and dry climate houses which have a central courtyard. This area has been designed extroverted and the view of the windows of the house and the yard has been around it. A construction holding company is considering new plans for housing based on advisers’ opinions. These plans include:

  1. (1)

    Access to public services.

  2. (2)

    Architectural style.

  3. (3)

    Infrastructure networks.

  4. (4)

    Auxiliary facilities.

  5. (5)

    Smart systems.

These plans and relations are more important for housing and employment in climate regions. In relation to different climatic regions, carrying out the above plans for the construction company is associated with avail, loss, or neutral states. Doing any of the above projects is directly or indirectly related to other projects. We showed the interactions of the above designs in terms of their location in different climatic points in the form of a PFDG. Therefore, we consider a PFDG at three climate regions of cold, temperate, and warm. Afterward, the energy, LE, and SLE of a PFDG are calculated for each of these regions.

The above plans are considered for the cold regions as a PFDG \(D_1\) shown in Fig. 6. The picture fuzzy numbers for the vertices are the percentage of the impact of each of the construction company’s plans for construction in different climatic zones. These percentages are determined by the opinion of consultants and based on the experiences and statistics available in the company. The relationships between the vertices are the edges that represent the most connection between the plans of the company, and therefore the edges are all strong.

Fig. 6
figure 6

PFDG \(D_1\)

Full size image

In this digraph, the plan degree \(x_3\) is (0.5, 0.2, 0.1). i.e. in cold regions in which for applying this plan \(50\%\) is avail, \(20\%\) is neutral and \(10\%\) is loss, too, and the degree of impact of infrastructure networks on intelligent systems \((x_3x_5)\) is (0.3, 0.2, 0.3). i.e. \(30\%\) is avail, \(20\%\) is neutral and \(40\%\) is loss.

We calculated energy, LE, and SLE.

$$\begin{aligned}&\mathrm{spec}(\mu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace -0.1651+0.2859i, -0.1651-0.2859i, 0.3301,0,0\rbrace ,\\&\mathrm{spec}(\eta _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 0.2,-0.1+0.1732i,-0.1-1732i,0,0\rbrace ,\\&\mathrm{spec}(\nu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace -0.1778+0.3080i,-0.1778-0.3080i,0.3557,0,0\rbrace . \end{aligned}$$

So, \(E(D_1)=(0.6603,0.4,0.7113)\). Also,

$$\begin{aligned}&\mathrm{Laplacian}\,\mathrm{spec}(\mu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 1.4,0.5+0.28282i,0.5-0.2828i,0.4,0\rbrace ,\\&\mathrm{Laplacian}\,\mathrm{spec}(\eta _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 0.5,0.3+0.1732i,0.3-0.1732i,0.1,0\rbrace ,\\&\mathrm{Laplacian}\,\mathrm{spec}(\nu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 1,0.55+0.2958i,0.55-0.2958i,0.4,0\rbrace . \end{aligned}$$

So, \(LE(D_1)=(1.8,0.72,1.18)\).

$$\begin{aligned}&\mathrm{skew}\,Lap\mathrm{Laplacian}\,lacian~\mathrm{spec}(\mu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 0.8239+0.6484i,0.8239-0.6434i,0.3829+0.6706i,\\&\quad 0.3829-0.6706i,0.3862\rbrace ,\\&\mathrm{skew}\,\mathrm{Laplacian}\,\mathrm{spec}(\eta _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 0.2378+0.3792i,0.2378-0.3792i,0.2853+0.2145i,\\&\quad 0.2853-0.2145i,0.1534\rbrace ,\\&\mathrm{skew}\,\mathrm{Laplacian}\,\mathrm{spec}(\nu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 0.4378+0.7807i,0.4378-0.7807i, 0.5729+0.3604i,\\&\quad 0.5729-0.3604i,0.4786\rbrace . \end{aligned}$$

So, the SLE \(D_1\) is equal to

$$\begin{aligned} \mathrm{SLE}(D_1)=(1.7990,0.2994,1.2688). \end{aligned}$$

The PFDG \(D_2\) shown in Fig. 7 is considered for temperate regions.

Fig. 7
figure 7

PFDG \(D_2\)

Full size image

After calculation, we have

$$\begin{aligned}&\mathrm{spec}(\mu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace -0.1817+0.3147i,-0.1817-0.3147i,0.3624,0,0\rbrace ,\\&\mathrm{spec}(\eta _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 0.1,-0.05+0.0866i,-0.05-0.0866i,0,0\rbrace ,\\&\mathrm{spec}(\nu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 0.3302,-0.1651+0.2859i,-0.1651-0.2859i,0,0\rbrace . \end{aligned}$$

So, \(E(D_2)=(0.7258,0.2,0.6604)\).

$$\begin{aligned}&\mathrm{Laplacian}\,\mathrm{spec}(\mu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 1.5,0.55+0.3122i,1.5-0.3122i,0.4,0\rbrace ,\\&\mathrm{Laplacian}\,\mathrm{spec}(\eta _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 0.5,0.2,0.15+0.0866i,0.15-0.0866i,0\rbrace ,\\&\mathrm{Laplacian}\,\mathrm{spec}(\nu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 0.7,0.5+0.2828i,0.5-0.2828i,0.2,0\rbrace . \end{aligned}$$

So, \(LE(D_2)=(2.04,0.72,1.14)\).

$$\begin{aligned}&\mathrm{skew}\,\mathrm{Laplacian}\,\mathrm{spec}(\mu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 0.8636+0.6533i, 0.8636-0.6533i,0.4128+0.7207i,\\&\quad 0.4128-0.7207i,0.4470\rbrace \\&\mathrm{skew}\,\mathrm{Laplacian}\,\mathrm{spec}(\eta _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 0.2888+0.2496i,0.2888-0.2496i,0.1371+2285i,\\&\quad 0.1371-2285i,0.1483\rbrace \\&\mathrm{skew}\,\mathrm{Laplacian}\,\mathrm{spec}(\nu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 0.3445+0.6250i,0.3445-0.6250i,0.4612+0.2595i,\\&\quad 0.4612-0.2595i,0.2885\rbrace . \end{aligned}$$

So, the SLE \(D_2\) is equal to

$$\begin{aligned} \mathrm{SLE}(D_2)=(2.0322,0.2263,0.7460). \end{aligned}$$

For warm regions, a PFDG \(D_3\) shown in Fig. 8 is considered.

Fig. 8
figure 8

PFDG \(D_3\)

Full size image

After calculation, we have

$$\begin{aligned}&\mathrm{spec}(\mu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 0.4309,-0.2154+0.3731i,-0.2154-0.3731i,0,0\rbrace ,\\&\mathrm{spec}(\eta _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace -0.0629+0.1091i,-0.0629-0.1091i,0.1260,0,0\rbrace ,\\&\mathrm{spec}(\nu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 0.3175,-0.1587+0.2749i,-0.1587-0.2749i,0,0\rbrace . \end{aligned}$$

So, \(E(D_3)=(0.8617,0.2518,0.6349)\).

$$\begin{aligned}&\mathrm{Laplacian}\,\mathrm{spec}(\mu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 1.4,0.8670,0.2165+0.3710i,0.2165-0.3710i,0.4\rbrace ,\\&\mathrm{Laplacian}\,\mathrm{spec}(\eta _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 0.3,0.2+0.1i,0.2-0.1i,0.1,0\rbrace ,\\&\mathrm{Laplacian}\,\mathrm{spec}(\nu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 1,0.5+0.2646i,0.5-0.2646i,0.4,0\rbrace . \end{aligned}$$

So, \(LE(D_3)=(2.154,0.4,1.16)\).

$$\begin{aligned}&\mathrm{skew}\,\mathrm{Laplacian}\,\mathrm{spec}(\mu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 0.9801+2.1042i, 0.9801-2.1042i,0.3650+0.6963i,\\&\quad 0.3650-0.6963i,0.4097\rbrace ,\\ \mathrm{Laplacian}\,&\quad =\lbrace 0.1540+0.2661i, 0.1540-0.2661i,0.1649+0.1151i\\&\quad 0.1649-0.1151i,0.1621\rbrace ,\\&\mathrm{skew}\,\mathrm{Laplacian}\,\mathrm{spec}(\nu _{\overrightarrow{B}}(x_ix_j))\\&\quad =\lbrace 0.3842+0.7014i,0.3842-0.7014i,0.6001+0.4569i ,\\&\quad 0.6001-0.4569i,0.4313\rbrace . \end{aligned}$$

So, the SLE \(D_3\) is equal to

$$\begin{aligned} \mathrm{SLE}(D_3)=(2.3555,0.1281,1.2015). \end{aligned}$$
Fig. 9
figure 9

Comparison of the energy of PFDGs

Full size image
Fig. 10
figure 10

Comparison of LEs of PFDGs

Full size image
Fig. 11
figure 11

Comparison of SLE of PFDGs

Full size image

The figures 9, 10 and 11 represent a comparison between the energy, LE, and SLE of applied plans in three regions corresponding to the avail, neutral, and loss memberships, respectively. From the above bar graphs, the energy, LE and SLE of avail membership for warm regions is high as compared to other regions. This shows that the above designs are more efficient in hot areas and more coherence between the designs can be considered. The energy, LE, and SLE of neutral membership for cold regions is high and the energy, LE, and SLE of loss membership for cold regions is high.

4.2 Choosing the Best Place to Start a Business

Everyone who works in business has heard that one of the most important factors in ensuring the profitability of a business is its location. Basic decision making in this field can transform a startup and lead to growing success. The size, layout, physical location and appearance of the workplace should all serve to develop your operations. The deeper you go into your business plan, the more realistic and tangible your budget for your property and business expenses will be, given the infrastructure, tax laws, facilities, regional laws, environmental regulations, and growth potential. In addition to physical businesses, this also affects Internet businesses.

Four different locations were considered for starting a new business, and a group of four consultants in finance, business, real estate and law were invited to decide on the appropriate location. Each expert made separate judgments based on their experiences and comparisons between the two different locations. These views were based on three factors: appropriate, neutral and inappropriate. The AMs related to picture fuzzy relations to each of the experts are as follows:

$$\begin{aligned}&M_1=\left[ \begin{array}{cccc} (0,0,0)&{}(0.4,0.1,0.3)&{}(0.4,0.1,0.3)&{}(0.3,0.1,0.4)\\ (0.3,0.1,0.4)&{}(0,0,0)&{}(0.5,0.1,0.2)&{}(0.5,0.1,0.3)\\ (0.3,0.1,0.3)&{}(0.4,0.1,0.4)&{}(0,0,0)&{}(0.6,0.1,0.2)\\ (0.4,0.1,0.3)&{}(0.4,0.1,0.2)&{}(0.5,0.1,0.3)&{}(0,0,0) \end{array}\right] , \\&M_2=\left[ \begin{array}{cccc} (0,0,0)&{}(0.4,0.1,0.3)&{}(0.4,0.1,0.3)&{}(0.5,0.1,0.2)\\ (0.6,0.1,0.2)&{}(0,0,0)&{}(0.5,0.1,0.3)&{}(0.5,0.1,0.3)\\ (0.5,0.2,0.3)&{}(0.5,0.1,0.3)&{}(0,0,0)&{}(0.5,0.1,0.4)\\ (0.4,0.1,0.3)&{}(0.7,0.1,0.2)&{}(0.4,0.1,0.4)&{}(0,0,0) \end{array}\right] , \\&M_3=\left[ \begin{array}{cccc} (0,0,0)&{}(0.4,0.1,0.2)&{}(0.3,0.1,0.2)&{}(0.5,0.2,0.3)\\ (0.5,0.1,0.3)&{}(0,0,0)&{}(0.3,0,0.2)&{}(0.5,0.1,0.3)\\ (0.4,0.1,0.3)&{}(0.4,0.1,0.3)&{}(0,0,0)&{}(0.4,0.1,0.3)\\ (0.4,0.1,0.3)&{}(0.5,0.1,0.4)&{}(0.3,0.1,0.4)&{}(0,0,0) \end{array}\right] , \\&M_4=\left[ \begin{array}{cccc} (0,0,0)&{}(0.6,0.1,0.3)&{}(0.5,0.1,0.3)&{}(0.6,0,0.3)\\ (0.5,0.1,0.3)&{}(0,0,0)&{}(0.5,0.1,0.3)&{}(0.6,0,0.3)\\ (0.4,0.2,0.4)&{}(0.3,0.1,0.4)&{}(0,0,0)&{}(0.5,0,0.4)\\ (0.5,0,0.4)&{}(0.5,0,0.4)&{}(0.4,0,0.3)&{}(0,0,0) \end{array}\right] . \end{aligned}$$

The PFDGs \(D_i\) corresponding to picture fuzzy preference relation (PFPR) given in matrices \(M_i\) are shown in Fig. 12\((i = 1, 2, 3, 4)\). The energy of each PFDG is calculated as:

$$\begin{aligned}&E(D_1)=(0.001,0.2,0.3) ,\quad E(D_2)=(0.001,0.1,0) , \\&E(D_3)=(0,0.2,0.001) , \quad E(D_4)=(0,0.1,-0.001). \end{aligned}$$

Then, the weight of each expert can be calculated as:

$$\begin{aligned}&w_i=( (w_{\mu })_i,(w_{\eta })_i,(w_r)_i) \\&\quad = \left( \frac{E(M_{\mu })_i}{\sum \limits _{j=1}^4E(M_{\mu })_j},\frac{E(M_{\eta })_i}{\sum \limits _{j=1}^4E(M_{\eta })_j},\frac{E(M_{r})_i}{\sum \limits _{j=1}^4E(M_r)_j}\right) \\&w_1=(0.5 ,0.33,1),\quad w_2=(0.5,0.16,0)\\&w_3=(0,0.33,0.0033),\quad w_4=(0,0.16,-0.0033). \end{aligned}$$

The collective picture fuzzy preference relation (PFPR), aggregated from the four PFPRs, is determined as:

$$\begin{aligned} M= & {} \sum \limits _{i=1}^4w_iM_i \\= & {} \left[ \begin{array}{cccc} (0,0,0)&{}(0.4,0.098,0.2)&{}(0.35,0.098,0.2)&{}(0.5,0.115,0.3003)\\ (0.55,0.098,0.3003)&{}(0,0,0)&{}(0.4,0.065,0.1997)&{}(0.5,0.066,0.3)\\ (0.45,0.13,0.2997)&{}(0.45,0.098,0.3)&{}(0,0,0)&{}(0.45,0.082,0.2993)\\ (0.4,0.082,0.2997)&{}(0.6,0.082,0.3993)&{}(0.15,0.082,0.4)&{}(0,0,0) \end{array}\right] . \end{aligned}$$
Fig. 12
figure 12

\(\mathrm{PFDG}_s\)

Full size image

A PFDG corresponding to a collective PFPRA above is drawn, as shown in Fig. 13. Then, under the \(\mu _{j_k} \ge 0.5~~(j,k=1,2,3,4)\) condition, a partial diagram is drawn, as shown in Fig. 14.

Fig. 13
figure 13

PFDG corresponding to PFPRA

Full size image
Fig. 14
figure 14

Partial PFDG of PFPRA

Full size image

Calculate the out-degrees \(d^+(x_i)~~(i=1,2,3,4)\) of all criteria in a partial PFDG as follow:

$$\begin{aligned}&d^+(x_1)=(1,05,0.213,0.6006),\quad d^+(x_2)=(0.5,0.066,0.3)\\&d^+(x_3)=(0,0,0),\quad d^+(x_4)=(0.6,0.082,0.3993). \end{aligned}$$

According to membership degrees of \(d^+(x_i)\,(i=1,2,3,4)\), we get the ranking of the places \(x_i\) as:

$$\begin{aligned} x_1 \succ x_4\succ x_2\succ x_3. \end{aligned}$$

Therefore, the best choice is \(x_1\).

The flowchart for the selection of the best choice in business is as follow:

figure a

5 Conclusions

The PFG can amplify flexibility and precision to model complex real-time problems better than an FG and IFG. They have several applications in many decision-making processes among solution choice, weather forecasting, prognosis risks in business, etc. In recent years, graph energy has been used in many fields. It is clear that the neutrality degree is associated with more flexibility for the graph energy. In this research, aiming at expanding the concept of energy on the PFG and using its results in modeling and solving the problems ahead, we studied some types of energy and their results in the PFG. We were able to show that neutrality, as part of total energy, is effective in energy-based decisions. Neutral energy is more pronounced in some types of energy and is not always very small compared to membership and non-membership degrees. We have obtained some properties and relations of lower and upper bounds of energy, LE, and SLE of PFG. Despite the energy fluctuations in different degrees of the PFG, no significant relationship was found between the degrees of energy. Finally, we presented applications of energy in decision-making based on superior choice. Examining the energy in the interval-valued intuitionistic fuzzy graph is an interval of the amount of our future work plans.