A Study on Linguistic Z-Graph and Its Application in Social Networks

Abstract

This paper presents a comprehensive study of the linguistic Z-graph, which is a novel framework designed to analyze linguistic structures within social networks. By integrating concepts from graph theory and linguistics, the linguistic Z-graph provides a detailed understanding of language dynamics in online communities. This study highlights the practical applications of linguistic Z-graphs in identifying central nodes within social networks, which are crucial for online businesses in market capture and information dissemination. Traditional methods for identifying central nodes rely on direct connections, but social network connections often exhibit uncertainty. This paper focuses on using fuzzy theory, particularly linguistic Z-graphs, to address this uncertainty, offering more detailed insights compared to fuzzy graphs. Our study introduces a new centrality measure using linguistic Z-graphs, enhancing our understanding of social network structures.

1. Introduction

Linguistic Z-numbers [1] provide a new way of expressing uncertainty and vagueness through the linguistic expressions of mathematical structures. They offer a powerful extension to traditional fuzzy numbers by integrating both uncertainty and reliability into a single framework. This comprehensive approach facilitates more accurate and reliable modeling, analysis, and decision making in environments characterized by complex uncertainties. Their use of linguistic terms [2] enhances interpretability and communication, making them particularly effective in collaborative and interdisciplinary settings.

Fuzzy graphs are used to represent uncertainty or vagueness in relationships between nodes or edges in a graph. Linguistic Z-graphs [3] extend the concept of fuzzy graphs by incorporating linguistic variables into the representation of uncertainty. Linguistic terms such as “very low”, “low”, “medium”, “high”, and “very high” are used to describe the degree of uncertainty associated with graph elements. Fuzzy graphs provide a quantitative representation of uncertainty but lack the linguistic granularity offered by linguistic variables [4].

In this study, we investigate the properties of linguistic Z-graphs, aiming to uncover the rules that govern word connections in language. The linguistic Z-graph visually represents these connections by linking related words. This research introduces an important class of uncertainty in graph theory and explores its properties. Additionally, examining centrality measures in social networks transcends academic interest. Centrality measures offer a unique perspective on the dynamics of influence, relationships, and information flow within a network. By studying these measures, we can identify key players, influencers, and potential communication pathways.

2. Literature Review

A graph is a powerful way to show connections between two items. The lines in the graph show how the things are connected, and the things themselves are shown as vertices. In fuzzy graphs, the lines use a number between 0 and 1 to show how strongly connected two things are. In 1973, Kaufmann [5] presented the concept of fuzzy graphs, and in 1975, Rosenfeld [6] expanded on it. Samanta et al. [7,8,9] provided several applications in fuzzy graphs. Mahapatra et al. [10,11,12,13,14,15] provided several applications and developed the coloring idea of fuzzy graphs. Gayathri et al. [16] extended connectivity concepts in DFIGs, introducing legal fuzzy incidence blocks (LFI-blocks), legal flow reduction nodes (LFR-nodes), and a DFIG-adapted Menger’s theorem. LFI-blocks, devoid of LFR-nodes, ensure uninterrupted flow between nodes via the widest legal di-paths. Influence graphs, representing natural flow networks with illegal flow conditions, are crucial in physical problems like networking and trafficking. Gayathri et al. [17] extended connectivity concepts in fuzzy influence graphs, examining fundamental properties such as fuzzy influence cycles and special influence vertices. Prabhath et al. [18] explored generalized cycle connectivity in fuzzy graphs, considering the strength of all cycles passing through vertex pairs. They discuss generalized cycle connectivity in fuzzy trees, fuzzy cycles, and complete fuzzy graphs, along with cyclic vertix-cuts and bridges. This study also introduces cyclically stable fuzzy graphs and establishes a condition for complete fuzzy graphs to achieve cyclic stability.

Linguistic Z-numbers extend the concept of Z-numbers, introduced by Zadeh [19], to incorporate linguistic terms for both the description of uncertainty and the reliability of information. This dual linguistic characterization allows for a more complex representation of expert knowledge, especially in situations involving subjective judgments and imprecise data. In the financial sector, linguistic Z-numbers [20] have been applied to model portfolio selection problems. By using linguistic-Z-number-weighted averaging operators, researchers have been able to reflect the various semantics of linguistic terms under different circumstances flexibly. This approach has been shown to increase the tendencies and capabilities of investors in the capital market, aiding them in managing their portfolios more efficiently.

The probabilistic linguistic Z number (PLZN) decision-making method [21,22,23] is another significant application. It effectively represents decision-making information and demonstrates its reliability, making it a valuable tool for handling complexity and uncertainty in real-life qualitative information indications [24,25,26]. The coloring of graphs is a very old issue. Fuzzy coloring was first proposed in fuzzy graphs by Samanta et al. [27]. Graph coloring was first presented by Hansen et al. [28]. The techniques of edge coloring and radio k coloring in fuzzy graphs were first presented by Mahapatra et al. [29,30,31]. Akram et al. [32,33,34,35] conducted a study on NGs proposed by Broumi, revealing a different definition of neutrosophic graphs incorporating concepts from both neutrosophic set theory and graph theory. The linguistic Z-graph was first proposed by Samanta et al. [3] using the concept of the linguistic Z-number.

A comparison of linguistic Z-graphs with existing graphs such as weighted graphs and fuzzy graphs is shown in

Table 1. Comparison of features of linguistic Z-graph with existing techniques.

Feature	Weighted Graphs	Fuzzy Graphs	Linguistic Z-Graph
Representation of Uncertainty	Precise weights	Fuzzy values	Linguistic Z-numbers
Handling of Reliability	No	Implicit	Explicit
Computational Complexity	Low	High	Medium
Applications	Network design, optimization	Social networks, decision making	Risk assessment, NLP
Transformation and Defuzzification	Not required	Required	Transformation functions

. Also, a comparison graph of the linguistic Z-graph with existing graphs is shown in Figure 1.

Social networks have surged in popularity, serving diverse purposes such as online business, news dissemination, and advertising. Platforms like Twitter, Facebook, Instagram, and LinkedIn have become integral parts of human life, representing interconnected social entities with various networks. Within these networks, individuals, organizations, and colleagues form attachments, contributing to the complexity of social connections. Identifying central nodes within these networks becomes crucial for tasks like spreading news and gathering information. Various methods have been proposed in the literature to compute centrality, starting from Bavelas’s [36] approach in 1950 to more recent advancements like Weighted Neighborhood Centrality and Structural Centrality. These methods aim to measure the importance of nodes based on different criteria, such as degree, closeness, betweenness, and subgraph centrality. However, traditional methods often overlook the uncertainties in social networks. Fuzzy graph theory [37,38,39,40,41,42,43] offers a more accurate approach to centrality measurement, considering relationship uncertainties and yielding potentially more accurate results.

Recent research has looked into innovative techniques for identifying influential nodes [44] in social networks. Lu et al. [45] proposed a method focusing on leadership networks, while Ling et al. [46] introduced the concept of gravity centrality, utilizing Newton’s gravity formula of node influence. Centrality measures traditionally evaluate node properties in deterministic social networks, but Zareie and Sakellariou [47] extended them to fuzzy graphs, incorporating uncertainty in edges using fuzzy set theory. It defines centrality as a fuzzy relation and adapts degree, h-index, and k-shell measures to calculate node importance in fuzzy graphs. The experimental results show that these proposed fuzzy centrality measures provide more accurate node importance evaluations compared to other methods. Curado et al. [48] introduced a new centrality measure combining the return random walk and the effective distance gravity model to identify influential nodes in networks. This measure considers both local and global information, as well as dynamic interactions, enhancing its relevance over traditional centrality models. Lu et al. [49] introduced the node centrality and similarity-based parameterized model for link prediction, addressing the limitations of graph convolutional networks, especially in short path and ego networks. They found that thermal shock from scram conditions significantly impacts the service life of central measuring shrouds, which can be protected by cladding. Zheng et al. [50] proposed and experimentally verified a cladding design procedure using an elastoplastic model, determining that the cladding thickness and gap distance should be at least 2 mm and 2.66 mm, respectively, for safety.

In this paper, we have introduced a new concept of the centrality measure of a network. Additionally, efforts have been made to incorporate collaboration and self-weight metrics into centrality calculations, acknowledging the importance of these factors in determining node influence within a network.

3. Preliminaries

3.1. Fuzzy Graphs

A fuzzy graph [6] is denoted by $G=(V,\eta ,\varpi )$, where V is a non-empty set, which is called a set of vertices. This structure consists of two functions: $\eta :V\to [0,1]$ and $\varpi :V\times V\to [0,1]$. These functions satisfy the condition $\varpi (x,y)\le min\left\{\eta \right(x),\eta (y\left)\right\}$ for all $x,y\in V$. Here, $\eta \left(x\right)$ and $\varpi (x,y)$ denote the vertex membership values of the vertex x and the edge membership value of the edge $(x,y)$ respectively in G.

3.2. Linguistic Set

A linguistic set is a collection of linguistic terms or categorized elements of a language. These terms are used to represent concepts or ideas within a specific context, and they may vary in their meaning and interpretation based on the context in which they are used. Linguistic sets are commonly used in fuzzy logic and fuzzy set theory to model vague information in natural language.

Examples of linguistic set is $\theta \left(z\right)=$ {Extremely low, Low, Moderately low, Moderate, Moderately high, High, Extremely high}.

3.3. Linguistic Term Set

A linguistic term set [1,2,51] is a mathematical representation of a linguistic set. It is a finite set of linguistic values of a linguistic variable. In the context of fuzzy logic, these linguistic values are often represented as fuzzy sets, with each linguistic term associated with a membership function that defines the degree to which an element belongs to the set. Consider the following linguistic term sets:

$$H=\{h_{-t_1}\le h_{\alpha }\le h_{t_1},\alpha \in [-t_1,t_1]\}$$

$$G=\{g_{-t_2}\le g_{\beta }\le h_{t_2},\beta \in [-t_2,t_2]\}$$

For example, a linguistic set is $\theta \left(z\right)=$ {Very unsure, Unsure, Moderately unsure, Neutral, Moderately sure, Sure, Very sure}.

Here, $t_1=3$, $t_{-3}=Veryunsure$, $t_{-2}=Unsure$, $t_{-1}=Moderatelyunsure$, $t_0=Neutral$, $t_1=Moderatelysure$, $t_2=Sure$, and $t_3=Verysure$

3.4. Transformation Function of Linguistic Terms

For a linguistic term set H or G, the transformation function T maps the set to the interval $[0,1]$.

For a linguistic term $h_{\alpha }$ in H, the transformation function [1,2,51] is defined as follows:

$$T:H\left(orG\right)\to [0,1]$$

$$T\left(h_{\alpha }\right)=0.5+{\displaystyle \frac{I\left(h_{\alpha }\right)}{2t_1}},whereI\left(h_{\alpha }\right)=\alpha$$

$$Similarly,T\left(g_{\beta }\right)=0.5+{\displaystyle \frac{I\left(g_{\beta }\right)}{2t_2}},whereI\left(g_{\beta }\right)=\beta$$

The transformation functions are used to map the linguistic terms to a numerical scale, which can be useful for computations and comparisons. The transformation function is a key component in the process of defuzzification in fuzzy logic, where fuzzy sets (linguistic terms) are converted into crisp values.

3.5. Linguistic Z-Number and Expected Value of the Linguistic Z-Number

A linguistic Z-number is represented as follows:

$$z=(A,B)=(h_{\alpha },g_{\beta })$$

Here, A is a linguistic term representing the membership value of z and B, represents the reliability of A.

The expected value of the linguistic Z-number [51] is calculated as follows:

$$E\left(z\right)=T\left(A\right)\times T\left(B\right)$$

4. Linguistic Z-Graph

The linguistic Z-graph is defined based on the linguistic Z-number. The formal definition of the linguistic Z-graph is as follows:

Definition 1. 

Let V be a set and R be a relation on $V\times V$.

A linguistic Z-graph $G=(V,\sigma ,\mu )$, where V is a non-empty set, and there exist two functions $\sigma :V\to \theta \left(z\right)$ and $\mu :V\times V\to \theta \left(z\right)$. $\theta \left(z\right)$ is a set of a linguistic Z-number such that $\mu (x,y)=\sigma \left(x\right)\ast \sigma \left(y\right)$, $\forall x,y\in V$, where $\sigma \left(x\right)=\theta \left(z_x\right)=(h_{\alpha },g_{\beta })$ and $\sigma \left(y\right)=\theta \left(z_y\right)=(h_{{\alpha }^{\prime }},g_{{\beta }^{\prime }})$; $\sigma \left(x\right)\ast \sigma \left(y\right)=(h_r,g_{\gamma })$, where $h_r\le min\{h_{\alpha },h_{{\alpha }^{\prime }}\}$, γ is an integer, and $\beta \le \gamma \le {\beta }^{\prime }$.

Example 1. 

Let us assume, $V=\{a,b,c,d,e,f\}$ and $V\times V=\left\{\right(a,b),(a,d),(a,e),(a,f),(b,c),(b,g),(c,d),(c,g),(d,g),(d,e),(d,f),(e,f\left)\right\}$. Also we consider the values of $t_1=5$ and $t_2=7$; therefore,

$$H=\{h_{-5}\le h_{\alpha }\le h_5,\alpha \in [-5,5]\}$$

$$G=\{g_{-7}\le g_{\beta }\le h_7,\beta \in [-7,7]\}.$$

It is displayed in Figure 2, a linguistic Z-graph with six vertices. The vertex and edge membership values are considered as in

Table 2. Vertex membership value of the linguistic graph of Figure 2.

Vertex	Membership Value	Vertex	Membership Value
a	$(h_5,g_{-5})$	d	$(h_3,g_{-6})$
b	$(h_{-4},g_{-6})$	e	$(h_3,g_3)$
c	$(h_2,g_{-1})$	f	$(h_4,g_{-3})$

and

Table 3. Edge membership value of the linguistic Z-graph of Figure 2.

Edge	Membership Value	Edge	Membership Value
(a,c)	$(h_1,g_{-3})$	(b,f)	$(h_{-4},g_{-5})$
(a,b)	$(h_{-4},g_{-5})$	(c,d)	$(h_2,g_{-5}$
(a,d)	$(h_{-4},g_{-6}$	(c,e)	$(h_1,g_2)$
(a,e)	$(h_2,g_2)$	(c,f)	$(h_2,g_{-2})$
(a,f)	$(h_4,g_{-4})$	(d,e)	$(h_3,g_2)$
(b,c)	$(h_{-4},g_{-5})$	(e,f)	$(h_3,g_2$)
(b,d)	$(h_{-4},g_{-6})$

, respectively. The expected value $E\left(z\right)$ of all vertices of the graph of Figure 2 has been shown in

Table 4. The value of $E\left(z\right)$ of all vertices of Figure 2.

Vertex	${\mathit{T}}_1$	$\mathit{\alpha }$	${\mathit{T}}_{\mathit{\alpha }}$	${\mathit{T}}_2$	$\mathit{\beta }$	${\mathit{T}}_{\mathit{\beta }}$	$\mathit{E}\left(\mathit{z}\right)$
a	5	5	1	7	−5	0.14	0.14
b	5	$-4$	0.1	7	−6	0.07	0.01
c	5	2	0.7	7	−1	0.43	0.3
d	5	3	0.8	7	−6	0.07	0.06
e	5	3	0.8	7	3	0.71	0.57
f	5	4	0.9	7	−3	0.29	0.26

. Also, the expected value $E\left(z\right)$ of all edges of the graph of Figure 2 is shown in

Table 5. The value of $E\left(z\right)$ of all edges of Figure 2.

EDGE	${\mathit{T}}_1$	$\mathit{\alpha }$	${\mathit{T}}_{\mathit{\alpha }}$	${\mathit{T}}_2$	$\mathit{\beta }$	${\mathit{T}}_{\mathit{\beta }}$	$\mathit{E}\left(\mathit{X}\right)$
(a,c)	5	1	0.6	7	−3	0.29	0.17
(a,b)	5	−4	0.1	7	−5	0.14	0.01
(a,d)	5	−4	0.1	7	−6	0.07	0.01
(a,e)	5	2	0.7	7	2	0.64	0.45
(a,f)	5	4	0.9	7	−4	0.21	0.19
(b,c)	5	−4	0.1	7	−5	0.14	0.01
(b,d)	5	−4	0.1	7	−6	0.07	0.01
(b,f)	5	−4	0.1	7	−5	0.14	0.01
(c,d)	5	2	0.7	7	−5	0.14	0.1
(c,e)	5	1	0.6	7	2	0.64	0.38
(c,f)	5	2	0.7	7	−2	0.36	0.25
(d,e)	5	3	0.8	7	2	0.64	0.51
(e,f)	5	3	0.8	7	2	0.64	0.51

.

Theorem 1. 

Let, $G=(V,\sigma ,\mu )$ be a linguistic Z-graph. If the expected value $E\left(z\right)$ for an edge or vertex is 1, then the membership value of the corresponding edge or vertex is $(h_{t_1},g_{t_2})$ where $h_{t_1}$ and $g_{t_2}$ are fixed linguistic terms.

Proof. 

Let $G=(V,\sigma ,\mu )$ be a linguistic Z-graph; then, the membership value of an edge or vertex is denoted by $(h_{\alpha },g_{\beta })$, where $\{h_{-t_1}\le h_{\alpha }\le h_{t_1},\alpha \in [-t_1,t_1]\}$, and $\{g_{-t_2}\le g_{\beta }\le h_{t_2},\beta \in [-t_2,t_2]\}$. Now, the formula for $E\left(z\right)$ is $E\left(z\right)=T\left(A\right)\times T\left(B\right)$, where, $T\left(A\right)=T\left(h_{\alpha }\right)=0.5+{\displaystyle \frac{I\left(h_{\alpha }\right)}{2t_1}},\{where,I\left(h_{\alpha }\right)=\alpha \}$, and $T\left(B\right)=T\left(g_{\beta }\right)=0.5+{\displaystyle \frac{I\left(g_{\beta }\right)}{2t_2}},\{where,I\left(g_{\beta }\right)=\beta \}$.

• Given that $E\left(z\right)=1$, and hence $T\left(A\right)\times T\left(B\right)=1$ is satisfied. Therefore, the value of $T\left(A\right)=1$, and $T\left(B\right)=1$. This leads to $T\left(A\right)=T\left(h_{\alpha }\right)=0.5+{\displaystyle \frac{I\left(h_{\alpha }\right)}{2t_1}})=1$, and $T\left(B\right)=T\left(g_{\beta }\right)=0.5+{\displaystyle \frac{I\left(g_{\beta }\right)}{2t_2}}=1$.
• Solving these equations, we obtain ${\displaystyle \frac{I\left(h_{\alpha }\right)}{2t_1}}=0.5$ and ${\displaystyle \frac{I\left(g_{\beta }\right)}{2t_2}}=0.5$.
• So, $2\times \alpha =2t_1$ and $\beta =2t_2$, $\{Giventhat,I\left(h_{\alpha }\right)=\alpha andI\left(g_{\beta }\right)=\beta \}$.
• So, $\alpha =t_1$ and $\beta =t_2$.
• Therefore, if the expected value $E\left(z\right)$ for an edge or vertex is 1, then the membership value of the corresponding edge or vertex is $(h_{t_1},g_{t_2})$. □

Theorem 2. 

Let, $G=(V,\sigma ,\mu )$ be a linguistic Z-graph. If the expected value $E\left(z\right)$ for an edge or vertex is 0, then the membership value of the corresponding edge or vertex is one of the following: $(h_{-t_1},g_{\beta })$, $(h_{\alpha },g_{-t_2})$, or $(h_{-t_1},g_{-t_2})$.

Proof. 

Let $G=(V,\sigma ,\mu )$ be a linguistic Z-graph; then, the membership value of an edge or vertex is denoted by $(h_{\alpha },g_{\beta })$, where $\{h_{-t_1}\le h_{\alpha }\le h_{t_1},\alpha \in [-t_1,t_1]\}$, and $\{g_{-t_2}\le g_{\beta }\le h_{t_2},\beta \in [-t_2,t_2]\}$.

• Given that $E\left(z\right)=0$,
• this means that the value of $T\left(A\right)\times T\left(B\right)=0$.
• Thus, one of the values $T\left(A\right)$ or $T\left(B\right)$ is 0, or both $T\left(A\right)$ and $T\left(B\right)$ are 0.
• If the value of $T\left(A\right)=0$ and $T\left(B\right)=0$,
• this means that $T\left(A\right)=T\left(h_{\alpha }\right)=0.5+{\displaystyle \frac{I\left(h_{\alpha }\right)}{2t_1}})=0$ and $T\left(B\right)=T\left(g_{\beta }\right)=0.5+{\displaystyle \frac{I\left(g_{\beta }\right)}{2t_2}}=0$.
• Solving these equations, we obtain ${\displaystyle \frac{I\left(h_{\alpha }\right)}{2t_1}}=-0.5$ and ${\displaystyle \frac{I\left(g_{\beta }\right)}{2t_2}}=-0.5$.
• So, $2\times \alpha =-2t_1$ and $\beta =-2t_2$, $\{where,I\left(h_{\alpha }\right)=\alpha andI\left(g_{\beta }\right)=\beta \}$.
• So, $\alpha =-t_1$ and $\beta =-t_2$.
• Thus, if $T\left(A\right)=0$ and $T\left(B\right)\ne 0$, then the membership value is $(h_{-t_1},g_{\beta })$.
• If $T\left(A\right)\ne 0$ and $T\left(B\right)=0$, then the membership value is $(h_{\alpha },g_{-t_2})$.
• If $T\left(A\right)=0$ and $T\left(B\right)=0$, then the membership value is $(h_{-t_1},g_{-t_2})$.
• Therefore, if the expected value $E\left(z\right)$ for an edge or vertex is 0, then the membership value of the corresponding edge or vertex is one of the following: $(h_{-t_1},g_{\beta })$, $(h_{\alpha },g_{-t_2})$, or $(h_{-t_1},g_{-t_2})$. □

4.1. Different Types of Linguistic Z-Graphs and Some Operations on Linguistic Z-Graphs

Now, we introduce the various types of linguistic Z-graphs. The proper definition and example of these graphs are given as follows.

4.1.1. Strong Linguistic Z-Graph

Definition 2. 

A linguistic Z-graph $G=(V,\sigma ,\mu )$ is said to be strong if $\mu (x,y)=\sigma \left(x\right)\ast \sigma \left(y\right)$, where $\sigma \left(x\right)=\theta \left(z_x\right)=(h_{\alpha },g_{\beta })$, and $\sigma \left(y\right)=\theta \left(z_y\right)=(h_{{\alpha }^{\prime }},g_{{\beta }^{\prime }})$; $\sigma \left(x\right)\ast \sigma \left(y\right)=(h_r,g_{\gamma })$, $h_r=min\{h_{\alpha },h_{{\alpha }^{\prime }}\}$, and $\gamma =max\{\beta ,{\beta }^{\prime }\}$.

Example 2. 

Consider $V=\{a,b,c,d,\}$ and $V\times V=\left\{\right(a,b),(b,c),(c,d\left)\right\}$. Also, let $t_1=5$ and $t_2=7$.

$$H=\{h_{-5}\le h_{\alpha }\le h_5,\alpha \in [-5,5]\}$$

$$G=\{g_{-7}\le g_{\beta }\le h_7,\beta \in [-7,7]\}$$

A strong linguistic Z-graph is shown in Figure 3.

4.1.2. $\alpha$-Strong Linguistic Z-Graph

Definition 3. 

A linguistic Z-graph $G=(V,\sigma ,\mu )$ is said to be α-strong if $\mu (x,y)=\sigma \left(x\right)\ast \sigma \left(y\right)$, where $\sigma \left(x\right)=\theta \left(z_x\right)=(h_{\alpha },g_{\beta })$, and $\sigma \left(y\right)=\theta \left(z_y\right)=(h_{{\alpha }^{\prime }},g_{{\beta }^{\prime }})$; $\sigma \left(x\right)\ast \sigma \left(y\right)=(h_r,g_{\gamma })$, $h_r=min\{h_{\alpha },h_{{\alpha }^{\prime }}\}$, and γ define integers satisfying $\beta \le \gamma \le {\beta }^{\prime }$.

Example 3. 

Consider $V=\{a,b,c,d,\}$ and $V\times V=\left\{\right(a,b),(b,c),(c,d\left)\right\}$. Also, let $t_1=5$ and $t_2=7$.

$$H=\{h_{-5}\le h_{\alpha }\le h_5,\alpha \in [-5,5]\}$$

$$G=\{g_{-7}\le g_{\beta }\le h_7,\beta \in [-7,7]\}$$

An α-strong linguistic Z-graph is shown in Figure 4.

4.1.3. $\beta$-Strong Linguistic Z-Graph

Definition 4. 

A linguistic Z-graph $G=(V,\sigma ,\mu )$ is said to be β-strong if $\mu (x,y)=\sigma \left(x\right)\ast \sigma \left(y\right)$, where $\sigma \left(x\right)=\theta \left(z_x\right)=(h_{\alpha },g_{\beta })$, and $\sigma \left(y\right)=\theta \left(z_y\right)=(h_{{\alpha }^{\prime }},g_{{\beta }^{\prime }})$; $\sigma \left(x\right)\ast \sigma \left(y\right)=(h_r,g_{\gamma })$, $h_r\le min\{h_{\alpha },h_{{\alpha }^{\prime }}\}$, and $\gamma =max\{\beta ,{\beta }^{\prime }\}$.

Example 4. 

Consider $V=\{a,b,c,d,\}$ and $V\times V=\left\{\right(a,b),(b,c),(c,d\left)\right\}$, and let $t_1=5$ and $t_2=7$.

$$H=\{h_{-5}\le h_{\alpha }\le h_5,\alpha \in [-5,5]\}$$

$$G=\{g_{-7}\le g_{\beta }\le h_7,\beta \in [-7,7]\}$$

A β-strong linguistic Z-graph is shown in Figure 5.

Note 1. 

Every strong linguistic Z-graph is an α-strong linguistic Z-graph and a β-strong linguistic Z-graph, but the converse is not true.

4.1.4. Complete Linguistic Z-Graph

Definition 5. 

A linguistic Z-graph $G=(V,\sigma ,\mu )$ is described as a complete linguistic Z-graph if the underlaying crisp graph is a complete graph and if $\mu (x,y)=\sigma \left(x\right)\ast \sigma \left(y\right)$, where $\sigma \left(x\right)=\theta \left(z_x\right)=(h_{\alpha },g_{\beta })$ and $\sigma \left(y\right)=\theta \left(z_y\right)=(h_{{\alpha }^{\prime }},g_{{\beta }^{\prime }})$; $\sigma \left(x\right)\ast \sigma \left(y\right)=(h_r,g_{\gamma })$, where $h_r=min\{h_{\alpha },h_{{\alpha }^{\prime }}\}$, and $\gamma =max\{\beta ,{\beta }^{\prime }\}$.

Example 5. 

Consider $V=\{a,b,c,d,\}$ and $V\times V=\left\{\right(a,b),(b,c),(c,d\left)\right\}$, and let $t_1=5$ and $t_2=7$.

$$H=\{h_{-5}\le h_{\alpha }\le h_5,\alpha \in [-5,5]\}$$

$$G=\{g_{-7}\le g_{\beta }\le h_7,\beta \in [-7,7]\}$$

A complete linguistic Z-graph is shown in Figure 6.

Note 2. 

Every complete linguistic Z-graph is a strong linguistic Z-graph, but the converse is not true.

4.1.5. Cycle in Linguistic Z-Graph

A linguistic Z-graph $G=(V,\sigma ,\mu )$ is described as a cycle linguistic Z-graph if the underlaying crisp graph of G is a cycle graph and if $\mu (x,y)=\sigma \left(x\right)\ast \sigma \left(y\right)$, where $\sigma \left(x\right)=\theta \left(z_x\right)=(h_{\alpha },g_{\beta })$ and $\sigma \left(y\right)=\theta \left(z_y\right)=(h_{{\alpha }^{\prime }},g_{{\beta }^{\prime }})$; $\sigma \left(x\right)\ast \sigma \left(y\right)=(h_r,g_{\gamma })$, where $h_r=min\{h_{\alpha },h_{{\alpha }^{\prime }}\}$ and $\gamma =max\{\beta ,{\beta }^{\prime }\}$.

Example 6. 

Consider $V=\{a,b,c,d,\}$ and $V\times V=\left\{\right(a,b),(b,c),(c,d),(d,a\left)\right\}$, and let $t_1=5$ and $t_2=7$.

$$H=\{h_{-5}\le h_{\alpha }\le h_5,\alpha \in [-5,5]\}$$

$$G=\{g_{-7}\le g_{\beta }\le h_7,\beta \in [-7,7]\}$$

A cycle in linguistic Z-graph is shown in Figure 7.

4.1.6. Star in Linguistic Z-Graph

A linguistic Z-graph $G=(V,\sigma ,\mu )$ is described as a star linguistic Z-graph if the underlaying crisp graph of G is a star graph.

Example 7. 

Consider $V=\{a,b,c,d,e\}$ and $V\times V=\left\{\right(a,e),(b,e),(c,e),(d,e\left)\right\}$, and let $t_1=5$ and $t_2=7$.

$$H=\{h_{-5}\le h_{\alpha }\le h_5,\alpha \in [-5,5]\}$$

$$G=\{g_{-7}\le g_{\beta }\le h_7,\beta \in [-7,7]\}$$

A star linguistic Z-graph is shown in Figure 8.

4.1.7. Degree and Total Degree of Linguistic Z-Graph

Definition 6. 

Let $G=(V,\sigma ,\mu )$ be a linguistic Z-graph; then, the degree of a vertex a is denoted by $D\left(a\right)$ and defined as $D\left(a\right)={\sum }_{a,a\ne b}E(a,b)$, where $E(a,b)$ is the expected value of an edge $(a,b)$ in G.

Also, the total degree of a vertex a is denoted as $tD\left(a\right)$ and defined by $tD\left(a\right)=D\left(a\right)+E\left(a\right)=\left\{{\sum }_{a,a\ne b}E(a,b)\right\}+E\left(a\right)$, where $E(a,b)$ is the expected value of an edge $(a,b)$, and $E\left(a\right)$ is the expected value of a vertex $\left(a\right)$ in G.

Example 8. 

Consider $V=\{a,b,c,d,e\}$ and $V\times V=\left\{\right(a,e),(b,e),(c,e),(d,e\left)\right\}$, and let $t_1=5$ and $t_2=7$.

$$H=\{h_{-5}\le h_{\alpha }\le h_5,\alpha \in [-5,5]\}$$

$$G=\{g_{-7}\le g_{\beta }\le h_7,\beta \in [-7,7]\}.$$

The degree and total degree of the linguistic Z-number graph of Figure 2 has been shown in

Table 6. Degree and total degree of the linguistic Z-number graph of Figure 2.

Vertex	Degree	Total Degree
a	0.83	0.97
b	0.04	0.05
c	0.91	1.21
d	0.63	0.69
e	1.85	2.42
f	0.96	1.22

.

4.1.8. Strength of an Edge

Definition 7. 

Let $G=(V,\sigma ,\mu )$ be a linguistic Z-graph; then, the strength of an edge $(a,b)$ is denoted by $S_{(a,b)}$ and defined by

$$S_{(a,b)}=min\{1,{\displaystyle \frac{E_{(a,b)}}{max\{E_a,E_b\}}}\},ifmax\{E_a,E_b\}\ne 0$$

$$=0,ifmax\{E_a,E_b\}=0,$$

where $E_{(a,b)}$ is the expected value of the edge $(a,b)$, and $E_a$ is the expected value of the vertex a in G.

The edge $(a,b)$ is said to be strong if the value $S_{(a,b)}\ge 0.5$; otherwise, the edge $(a,b)$ is weak.

Example 9. 

Consider $V=\{a,b,c,d,e,f\}$ and $V\times V=\left\{\right(a,b),(a,d),(a,e),(a,f),(b,c),(b,g),(c,d),(c,g),(d,g),(d,e),(d,f),(e,f\left)\right\}$, and let $t_1=5$ and $t_2=7$.

$$H=\{h_{(-5)}\le h_{\alpha }\le h_5,\alpha \in [-5,5]\}$$

$$G=\{g_{(-7)}\le g_{\beta }\le h_7,\beta \in [-7,7]\}$$

The graph of Figure 2 and the corresponding Table 2 and Table 3 are considered here. The value of $E\left(z\right)$ of all vertices of the graph of Figure 2 has been shown in Table 4. Also, the value of $E\left(z\right)$ of all edges of the graph of Figure 2 has been shown in Table 5.

The strength of an edge $(a,c)$ is $S_{(a,c)}={\displaystyle \frac{0.17}{max\{0.14,0.3\}}}={\displaystyle \frac{0.17}{0.3}}=0.57$. Also, the strength of all edges of the graph in Figure 2 has been shown in

Table 7. Strength of all edges of Figure 2.

EDGE	${\mathit{T}}_1$	$\mathit{\alpha }$	${\mathit{T}}_{\mathit{\alpha }}$	${\mathit{T}}_2$	$\mathit{\beta }$	${\mathit{T}}_{\mathit{\beta }}$	E(X)	Strength
(a,c)	5	1	0.6	7	−3	0.29	0.17	0.57
(a,b)	5	−4	0.1	7	−5	0.14	0.01	0.07
(a,d)	5	−4	0.1	7	−6	0.07	0.01	0.07
(a,e)	5	2	0.7	7	2	0.64	0.45	0.79
(a,f)	5	4	0.9	7	−4	0.21	0.19	0.73
(b,c)	5	−4	0.1	7	−5	0.14	0.01	0.03
(b,d)	5	−4	0.1	7	−6	0.07	0.01	0.17
(b,f)	5	−4	0.1	7	−5	0.14	0.01	0.04
(c,d)	5	2	0.7	7	−5	0.14	0.1	0.33
(c,e)	5	1	0.6	7	2	0.64	0.38	0.67
(c,f)	5	2	0.7	7	−2	0.36	0.25	0.83
(d,e)	5	3	0.8	7	2	0.64	0.51	0.89
(e,f)	5	3	0.8	7	2	0.64	0.51	0.89

. So, the edges $(a,c),(a,e),(a,f),(c,e),$$(c,f),(d,e),(e,f)$ are strong, and other edges are weak.

Theorem 3. 

Let $G=(V,\sigma ,\mu )$ be a strong linguistic Z-graph, where $E(a,b)$ is the expected value of the edge $(a,b)$ and $E\left(a\right)$. $E\left(b\right)$ defines the expected value of the vertices a and b, respectively. Then, $E(a,b)\le max\left\{E\right(a),E(b\left)\right\}$ is true.

Proof. 

Let $G=(V,\sigma ,\mu )$ be a linguistic Z-graph; then, the membership value of an edge or vertex is denoted by $(h_{\alpha },g_{\beta })$, where $\{h_{-t_1}\le h_{\alpha }\le h_{t_1},\alpha \in [-t_1,t_1]\}$, and $\{g_{-t_2}\le g_{\beta }\le h_{t_2},\beta \in [-t_2,t_2]\}$. Now, the formula of $E\left(z\right)$ is $E\left(z\right)=T\left(A\right)\times T\left(B\right)$, where $T\left(A\right)=T\left(h_{\alpha }\right)=0.5+{\displaystyle \frac{I\left(h_{\alpha }\right)}{2t_1}},\{where,I\left(h_{\alpha }\right)=\alpha \}$, and $T\left(B\right)=T\left(g_{\beta }\right)=0.5+{\displaystyle \frac{I\left(g_{\beta }\right)}{2t_2}},\{where,I\left(g_{\beta }\right)=\beta \}$.

• Consider that the membership values of the vertices a and b are $(h_{{\alpha }_1},h_{{\beta }_1})$ and $(h_{{\alpha }_2},h_{{\beta }_2})$, respectively, and the membership value of the edge $(a,b)$ is $(h_{{\alpha }_3},h_{{\beta }_3})$. Since the graph G is strong, $h_{{\alpha }_3}=min\{h_{{\alpha }_1},h_{{\alpha }_2}\}$, and $h_{{\beta }_3}=max\{h_{{\beta }_1},h_{{\beta }_2}\}$.
• Now, $E\left(a\right)=T\left(h_{{\alpha }_1}\right)\times T\left(h_{{\beta }_1}\right)$, and $E\left(b\right)=T\left(h_{{\alpha }_2}\right)\times T\left(h_{{\beta }_2}\right)$.
• Since the graph G is strong, this means that the value of $E(a,b)$ is
• $E(a,b)=min\{T\left(h_{{\alpha }_1}\right),T\left(h_{{\alpha }_2}\right)\}\times max\{T\left(h_{{\beta }_1}\right),T\left(h_{{\beta }_2}\right)\}$.
• So, $E(a,b)\le max\left\{E\right(a),E(b\left)\right\}$.
• Also, if any one of the vertex membership values of a or b is $(h_{t_1},g_{t_2})$, then the value of $E(a,b)$ is $E(a,b)=max\left\{E\right(a),E(b\left)\right\}=1$.
• Therefore, $E(a,b)\le max\left\{E\right(a),E(b\left)\right\}$ is true. □

Theorem 4. 

Let $G=(V,\sigma ,\mu )$ be a linguistic Z-graph. If the strength of an edge $(a,b)$ is $S_{(a,b)}$, then $0\le S_{(a,b)}\le 1$.

Proof. 

Let $G=(V,\sigma ,\mu )$ be a linguistic Z-graph; then, the membership value of an edge or vertex is denoted by $(h_{\alpha },g_{\beta })$, where $\{h_{-t_1}\le h_{\alpha }\le h_{t_1},\alpha \in [-t_1,t_1]\}$, and $\{g_{-t_2}\le g_{\beta }\le h_{t_2},\beta \in [-t_2,t_2]\}$. Now, the formula of $E\left(z\right)$ is $E\left(z\right)=T\left(A\right)\times T\left(B\right)$, where $T\left(A\right)=T\left(h_{\alpha }\right)=0.5+{\displaystyle \frac{I\left(h_{\alpha }\right)}{2t_1}},\{where,I\left(h_{\alpha }\right)=\alpha \}$, and $T\left(B\right)=T\left(g_{\beta }\right)=0.5+{\displaystyle \frac{I\left(g_{\beta }\right)}{2t_2}},\{where,I\left(g_{\beta }\right)=\beta \}$.

• Now, from Theorems 1 and 2 we obtain that the value of $E\left(a\right)$ is $0\le E\left(a\right)\le 1$. Also, the formula of the strength of an edge $(a,b)$ is

  $$S_{(a,b)}=min\{1,{\displaystyle \frac{E_{(a,b)}}{max\{E_a,E_b\}}}\},ifmax\{E_a,E_b\}\ne 0$$

  $$=0,ifmax\{E_a,E_b\}=0$$

So, $0\le S_{(a,b)}\le 1$ is true. □

Theorem 5. 

Let $G=(V,\sigma ,\mu )$ be a strong linguistic Z-graph. Then, the strength of all edges are less or equal to 1.

Proof. 

Let $G=(V,\sigma ,\mu )$ be a strong linguistic Z-graph; then, the membership value of an edge or vertex is denoted by $(h_{\alpha },g_{\beta })$, where $\{h_{-t_1}\le h_{\alpha }\le h_{t_1},\alpha \in [-t_1,t_1]\}$, and $\{g_{-t_2}\le g_{\beta }\le h_{t_2},\beta \in [-t_2,t_2]\}$. Now, the formula of $E\left(z\right)$ is $E\left(z\right)=T\left(A\right)\times T\left(B\right)$, where $T\left(A\right)=T\left(h_{\alpha }\right)=0.5+{\displaystyle \frac{I\left(h_{\alpha }\right)}{2t_1}}),\{where,I\left(h_{\alpha }\right)=\alpha \}$, and $T\left(B\right)=T\left(g_{\beta }\right)=0.5+{\displaystyle \frac{I\left(g_{\beta }\right)}{2t_2}},\{where,I\left(g_{\beta }\right)=\beta \}$.

• Given that the graph G is strong based on Theorem 3, we obtain that $E(a,b)\le max\left\{E\right(a),E(b\left)\right\}$.
• Also, the formula of strength of an edge $(a,b)$ is

  $$S_{(a,b)}=min\{1,{\displaystyle \frac{E_{(a,b)}}{max\{E_a,E_b\}}}\},ifmax\{E_a,E_b\}\ne 0$$

  $$=0,ifmax\{E_a,E_b\}=0.$$
• Therefore, the strength of all edges of a strong linguistic Z-graph are less or equal to 1. □

5. Centrality Measure in a Linguistic Z-Graph

The degree centrality of a linguistic Z-graph is defined as follows.

Definition 8. 

Let $G=(V,\sigma ,\mu )$ be a linguistic Z-graph with $\left|V\right|=n$; then, the centrality of a vertex $v_i$ is denoted by $C_{V_i}$ and defined as

$$C_{V_i}=\sum _{k=1}^p{\displaystyle \frac{S_{(v_i,v_k)}}{(n-1)}},$$

where k is the number of vertices connected with the vertex $v_i$ in G, and $S_{(v_i,v_k)}$ is the strength of an edge $(v_i,v_k)$.

Now, the total degree centrality of a vertex of a linguistic Z-graph is defined as follows:

Definition 9. 

Let $G=(V,\sigma ,\mu )$ be a linguistic Z-graph with $\left|V\right|=n$. The total degree centrality of a node $v_i$, denoted by $P_{V_i}$, is defined as follows:

$$P_{V_i}=\lambda \times C_{V_i}+(1-\lambda )E\left(z_{V_i}\right),$$

where $\lambda \in [0,1]$, and the centrality of a vertex $v_i$ is $C_{V_i}$.

Then, the central node of this network is this node whose total degree centrality is maximum, i.e., $max\left\{P_{V_i}\right\},i=1,2,3,\dots ,n.$

Algorithm to Find the Central Node of a Linguistic Z-Graph

Input: $G=(V,\sigma ,\mu )$ be a linguistic Z-graph and $\left|V\right|=n$.

• Output: Central node of a linguistic Z-graph G.

• Label all vertices as $V_1,V_2,V_3,\dots ,V_n$.
• Calculate the centrality of vertex $V_1$ using the formula:

  $$C_{V_1}=\sum _{k=1}^p{\displaystyle \frac{S_{(v_1,v_k)}}{(n-1)}},$$

  where k is the number of vertices connected with vertex $V_1$ in G.
• Calculate the total degree centrality of vertex $V_1$ using the formula:

  $$P_{V_1}=\{\lambda \times C_{V_1}+(1-\lambda )E\left(z_{V_1}\right)\},$$

  where $\lambda \in [0,1]$.
• Repeat Steps 2 and 3 for all vertices $V_1,V_2,V_3,\dots ,V_n$.
• The central node of the network is the node with the maximum total degree centrality, i.e., $max\left\{P_{V_i}\right\},i=1,2,3,\dots ,n$.

Example 10. 

Consider $V=\{a,b,c,d,e,f\}$ and $V\times V=\left\{\right(a,b),(a,d),(a,e),(a,f),(b,c),$$(b,g),(c,d),(c,g),(d,g),(d,e),(d,f),(e,f)\}$. Also, consider the value of $t_1=5$ and $t_2=7$.

$$H=\{h_{(-5)}\le h_{\alpha }\le h_5,\alpha \in [-5,5]\}$$

$$G=\{g_{(-7)}\le g_{\beta }\le h_7,\beta \in [-7,7]\}$$

Consider in Figure 2 a linguistic Z-graph with six vertices. The vertex and edge membership values are shown in Table 2 and Table 3, respectively. The value of $E\left(z\right)$ of all vertices of the graph of Figure 2 is shown in Table 4. Also, the value of $E\left(z\right)$ of all edges of the graph of Figure 2 is shown in Table 5.

The strength of an edge $(a,c)$ was calculated as $S_{(a,c)}={\displaystyle \frac{0.17}{max\{0.14,0.3\}}}={\displaystyle \frac{0.17}{0.3}}=0.57$. Also, the strength of all the edges of the graph in Figure 2 has been shown in Table 7. Thus, the edges $(a,c),(a,e),(a,f),(c,e),(c,f),(d,e),(e,f)$ are strong, and other edges are weak.

Now, the centrality of a vertex a came out to $C_a={\displaystyle \frac{(0.57+0.07+0.07+0.79+0.73)}{(6-1)}}={\displaystyle \frac{2.23}{5}}=0.446\approx 0.45$. Also, the centrality of all the vertices of this linguistic Z-graph of Figure 2 is shown in

Table 8. Centrality of all vertices of linguistic Z-graph of Figure 2.

Vertex	Centrality	E(z)	Total Centrality
a	0.45	0.26	0.37
b	0.06	0.45	0.22
c	0.49	0.11	0.34
d	0.29	0.34	0.31
e	0.65	0.55	0.61
f	0.5	0.21	0.38

. Consider that the value λ was $0.6$ for this calculation. So, the total degree centrality of the vertex a was $P_a=0.45\times 0.6+0.26\times 0.4=0.27+0.104=0.374\approx 0.37$. The total degree centrality of all vertices of this linguistic Z-graph of Figure 2 is shown in Table 8. Thus, the central node of this network came out to $max\{P_a,P_b,P_c,P_d,P_e,\}=P_e$. So, the node e is the central node of this network.

Theorem 6. 

Let $G=(V,\sigma ,\mu )$ be a linguistic z-graph. If the degree centrality of a vertex a is $C_a$, then $0\le C_a\le 1$.

Proof. 

Let $G=(V,\sigma ,\mu )$ be a linguistic z-graph.

Also, the degree centrality of a vertex a is $C_{V_i}$ is

$$C_{V_i}=\sum _{k=1}^p{\displaystyle \frac{S_{(v_i,v_k)}}{(n-1)}},$$

where k is the number of vertices connected with the vertex $v_i$ in G, and $S_{(v_i,v_k)}$ is the strength of an edge $(v_i,v_k)$. Now, from Theorem 4 we obtain $0\le S_{(v_i,v_k)}\le 1$. So, $0\le C_{V_i}\le 1$. So, $0\le C_a\le 1$ is true. □

Theorem 7. 

Let $G=(V,\sigma ,\mu )$ be a complete linguistic Z-graph; then, the degree centrality of all vertices are equal, and the value is 1.

Proof. 

Let $G=(V,\sigma ,\mu )$ be a complete linguistic Z-graph. From Note 2 we obtain that every complete linguistic Z-graph is a strong linguistic Z-graph. So, G is also a strong linguistic Z-graph.

• Also, from Theorem 5 we obtain that if G is a strong linguistic Z-graph, then the strength of all edges is 1.
• Now, the degree centrality of a vertex a is $C_{V_i}$ is

  $$C_{V_i}=\sum _{k=1}^p{\displaystyle \frac{S_{(v_i,v_k)}}{(n-1)}},$$

  where k is the number of vertices connected with the vertex $v_i$ in G, and $S_{(v_i,v_k)}$ is the strength of an edge $(v_i,v_k)$.
• Given that G is a complete linguistic Z-graph, this means that every vertex is connected with $n-1$ vertices. So, the value of k is $n-1$. Therefore, the value comes out to $C_{V_i}=1$.
• Thus, the degree centrality of all vertices of a complete linguistic z-graph is equal, and the value is 1. □

Lemma 1. 

Let $G=(V,\sigma ,\mu )$ be a linguistic Z-graph. If the total degree centrality of a vertex a is $P_a$, then $0\le P_a\le 1$.

Proof. 

Let $G=(V,\sigma ,\mu )$ be a linguistic Z-graph with $\left|V\right|=n$; then, the degree centrality of a vertex $v_i$ is $C_{V_i}$, and the total degree centrality of a vertex $v_i$ is

$$P_{V_i}=\{\lambda \times C_{V_i}+(1-\lambda )E\left(z_{V_i}\right)\},$$

where $\lambda \in [0,1].$

Now, from Theorem 6 we obtain that $0\le C_a\le 1$.

• Also, from Theorems 1 and 2 we obtain that the value of $E\left(a\right)$ is $0\le E\left(a\right)\le 1$, and $\lambda \in [0,1]$.
• So, $0\le P_a\le 1$ is true. □

Lemma 2. 

Let $G=(V,\sigma ,\mu )$ be a complete linguistic Z-graph; then, the degree centrality of all vertices are the same, but the total degree centrality values of all vertices are not the same.

Proof. 

Let $G=(V,\sigma ,\mu )$ be a complete linguistic Z-graph with $\left|V\right|=n$; then, the degree centrality of a vertex $v_i$ is $C_{V_i}$, and the total degree centrality of a vertex $v_i$ is

$$P_{V_i}=\{\lambda \times C_{V_i}+(1-\lambda )E\left(z_{V_i}\right)\},$$

where $\lambda \in [0,1].$

Now, from Theorem 7 we obtain that the degree centrality values of all vertices are same, and the value is 1.

However, according to the definition of total degree centrality, it is easy to observe that the degree centrality of all vertices is the same, but the total degree centrality of all vertices is not the same. □

Lemma 3. 

Let $G=(V,\sigma ,\mu )$ be a star linguistic Z-graph; then, the central node is the center node of G.

Proof. 

Let $G=(V,\sigma ,\mu )$ be a star linguistic Z-graph with $\left|V\right|=n$; then, the degree centrality of a vertex $v_i$ is $C_{V_i}$, and the total degree centrality of a vertex $v_i$ is

$$P_{V_i}=\{\lambda \times C_{V_i}+(1-\lambda )E\left(z_{V_i}\right)\},$$

where $\lambda \in [0,1].$

• The central node of this graph G is the node whose total degree centrality is maximum, i.e., $max\left\{P_{V_i}\right\},i=1,2,3,\dots ,n.$
• Now, from the definition of a star linguistic Z-graph, we see that the vertex with the maximum degree centrality becomes the center vertex.
• So, according to the definition of total degree centrality, it is easy to observe that central node is the center node of G. □

6. Application

Selecting the head of an institution is a pivotal decision that significantly impacts its success and trajectory. The process requires careful consideration and thorough evaluation to ensure that the chosen individual possesses the necessary qualifications, experience, and leadership qualities to effectively steer the institution towards its goals.

The key factors to consider in the selection process include the following:

• Qualifications: The candidate should have relevant academic qualifications and expertise in the field pertinent to the institution’s mission and objectives. This ensures that they possess the necessary knowledge to understand the difficulties of the institution’s operations.
• Experience: Prior experience in leadership roles, particularly within similar institutions or industries, provides valuable insight into managing challenges and driving growth. A candidate with a proven track record of success demonstrates their capability to lead effectively.
• Vision and Strategy: The ability to articulate a clear vision for the institution’s future and develop strategic plans to achieve goals is essential. The candidate should demonstrate innovative thinking and the capacity to adapt to evolving trends and challenges in the sector.
• Leadership and Management Skills: Strong leadership qualities such as communication, decision making, problem solving, and team building are imperative. The head of the institution should inspire trust and confidence among staff, stakeholders, and the broader community.
• Integrity and Ethical Conduct: Integrity and ethical behavior are non-negotiable qualities for any leader. The candidate must demonstrate honesty, transparency, and a commitment to upholding the values and standards of the institution.
• Cultural Fit: Consideration should be given to how well the candidate aligns with the institution’s culture, values, and ethos. A good fit fosters harmony, collaboration, and a sense of shared purpose among staff and stakeholder.

The selection of a head of an institution is a critical decision that can greatly impact its success and effectiveness. However, there are potential negative aspects associated with the selection process that need to be carefully considered.

• Bias and Nepotism: One of the most significant drawbacks of the selection process is the potential for nepotism to influence the decision. If the selection is based on personal connections, favoritism, or political considerations rather than merit and qualifications, it can undermine trust and credibility within the institution.
• Lack of Diversity: Inadequate efforts to promote diversity in the selection process can result in a lack of representation among leadership positions. This can lead to homogeneity in perspectives, limited innovation, and the exclusion of underrepresented groups, ultimately hindering the institution’s ability to effectively address diverse challenges and opportunities.
• Skill Gaps: If the selected candidate lacks the essential skills or experience required for effective leadership, it can impede the institution’s progress and performance. Inadequate leadership abilities, poor decision making, and an inability to inspire or motivate others can lead to inefficiencies, conflicts, and missed opportunities for growth.
• Resistance to Change: A head of an institution who is resistant to change or unwilling to adapt to new trends, technologies, or methodologies may hinder the institution’s ability to evolve and remain competitive. This can result in stagnation, missed opportunities for innovation, and a failure to meet the evolving needs of stakeholder.
• Lack of Transparency: A lack of transparency in the selection process can foster mistrust among staff, students, and other stakeholders. When decision-making processes are not open and inclusive, it can lead to speculation, rumors, and decreased morale, negatively impacting organizational culture and effectiveness. Short-Term Focus: If the selection process prioritizes candidates who focus solely on short-term goals or immediate results, it may neglect the long-term stainability and strategic vision of the institution.
• Inadequate Governance: Weak governance structures or ineffective oversight mechanisms in the selection process can result in poor decision making, a lack of accountability, and potential conflicts of interest. Without proper checks and balances, the institution may be vulnerable to mismanagement, corruption, or ethical lapses.

So, the selection of the head of an institution is a very important task for the future success of this institution. This process often begins with the establishment of selection criteria, which may include leadership experience, academic qualifications, managerial skills, and a demonstrated commitment to the institution’s mission and values. Following this, a search committee or panel is formed to review applications, conduct interviews, and assess candidates against the established criteria. Ultimately, the candidate who best aligns with the institution’s goals and vision and who demonstrates the ability to effectively lead and inspire others is chosen to serve as the head of the institution.

Consider a network of all the applicants. Here, the applicants are considered to be a vertex, and there exists an edge if they are collaborating on any kind of work like research, social work, etc. So, we can design this selection procedure using a network. In particular, consider that 10 applicants are registering for this post. Suppose that a neutral expert team can verify all the information of all applicants. Also, the expert team can give a linguistic number to the applicants. So, they can be designed using a linguistic Z-graph. As a result, the membership values of the corresponding vertices of the applicants may depend on the following issues:

(1)

Academic;

(2)

Age;

(3)

Management skill;

(4)

Speaking skill;

(5)

Cultural skill;

(6)

Leadership skill;

(7)

Computer knowledge;

(8)

Behavior, etc.

Now, the membership values of the corresponding edges of the network may depend on the following issues.

(1)

Any collaboration work in this institution;

(2)

Any collaboration works;

(3)

Collaboration research works;

(4)

Communication in social networks, etc.

Consider $V=\{v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8,v_9,v_{10}\}$ and $V\times V=\{(v_1,v_4),(v_1,v_5),(v_1,v_6),$$(v_2,v_3),(v_2,v_6),(v_2,v_{10}),(v_3,v_6),(v_3,v_4),(v_3,v_7),(v_3,v_8),(v_4,v_8),(v_5,v_9),(v_5,v_{10}),(v_6,v_8),(v_6,v_9),(v_7,v_8),(v_7,v_{10}),(v_8,v_9)\}$. Also, consider the values of $t_1=5$ and $t_2=7$.

$$H=h_{-5}\le h_{\alpha }\le h_5,\alpha \in [-5,5]$$

$$G=g_{-7}\le g_{\beta }\le h_7,\beta \in [-7,7]$$

Consider Figure 9, which depicts a linguistic Z-graph with ten vertices of some employees. Also, a graph with a numerical edge and a graph with linguistic edge are shown in Figure 10. The vertex and edge membership values are shown in

Table 9. Vertex membership value of all vertices of Figure 9.

Vertex	Membership Value	Vertex	Membership Value
$v_1$	$(h_{-4},g_{-6})$	$v_6$	$(h_5,g_{-5})$
$v_2$	$(h_4,g_3)$	$v_7$	$(h_3,g_3)$
$v_3$	$(h_2,g_{-1})$	$v_8$	$(h_1,g_1)$
$v_4$	$(h_3,g_3)$	$v_9$	$(h_2,g_1)$
$v_5$	$(h_3,g_{-6})$	$v_{10}$	$(h_{-3},g_2)$

and

Table 10. Edge membership value of all edges of Figure 9.

Edge	Membership Value	Edge	Membership Value
$(v_1,v_4)$	$(h_{-4},g_3)$	$(v_4,v_7)$	$(h_1,g_2)$
$(v_1,v_5)$	$(h_{-4},g_3)$	$(v_4,v_8)$	$(h_2,g_2)$
$(v_1,v_6)$	$(h_{-4},g_{-5})$	$(v_5,v_9)$	$(h_3,g_{-2})$
$(v_2,v_3)$	$(h_1,g_1)$	$(v_5,v_{10})$	$(h_{-3},g_1)$
$(v_2,v_6)$	$(h_4,g_2)$	$(v_6,v_8)$	$(h_2,g_{-4})$
$(v_2,v_{1}0)$	$(h_{-3},g_2)$	$(v_6,v_9)$	$(h_5,g_{-2})$
$(v_3,v_6)$	$(h_1,g_{-2})$	$(v_7,v_8)$	$(h_1,g_{-1})$
$(v_3,v_4)$	$(h_2,g_2)$	$(v_7,v_{10})$	$(h_{-3},g_1)$
$(v_3,v_7)$	$(h_1,g_{-1})$	$(v_8,v_9)$	$(h_{-3},g_{-2})$
$(v_3,v_8)$	$(h_2,g_{-1})$

, respectively. The values of $E\left(z\right)$ of all vertices of the graph in Figure 9 are shown in

Table 11. The values of $E\left(z\right)$ of all vertices of Figure 9.

Vertex	${\mathit{T}}_1$	$\mathit{\alpha }$	${\mathit{T}}_{\mathit{\alpha }}$	${\mathit{T}}_2$	$\mathit{\beta }$	${\mathit{T}}_{\mathit{\beta }}$	$\mathit{E}\left(\mathit{z}\right)$
$V_1$	5	−4	0.1	7	−6	0.07	0.01
$V_2$	5	4	0.9	7	3	0.71	0.64
$V_3$	5	2	0.7	7	−1	0.43	0.3
$V_4$	5	3	0.8	7	3	0.71	0.57
$V_5$	5	3	0.8	7	−6	0.07	0.06
$V_6$	5	5	1	7	−5	0.14	0.14
$V_7$	5	1	0.6	7	1	0.57	0.34
$V_8$	5	2	0.7	7	−2	0.36	0.25
$V_9$	5	5	1	7	1	0.57	0.57
$V_{10}$	5	−3	0.2	7	2	0.64	0.13

. Also, the values of $E\left(z\right)$ of all edges of the graph in Figure 9 are shown in

Table 12. The values of $E\left(z\right)$ of all edges of Figure 9.

Edge	${\mathit{T}}_1$	$\mathit{\alpha }$	${\mathit{T}}_{\mathit{\alpha }}$	${\mathit{T}}_2$	$\mathit{\beta }$	${\mathit{T}}_{\mathit{\beta }}$	$\mathit{E}\left(\mathit{X}\right)$
$(V_1,V_4)$	5	−4	0.1	7	3	0.71	0.07
$(V_1,V_5)$	5	−4	0.1	7	2	0.64	0.06
$(V_1,V_6)$	5	−4	0.1	7	−5	0.14	0.01
$(V_2,V_3)$	5	1	0.6	7	1	0.57	0.34
$(V_2,V_6)$	5	4	0.9	7	2	0.64	0.58
$(V_2,V_{10})$	5	−3	0.2	7	2	0.64	0.13
$(V_3,V_6)$	5	1	0.6	7	−2	0.36	0.22
$(V_3,V_4)$	5	2	0.7	7	2	0.64	0.45
$(V_3,V_7)$	5	1	0.6	7	−1	0.43	0.26
$(V_3,V_8)$	5	2	0.7	7	−1	0.43	0.3
$(V_4,V_7)$	5	1	0.6	7	−2	0.36	0.22
$(V_4,V_8)$	5	2	0.7	7	2	0.64	0.45
$(V_5,V_9)$	5	3	0.8	7	−2	0.36	0.29
$(V_5,V_{10})$	5	−3	0.2	7	1	0.57	0.11
$(V_6,V_8)$	5	2	0.7	7	−4	0.21	0.15
$(V_6,V_9)$	5	5	1	7	−2	0.36	0.36
$(V_7,V_8)$	5	−3	0.2	7	−1	0.43	0.09
$(V_7,V_{10})$	5	−3	0.2	7	1	0.57	0.11
$(V_8,V_9)$	5	−3	0.2	7	−2	0.36	0.07

. The strength of an edge $(v_1,v_4)$ is $S_{(v_1,v_4)}={\displaystyle \frac{0.07}{max\{0..01,0.57\}}}={\displaystyle \frac{0.07}{0.57}}=0.12$. Also, the strengths of all edges of the graph in Figure 9 are shown in

Table 13. Strength values of all edges of Figure 9.

Edge	${\mathit{T}}_1$	$\mathit{\alpha }$	${\mathit{T}}_{\mathit{\alpha }}$	${\mathit{T}}_2$	$\mathit{\beta }$	${\mathit{T}}_{\mathit{\beta }}$	$\mathit{E}\left(\mathit{X}\right)$	Strength
$(V_1,V_4)$	5	−4	0.1	7	3	0.71	0.07	0.12
$(V_1,V_5)$	5	−4	0.1	7	2	0.64	0.06	1
$(V_1,V_6)$	5	−4	0.1	7	−5	0.14	0.01	0.07
$(V_2,V_3)$	5	1	0.6	7	1	0.57	0.34	0.53
$(V_2,V_6)$	5	4	0.9	7	2	0.64	0.58	0.91
$(V_2,V_{10})$	5	−3	0.2	7	2	0.64	0.13	0.2
$(V_3,V_6)$	5	1	0.6	7	−2	0.36	0.22	0.73
$(V_3,V_4)$	5	2	0.7	7	2	0.64	0.45	0.79
$(V_3,V_7)$	5	1	0.6	7	−1	0.43	0.26	0.76
$(V_3,V_8)$	5	2	0.7	7	−1	0.43	0.3	1
$(V_4,V_7)$	5	1	0.6	7	−2	0.36	0.22	0.39
$(V_4,V_8)$	5	2	0.7	7	2	0.64	0.45	0.79
$(V_5,V_9)$	5	3	0.8	7	−2	0.36	0.29	0.51
$(V_5,V{1}_0)$	5	−3	0.2	7	1	0.57	0.11	0.85
$(V_6,V_8)$	5	2	0.7	7	−4	0.21	0.15	0.6
$(V_6,V_9)$	5	5	1	7	−2	0.36	0.36	0.63
$(V_7,V_8)$	5	−3	0.2	7	−1	0.43	0.09	0.26
$(V_7,V_{10})$	5	−3	0.2	7	1	0.57	0.11	0.32
$(V_8,V_9)$	5	−3	0.2	7	−2	0.36	0.07	0.12

. So, the edges $(v_1,v_5),(v_2,v_3),(v_2,v_6),(v_3,v_6),(v_3,v_4),(v_3,v_7),(v_3,v_8),(v_4,v_8),(v_5,v_9),(v_5,v_{10}),(v_6,v_8),(v_6,v_9)$ are strong, and other edges are weak. Now, the centrality of a vertex $v_1$ is $C_{v_1}={\displaystyle \frac{(0.12+1.0+0.07)}{(10-1)}}={\displaystyle \frac{1.19}{9}}=0.1322\approx 0.13$. Also, the centrality values of all vertices of this linguistic Z-graph of Figure 9 are shown in

Table 14. Centrality values of all vertices of linguistic Z-graph of Figure 9.

Vertex	Centrality	$\mathit{E}\left(\mathit{z}\right)$	Total Centrality
$V_1$	0.13	0.01	0.06
$V_2$	0.18	0.64	0.46
$V_3$	0.6	0.3	0.42
$V_4$	0.33	0.57	0.47
$V_5$	0.37	0.06	0.18
$V_6$	0.52	0.14	0.29
$V_7$	0.32	0.34	0.33
$V_8$	0.31	0.25	0.27
$V_9$	0.14	0.57	0.4
$V_{10}$	0.15	0.13	0.14

. Consider that the value $\lambda$ is $0.4$ for this particular case. So, the total degree centrality of the vertex $v_1$ comes out to $P_{v_1}=0.13\times 0.4+0.01\times 0.6=0.052+0.006=0.058\approx 0.06$. The total degree centrality values of all vertices of this linguistic Z-graph of Figure 9 are shown in Table 14. Now, the central node of this network becomes $max\{P_{v_1},P_{v_2},\dots ,P_{v_{1}0}\}=P_{v_4}$. So, the node $v_4$ is the central node of this network. So, the applicant $v_4$ can be selected for the post. Here, this network can be designed using linguistic numbers. Additionally, this selection procedure is designed based on self-weight and collaboration. Therefore, this method yields better results.

7. Conclusions

This study advances the understanding of linguistic Z-graphs within social networks by incorporating both node weight and collaboration. The centrality measure in a linguistic Z-graph provides an insightful method for identifying influential individuals within institutions. The application of linguistic Z-graphs holds significant potential for improving management and decision-making processes within organizations, offering a powerful framework for identifying key influencers and enhancing our understanding of institutional leadership.

In the future, the theory of Z-graphs can be further developed for specific applications in decision making and predictive analysis. By utilizing Z-graphs for predictive analytics, it will be possible to forecast future trends/links in organizational behavior and decision-making processes, providing valuable tools for strategic planning and management. These advancements will enhance the practical applications of Z-graphs, improving our understanding of structural dynamics.