Uncertainty measure for Z-soft covering based rough graphs with application

Abstract

Soft graphs are an interesting way to represent specific information. In this paper, a new form of graphs called Z-soft covering based rough graphs using soft adhesion is defined. Some important properties are explored for the newly constructed graphs. The aim of this study is to investigate the uncertainty in Z-soft covering based rough graphs. Uncertainty measures such as information entropy, rough entropy and granularity measures related to Z-soft covering-based rough graphs are discussed. In addition, we develop a novel Multiple Attribute Group Decision-Making (MAGDM) model using Z-soft covering based rough graphs in medical diagnosis to identify the patients at high risk of chronic kidney disease using the collected data from the UCI Machine Learning Repository.

Keywords

Soft graphs soft covering rough set uncertainty measures decision making

1 Introduction

Different theories have been developed to handle uncertainty in data mining. Various theories such as probability, fuzzy sets, rough sets, soft sets and a combination of these theories have been used to solve issues involving uncertainty. Each of these concepts seems to have its own set of limits and restrictions. Zadeh [28] pioneered fuzzy sets as an effective mathematical tool for dealing with uncertainty. Pawlak [19] proposed the notion of rough sets in 1982. This theory is utilized in data analysis to find fundamental patterns in data, reduce redundancies and create decision rules. Rough sets are mainly used in decision making problems [13]. Several researchers have recently proposed various models to generalize Pawlak’s rough set. Some important models are mentioned in this study. Abd El-Monsef [1] introduced j-neighborhood space and generated eight topologies to obtain eight methods for approximating rough sets. Atef et al. [7] generalized the concepts of three types of rough set models based on j-neighborhood space defined in [1] and they proposed another three types of rough set models based on j-adhesion neighborhood space. El Sayed [9] proposed a novel neighborhood that satisfies all the properties of classical rough sets without adding any additional constraints. El-Bably and Abo-Tabl [10] proposed a new type of rough sets with different neighborhoods and applied their model to the medical application of lung cancer disease to identify the most risk factors for this disease. As an extension of rough sets, covering rough set models is an essential subject for research. Covering rough sets is a valuable technique that allows researchers to look at uncertainty and roughness in a wider sense. Uncertainty measures can provide new perspectives on data analysis. They can assist us in revealing data’s substantive properties. The uncertainty measure is an important problem in rough set theory. Some uncertainty measures like information entropy, rough entropy and granularity measure are introduced by Liang in [14]. The uncertainty for covering rough sets are measured in [8]. Molodtsov [17] coined the term soft sets to describe a valuable mathematical tool to deal with multi-attribute ambiguity. Soft set theory provides a variety of information descriptions and computing operations. Maji et al. created various operations on soft sets [15, 16]. Ali [6] investigated the interconnections among rough set, fuzzy set and soft set. Feng [12] coined the term soft P-rough set and applied it to a real-life problem. Shabir et al. [22] created modified soft rough sets (MSR-sets) using the concept of a soft P-rough set. MSR-sets exceeds the soft P-rough set in terms of precision and soft P-rough set constructions need additional criteria than MSR-sets. Yüksel et al. [26, 27] suggested soft covering based rough sets (SCRS) and provided an application by merging covering soft sets and rough sets. Zhan et al. [29] presented five types of SCRS and explained the relation between soft rough sets and SCRS. In this paper, we use the third type of SCRS as Z-soft covering based rough set (Z-SCRS). Based on an outranking relation, Zhan et al. [30] implemented three-way decisions into multi attribute decision making. They developed the outranked set for each alternative and presented a hybrid information table that an multi attribute decision making matrix with a loss function table. Zhan et al. [31] established group decision-making ideas on Wu-Leung multi-scale information systems from the perspective of multi-expert group decision-making.

Graphs are used in a variety of real-world scenarios, including physical sciences, computer sciences, and several other fields. Computer scientists have spent considerable time researching graphs and graphical operations. Leonhard Euler [11] is widely credited with publishing the first article in 1736, when he constructed the Eulerian graph to solve a Königsburg Bridge problem. Akram and Nawaz [2] introduced the term soft graphs. Various extensions of soft graphs are presented in the following literature [3–5 , 25]. Noor et al. [18] proposed a soft rough graph by combining rough sets, graphs and soft sets. Park et al. [20] established a new form of graphs known as soft covering based rough graphs. To study the uncertainty in soft graphs, Rehman et al. [21] introduced neighborhood based soft covering rough graphs (NSCRG). Uncertainty measure for modified soft rough graphs (MSR-graphs) is explored in [23].

Rough sets and soft sets are two different tools that represent uncertainty and vagueness. Soft graphs are an attractive way to represent information. To strengthen and improve the applicability of soft graphs, an innovative approach is introduced here by combining rough sets with soft graphs called Z-soft covering based rough graphs. We have discussed the uncertainty measures associated with Z-soft covering based rough graphs such as roughness measure, entropy measure and granularity. Some important properties of these uncertainty measures are explored and the relationships between such measures are established. In multiple attribute group decision (MAGDM) methods, decisions are made based on information from individuals. Using this graph-related technique, we have considered individuals as vertices and connections between individuals as edges that increase the accuracy of the results.

The motivation of this study is to discuss the uncertainty in soft graphs. Rehman et al. [21] proposed neighborhood based soft covering rough graphs (NSCRG). From the analysis, we see that NSCRG does not satisfy some of Pawlak’s rough approximations properties. To find an effective model, we have introduced a Z-soft covering based rough graph that satisfies Pawlak’s rough approximation properties.

This paper is organized in the following manner: In Section 2, the basic definitions needed to understand the following sections are provided. In Section 3, soft rough vertex covering graph and soft rough edge covering graph are defined which forms unique graphs called Z-SCRG. Section 4 is devoted to uncertainty measures such as information entropy, rough entropy and granularity associated with Z-SCRG. In Section 5, the relationships between NSCRG and Z-SCRG are analyzed. In Section 6, an application of Z-SCRG in the medical field which helps to identify patients at high risk of chronic kidney disease is provided. The conclusion is discussed in Section 7.

The symbols and abbreviations used in the study are listed in Tables 2, respectively.

Table 1
List of symbols

Symbol Description

Ω Universal set

B Set of parameters

K Soft set

C _K Covering soft set

S Soft covering approximation space

G ^* Simple graph

G ^** Subgraph of G^*

$G$ Soft graph

C _P Covering soft set over V

C _T Covering soft set over E

Q Soft vertex covering approximation space

G _Q Q-soft rough vertex covering graph

L Soft edge covering approximation space

G _L L-soft rough edge covering graph

G = (G_Q, G_L) Z-soft covering rough graph

G_n = (G_{Q
_n}, G_{L
_n}) Neighborhood soft covering rough graph

Δ_Ent (G) Soft adhesion information entropy

Δ_Gran (G) Naive granularity measure

Δ_SA-Ent (G) Soft adhesion elementary entropy

Δ_SA-R-Ent (G) Soft adhesion rough entropy

Symbol	Description
Ω	Universal set
B	Set of parameters
K	Soft set
C _K	Covering soft set
S	Soft covering approximation space
G ^*	Simple graph
G ^**	Subgraph of G^*
$G$	Soft graph
C _P	Covering soft set over V
C _T	Covering soft set over E
Q	Soft vertex covering approximation space
G _Q	Q-soft rough vertex covering graph
L	Soft edge covering approximation space
G _L	L-soft rough edge covering graph
G = (G_Q, G_L)	Z-soft covering rough graph
G_n = (G_{Q _n}, G_{L _n})	Neighborhood soft covering rough graph
Δ_Ent (G)	Soft adhesion information entropy
Δ_Gran (G)	Naive granularity measure
Δ_SA-Ent (G)	Soft adhesion elementary entropy
Δ_SA-R-Ent (G)	Soft adhesion rough entropy

Table 2

List of abbreviations

Abbreviation	Name
MSR-sets	Modified soft rough sets
SCRS	Soft covering based rough sets
Z-SCRS	Z-soft covering based rough sets
CSS	Covering soft set
SCA	Soft covering approximation space
SVCA	Soft vertex covering approximation space
SECA	soft edge covering approximation space
CSG	Covering soft graph
NSCRG	Neighborhood based soft covering rough graphs
MSR-graphs	Modified soft rough graphs
Z-SCRG	Z-soft covering based rough graphs
MAGDM	Multi-attribute group decision making

2 Preliminaries

This section describes the basic definitions needed to understand the following sections. Also, throughout this paper, let Ω be a finite universal set.

Definition 1. A graph is defined as G^* = (V, E), where V is a finite non-empty set whose elements are called vertices and E is a set of unordered pairs of distinct vertices called edges.

Definition 2. [15] Let B be the set of all parameters. An ordered pair K = (N, B) is a soft set over Ω where N is a mapping defined by N : B → P (Ω) and P (Ω) denotes the power set of Ω.

Definition 3. [12] If $⋃_{b \in B} N (b) = Ω$ , then K = (N, B) over Ω is called a full soft set.

Definition 4. [26] A full soft set K = (N, B) over Ω is called a covering soft set (CSS) if N (b)≠ ∅, for all b ∈ B. We denote the covering soft set by C_K.

Definition 5. [26] Let K = (N, B) be CSS over Ω. We call the ordered pair S = (Ω, C_K) a soft covering approximation space (SCA).

Definition 6. [29] Let S = (Ω, C_K) be a SCA. For all u ∈ Ω, the soft adhesion of u is defined as

SA_S (u) ={ v ∈ Ω : ∀ b ∈ B (u ∈ N (b) ↔ v ∈ N (b)) }.

Definition 7. [29] Let S = (Ω, C_K) be a SCA. The soft covering lower and upper approximations of any M ⊆ Ω are defined as

${\underline{SC}}_{S} (M) = {u \in Ω : {SA}_{S} (u) \subseteq M}$ and

${\bar{SC}}_{S} (M) = {u \in Ω : {SA}_{S} (u) \cap M \neq \emptyset}$ respectively. If ${\underline{SC}}_{S} (M) \neq {\bar{SC}}_{S} (M)$ , then M is said to be Z-SCRS; otherwise M is called a Z-soft covering based definable.

Definition 8. [2] Let G^* = (V, E) be a simple graph. Then $G$ = (G^*, ρ, σ, B) is called a soft graph if,

(1) (ρ, B) is a soft set over V,

(2) (σ, B) is a soft set over E,

(3) (ρ (b) , σ (b)) is a subgraph of G^*, for each b ∈ B.

Definition 9. [20] Let $G$ = (G^*, ρ, σ, B) be a soft graph. Then $G$ is called

(1) Full soft vertex graph if $⋃_{b \in B} ρ (b)$ = V.

(2) Full soft edge graph if $⋃_{b \in B} σ (b)$ = E.

(3) Full soft graph if G^* = $(⋃_{b \in B} ρ (b), ⋃_{b \in B} σ (b))$ .

(4) Soft covering vertex graph if ρ (b)≠ ∅, for all b ∈ B.

(5) Soft covering edge graph if σ (b) ≠ ∅, for all b ∈ B. Accordingly, let C_P = (ρ, B) and C_T = (σ, B) be CSS over V and E respectively.

3 Z-soft covering based rough graphs

In this section, three different concepts such as rough sets, soft sets and graphs are combined to introduce Z-soft covering based rough graphs to deal with uncertainty in soft graphs. Examples are provided in order to clarify the concepts. Several basic properties of Z-soft covering based rough graphs are investigated.

Definition 10. Let Q = (V, C_P) be a soft vertex covering approximation space (SVCA) and v_i ∈ V. The soft adhesion for each vertex v_i is defined as, SA_Q (v_i) ={ v_j ∈ V : ∀ b ∈ B (v_i ∈ ρ (b) ↔ v_j ∈ ρ (b)) }.

Definition 11. Let Q = (V, C_P) be an SVCA. For all M ⊆ V, the lower and upper Q-soft vertex covering approximations of M are defined as

${\underline{SC}}_{Q} (M) = {v_{i} \in M : {SA}_{Q} (v_{i}) \subseteq M}$ and

${\bar{SC}}_{Q} (M) = {v_{i} \in M : {SA}_{Q} (v_{i}) \cap M \neq \emptyset}$ respectively. If ${\underline{SC}}_{Q} (M) \neq {\bar{SC}}_{Q} (M)$ , then M is called Q-soft vertex covering based rough set. Let G_Q be a Q-soft rough vertex covering graph. Lower and Upper Q-soft vertex covering approximations for G_Q are defined as $\underline{G_{Q}} = ({\underline{SC}}_{Q} (M), E)$ and $\bar{G_{Q}} = ({\bar{SC}}_{Q} (M), E)$ .

Definition 12. Let G_Q be a Q-soft rough vertex covering graph. The Q-roughness membership function for M ⊆ V is defined as

$δ_{G_{Q}} (M) = 1 - \frac{1}{2} [1 + \frac{| {\underline{SC}}_{Q} (M) |}{| {\bar{SC}}_{Q} (M) |}] .$ G_Q is called Q-soft vertex covering definable if ${\underline{SC}}_{Q} (M)$ = ${\bar{SC}}_{Q} (M)$ , then δ_{G
_Q} (M) = 0.

Example 1. Let G^* = (V, E) be a simple graph where V = {v₁, v₂, v₃, v₄, v₅, v₆, v₇} is a vertex set and E = {e₁, e₂, e₃, e₄, e₅, e₆, e₇, e₈, e₉, e₁₀, e₁₁, e₁₂} is an edge set as shown in Fig. 1. Let B = {b₁, b₂, b₃, b₄, b₅} be the parameter set. Let (ρ, B) be a CSS over V as shown in Table 3. Let Q = (V, C_P) be an SVCA. Then, the soft adhesion for each v_i are SA_Q (v₁) ={ v₁ }, SA_Q (v₂) ={ v₂ }, SA_Q (v₃) ={ v₃ }, SA_Q (v₄) ={ v₄ } SA_Q (v₅) ={ v₅ }, SA_Q (v₆) ={ v₆, v₇ } and SA_Q (v₇) ={ v₆, v₇ }.

Fig. 1

G^* = (V, E).

Table 3

Tabular representation of (ρ, B)

(ρ, B)	v ₁	v ₂	v ₃	v ₄	v ₅	v ₆	v ₇
b ₁	1	0	1	1	0	0	0
b ₂	1	1	0	0	1	0	0
b ₃	0	0	0	0	1	1	1
b ₄	0	0	1	0	1	0	0
b ₅	0	0	0	0	0	1	1

(1) Let M = {v₁, v₃, v₅, v₇ } ⊆ V.

${\underline{SC}}_{Q} (M) = {v_{1}, v_{3}, v_{5}}$ and

${\bar{SC}}_{Q} (M) = {v_{1}, v_{3}, v_{5}, v_{6}, v_{7}} .$

${\underline{G}}_{Q} = ({\underline{SC}}_{Q} (M), E)$ = ({ v₁, v₃, v₅ } , E) and

${\bar{G}}_{Q}$ = $({\bar{SC}}_{Q} (M), E)$ = ({ v₁, v₃, v₅, v₆, v₇ } , E) .

δ_{G
_Q} (M)= 0.2.

(2) Let M = {v₁, v₆, v₇ } ⊆ V .

${\underline{SC}}_{Q} (M) = {v_{1}, v_{6}, v_{7}}$ and ${\bar{SC}}_{Q} (M) = {v_{1}, v_{6}, v_{7}}$ .

where M is Q-soft vertex covering based definable. Then, G_Q is Q-soft vertex covering based definable graph where, ${\underline{G}}_{Q}$ = ({ v₁, v₆, v₇ } , E) = ${\bar{G}}_{Q} .$

The Q-roughness membership function for M is 0.

Definition 13. Let L = (E, C_T) be a soft edge covering approximation space (SECA) and e_m ∈ E. The soft adhesion for each edge e_m is defined as SA_L (e_m) ={ e_n ∈ E : ∀ b ∈ B (e_m ∈ σ (b) ↔ e_n ∈ σ (b)) }.

Definition 14. Let L = (E, C_T) be a SECA. For all O ⊆ E, the lower and upper L-soft edge covering approximations of O are defined as

${\underline{SC}}_{L} (O) = {e_{m} \in O : {SA}_{L} (e_{m}) \subseteq O}$ and

${\bar{SC}}_{L} (O) = {e_{m} \in O : {SA}_{L} (e_{m}) \cap O \neq \emptyset}$ respectively. If ${\underline{SC}}_{L} (O) \neq {\bar{SC}}_{L} (O)$ , then O is called L-soft edge covering based rough sets and G_L is called an L-soft rough edge covering graph. Lower and Upper L-soft edge covering approximations of G_L are denoted by $\underline{G_{L}} = (V, {\underline{SC}}_{L} (O))$ and $\bar{G_{L}} = (V, {\bar{SC}}_{L} (O))$ , for each O ⊆ E.

Definition 15. Let G_L be an L-soft rough edge covering graph. The L-roughness membership function for O is given by, $δ_{G_{L}} (O) = 1 - \frac{1}{2} [1 + \frac{| {\underline{SC}}_{L} (O) |}{| {\bar{SC}}_{L} (O) |}] .$ G_L is called L-soft edge covering definable if ${\underline{SC}}_{L} (O) = {\bar{SC}}_{L} (O)$ , then δ_{G
_L} (O) = 0.

Definition 16. A full soft graph $G = (G^{*}, ρ, σ, B)$ is called a covering soft graph (CSG), if ρ (b)≠ ∅ and σ (b)≠ ∅, for all b ∈ B.

Example 2. Let G^* be a simple graph as shown in Fig. 1. Let (σ, B) be a CSS over E as shown in Table 4. Then, the soft adhesion for each edge is SA (e₁) ={ e₁, e₃, e₅ }, SA_L (e₂) ={ e₂, e₇ }, SA_L (e₃) ={ e₃, e₅ }, SA_L (e₄) ={ e₄, e₉, e₁₂ }, SA_L (e₅) ={ e₃, e₅ }, SA_L (e₆) ={ e₆ }, SA_L (e₇) ={ e₂, e₇ }, SA_L (e₈) ={ e₈, e₁₀ }, SA_L (e₉) ={ e₄, e₉, e₁₂ }, SA_L (e₁₀) ={ e₈, e₁₀ }, SA_L (e₁₁) ={ e₁₁ } and SA_L (e₁₂) ={ e₄, e₉, e₁₂ }.

Let O = {e₁, e₂, e₃, e₅, e₆, e₈} ⊆ E. Then

${\underline{SC}}_{L} (O) = {e_{1}, e_{3}, e_{5}, e_{6}}$ and

${\bar{SC}}_{L} (O) = {e_{1}, e_{2}, e_{3}, e_{5}, e_{6}, e_{7}, e_{8}, e_{10}}$ .

Since, ${\underline{SC}}_{L} (O) \neq {\bar{SC}}_{L} (O)$ . Hence, O is the soft edge covering based rough set and G_L is an L-soft rough edge covering graph where, ${\underline{G}}_{L} = (V, {e_{1}, e_{3}, e_{5}, e_{6}})$ ,

${\bar{G}}_{L} = (V, {e_{1}, e_{2}, e_{3}, e_{5}, e_{6}, e_{7}, e_{8}, e_{10}})$ . The L-roughness membership function for O is 0.25.

Table 4

Tabular representation of (σ, B)

(σ, B)	e ₁	e ₂	e ₃	e ₄	e ₅	e ₆	e ₇	e ₈	e ₉	e ₁₀	e ₁₁	e ₁₂
b ₁	0	1	1	0	1	0	1	0	0	0	0	0
b ₂	1	0	1	0	1	0	0	1	0	1	0	0
b ₃	1	0	1	1	1	1	0	0	1	0	0	1
b ₄	0	0	1	1	1	0	0	0	1	0	1	1
b ₅	0	0	0	1	0	0	0	0	1	0	0	1

Definition 17. Let G^** = (M, O) be a subgraph of G^* then $δ_{G^{*}} (G^{* *}) = 1 - \frac{1}{2} [\frac{| {\underline{SC}}_{Q} (M) |}{| {\bar{SC}}_{Q} (M) |} + \frac{| {\underline{SC}}_{L} (O) |}{| {\bar{SC}}_{L} (O) |}]$ is the roughness membership function of G^**.

Definition 18. Let $G = (G^{*}, ρ, σ, B)$ is called Z-soft covering based definable if

(1) M is Q-soft vertex covering definable, i.e.,

${\underline{SC}}_{Q} (M) = {\bar{SC}}_{Q} (M)$ for all M ⊆ V and

(2) O is L-soft edge covering definable, i.e.,

${\underline{SC}}_{L} (O) = {\bar{SC}}_{L} (O)$ for all O ⊆ E.

Definition 19. Let $G = (G^{*}, ρ, σ, B)$ is called Z-SCRG if

(1) M is Q-soft vertex covering based rough set, i.e., ${\underline{SC}}_{Q} (M) \neq {\bar{SC}}_{Q} (M)$ for all M ⊆ V and

(2) O is L-soft edge covering based rough set, i.e., ${\underline{SC}}_{L} (O) \neq {\bar{SC}}_{L} (O)$ for all O ⊆ E.

A Z-SCRG is denoted by G = (G_Q, G_L).

Definition 20. The upper and lower approximations of G = (G_Q, G_L) are defined as

$\bar{SC} (G) = ({\bar{SC}}_{Q} (M), {\bar{SC}}_{L} (O))$ and

$\underline{SC} (G) = ({\underline{SC}}_{Q} (M), {\underline{SC}}_{L} (O))$ respectively, for all M ⊆ V and O ⊆ E.

Proposition 1. Let $G = (G^{*}, ρ, σ, B)$ be a CSG. Let Q = (V, C_P) and L = (E, C_T) be SVCA and SECA respectively. Let G₁ and G₂ be subgraphs of G^*. Then

(1) $\underline{SC} (G^{*}) = G^{*} = \bar{SC} (G^{*})$ .

(2) $\underline{SC} (\emptyset) = \emptyset = \bar{SC} (\emptyset)$ .

(3) ${\underline{SC}}_{Q} (M) = M = {\bar{SC}}_{Q} (M)$ .

(4) ${\underline{SC}}_{L} (O) = O = {\bar{SC}}_{L} (O)$ .

(5) If G₁ ⊆ G₂ then $\underline{SC} (G_{1})$ ⊆ $\underline{SC} (G_{2})$ and $\bar{SC} (G_{1})$ ⊆ $\bar{SC} (G_{2})$ .

(6) $\bar{SC} (G_{1} \cap G_{2}) \subseteq \bar{SC} (G_{1}) \cap \bar{SC} (G_{2})$ .

(7) $\bar{SC} (G_{1} \cup G_{2}) = \bar{SC} (G_{1}) \cup \bar{SC} (G_{2})$ .

(8) $\underline{SC} (G_{1} \cup G_{2}) \supseteq \underline{SC} (G_{1}) \cup \underline{SC} (G_{2}) .$

Proof. (1), (2), (3) and (4) can be verified directly.

(5) If G₁ ⊆ G₂, so (M₁, O₁) ⊆ (M₂, O₂) or M₁ ⊆ M₂ and O₁ ⊆ O₂. Let $(v_{i}, e_{m}) \in \underline{SC} (G_{1})$ = $({\underline{SC}}_{Q} (M_{1}), {\underline{SC}}_{L} (O_{1}))$ . i.e. v_i ∈ M₁ with SA_Q (v_i) ⊆ M₁ and e_m ∈ O₁ with SA_L (e_m) ⊆ O₁ since M₁ ⊆ M₂ and O₁ ⊆ O₂, so v_i ∈ M₂ with SA_Q (v_i) ⊆ M₂ and e_m ∈ O₂ with SA_L (e_m) ⊆ O₂. Thus $v_{i} \in {\underline{SC}}_{Q} (M_{2})$ and $e_{m} \in {\underline{SC}}_{L} (O_{2})$ showing that $(v_{i}, e_{m}) \in ({\underline{SC}}_{Q} (M_{2})$ , ${\underline{SC}}_{L} (O_{2})) = \underline{SC} (G_{2})$ . Hence, $\underline{SC} (G_{1}) \subseteq \underline{SC} (G_{2})$ . Similarly, we can prove $\bar{SC} (G_{1}) \subseteq \bar{SC} (G_{2})$ .

(6) Since G₁ ∩ G₂ ⊆ G₁ and G₁ ∩ G₂ ⊆ G₂.

From (5), $\bar{SC} (G_{1} \cap G_{2}) \subseteq \bar{SC} (G_{1})$ and $\bar{SC} (G_{1} \cap G_{2}) \subseteq \bar{SC} (G_{2})$ . Therefore, $\bar{SC} (G_{1} \cap G_{2}) \subseteq \bar{SC} (G_{1}) \cap \bar{SC} (G_{2})$ .

(7) By using (5), we can say that $\bar{SC} (G_{1}) \cup \bar{SC} (G_{2}) \subseteq \bar{SC} (G_{1} \cup G_{2})$ . Now, we have to prove that $\bar{SC} (G_{1} \cup G_{2}) \subseteq \bar{SC} (G_{1}) \cup \bar{SC} (G_{2})$ . If $(v_{i}, e_{m}) \in \bar{SC} (G_{1} \cup G_{2})$ = $({\bar{SC}}_{Q} (M_{1} \cup M_{2}), {\bar{SC}}_{L} (O_{1} \cup O_{2}))$ then $v_{i} \in {\bar{SC}}_{Q} (M_{1} \cup M_{2})$ and $e_{m} \in {\bar{SC}}_{L} (O_{1} \cup O_{2})$ where v_i ∈ (M₁ ∪ M₂) with SA_Q (v_i)∩ (M₁ ∪ M₂) ≠ ∅ and e_m ∈ (O₁ ∪ O₂) with SA_L (e_m)∩ (O₁ ∪ O₂) ≠ ∅. So v_i ∈ M₁ such that SA_Q (v_i)∩ M₁ ≠ ∅ or v_i ∈ M₂ with SA_Q (v_i)∩ M₂ ≠ ∅. Both the conditions says that $v_{i} \in {\bar{SC}}_{Q} (M_{1})$ or ${\bar{SC}}_{Q} (M_{2})$ . So $v_{i} \in ({\bar{SC}}_{Q} (M_{1}) \cup {\bar{SC}}_{Q} (M_{2}))$ . Similarly, we can prove that $e_{m} \in ({\bar{SC}}_{L} (O_{1}) \cup {\bar{SC}}_{L} (O_{2}))$ , which means that $(v_{i}, e_{m}) \in \bar{SC} (G_{1})$ or $\bar{SC} (G_{2})$ . i.e. $(v_{i}, e_{m}) \in \bar{SC} (G_{1}) \cup \bar{SC} (G_{2})$ . Thus, $\bar{SC} (G_{1} \cup G_{2}) \subseteq \bar{SC} (G_{1}) \cup \bar{SC} (G_{2})$ . Hence, $\bar{SC} (G_{1} \cup G_{2})$ = $\bar{SC} (G_{1}) \cup \bar{SC} (G_{2})$ .

(8) Since G₁ ⊆ G₁ ∪ G₂ and G₂ ⊆ G₁ ∪ G₂ by using (5) $\underline{SC} (G_{1}) \subseteq \underline{SC} (G_{1} \cup G_{2})$ and $\underline{SC} (G_{2}) \subseteq \underline{SC} (G_{1} \cup G_{2})$ . Thus $\underline{SC} (G_{1} \cup G_{2}) \supseteq \underline{SC} (G_{1}) \cup \underline{SC} (G_{2}) .$ □

In the above proposition, the lower and upper approximations of the newly constructed graphs satisfy the properties of Pawlak’s rough approximations. This proves that our proposed model Z-soft covering-based rough graph is significant and effective.

4 Uncertainty measurement for Z-soft covering based rough graphs

In this section, uncertainty measures in Z-soft covering based rough graphs are explored.

Definition 21. Let Q = {V, C_P} be an SVCA. For all v_i ∈ V, the soft association for each v_i is defined as C_Q (v_i) ={ ρ (b) ∈ C_P : v_i ∈ ρ (b) }.

Definition 22. Let Q = {V, C_P} be a SVCA and v_i ∈ V, then the other form of soft adhesion for each v_i ∈ V is defined as

SA_Q (v_i) ={ v_j ∈ V : C_Q (v_i) = C_Q (v_j) } for all v_i, v_j ∈ V.

Definition 23. Let L = {E, C_T} be a SECA and e_m ∈ E, then the soft association for each e_m is defiend as

C_L (e_m) ={ σ (b) ∈ C_T : e_m ∈ σ (b) }.

Definition 24. Let L = {E, C_T} be a SECA and e_m ∈ E, then the other form of soft adhesion for each e_m ∈ E is defined as

SA_L (e_m) ={ e_n ∈ E : C_L (e_m) = C_L (e_n) } for all e_m, e_n ∈ E.

Definition 25. Let Q = {V, C_P} and Q^* = ${V, C_{P}^{*}}$ be any two SVCA with

(1) for any ρ (b) ∈ C_Q (v), there exists $ρ^{*} (b) \in C_{Q}^{*} (v)$ such that ρ (b) ⊆ ρ^* (b) and

(2) for any $ρ^{*} (b) \in C_{Q}^{*} (v)$ , there exists ρ (b) ∈ C_Q (v) such that ρ (b) ⊆ ρ^* (b).

Thus, we can say $C_{P}^{*}$ is finer than C_P and denoted by C_P ⪯ $C_{P}^{*}$ .

Proposition 2. Let Q = {V, C_P} and Q^* = ${V, C_{P}^{*}}$ be two SVCA with C_P ⪯ $C_{P}^{*}$ . Then SA_Q (v_i) ⊆ SA_{Q
^*} (v_i).

Proof. Suppose v_j ∈ SA_Q (v_i). Then v_j ∈ ρ (b) for any ρ (b) ∈ C_Q (v_i) = v_j ∈ ρ (b) for any ρ (b) ∈ C_Q (v_j). By definition, for any ρ^* (b) ∈ $C_{Q}^{*} (v_{i})$ there exist ρ (b) ∈ $C_{Q}^{*} (v_{i})$ such that v_j ∈ ρ (b) ⊆ ρ^* (b). Hence v_j ∈ ρ^* (b) for any ρ^* (b) ∈ $C_{Q}^{*} (v_{i})$ = v_j ∈ ρ^* (b) for any ρ^* (b) ∈ $C_{Q}^{*} (v_{j})$ . Therefore, $C_{Q}^{*} (v_{i})$ = $C_{Q}^{*} (v_{j})$ which implies v_j ∈ SA_{Q
^*} (v_i). Thus SA_Q (v_i) ⊆ SA_{Q
^*} (v_i). □

Definition 26. Let L = {E, C_T} and L^* = ${E, C_{T}^{*}}$ be two SECA with

(1) for any σ (b) ∈ C_L (e_m), there exists $σ^{*} (b) \in C_{L}^{*} (e_{m})$ such that σ (b) ⊆ σ^* (b) and

(2) for any $σ^{*} (b) \in C_{L}^{*} (e_{m})$ , there exists σ (b) ∈ C_L (e_m) such that σ (b) ⊆ σ^* (b).

Thus, we can say $C_{T}^{*}$ is finer than C_T and denoted by C_T ⪯ $C_{T}^{*}$ .

Proposition 3. Let L = {E, C_T} and L^* = ${E, C_{T}^{*}}$ be two SECA with C_T ⪯ $C_{T}^{*}$ . Then SA_L (e_m) ⊆ SA_{L
^*} (e_m).

Proof. The proof is similar to the proof of Proposition 2. □

Definition 27. Let G = (G_Q, G_L) be a Z-SCRG. Then $\begin{matrix} Δ_{Ent} (G) & = \sum_{v_{i} \in V} \frac{1}{| V |} [1 - \frac{| {SA}_{Q} (v_{i}) |}{| V |}] \\ + \sum_{e_{m} \in E} \frac{1}{| E |} [1 - \frac{| {SA}_{L} (e_{m}) |}{| E |}] \end{matrix}$ is called a soft adhesion information entropy.

Proposition 4. Suppose G = (G_Q, G_L) and G^* = (G_{Q
^*}, G_{L
^*}) are two Z-SCRG such that C_P ⪯ $C_{P}^{*}$ and C_T ⪯ $C_{T}^{*}$ then Δ_Ent (G) ≥ Δ_Ent (G^*).

Proof. By Propositions 2 and 3, we get SA_Q (v_i) ⊆ SA_{Q
^*} (v_i) and SA_L (e_m) ⊆ SA_{L
^*} (e_m) which implies $\begin{matrix} \sum_{v_{i} \in V} \frac{1}{| V |} [1 - \frac{| {SA}_{Q} (v_{i}) |}{| V |}] & + \sum_{e_{m} \in E} \frac{1}{| E |} [1 - \frac{| {SA}_{L} (e_{m}) |}{| E |}] \\ \geq \sum_{v_{i} \in V} \frac{1}{| V |} [1 - \frac{| {SA}_{Q^{*}} (v_{i}) |}{| V |}] \\ + \sum_{e_{m} \in E} \frac{1}{| E |} [1 - \frac{| {SA}_{L^{*}} (e_{m}) |}{| E |}] \end{matrix}$ Hence, Δ_Ent (G) ≥ Δ_Ent (G^*). □

Remark 1. Δ_Ent (G) attains its maximal value 2 - $\frac{1}{| V |}$ - $\frac{1}{| E |}$ for |SA_Q (v_i) | = 1 = |SA_L (e_m) | and Δ_Ent (G) attains its minimal value 0 for each |SA_Q (v_i) | = |V| and |SA_L (e_m) | = |E| for v_i ∈ V and e_m ∈ E.

Definition 28. Let G = (G_Q, G_L) be Z-SCRG. Then $\begin{matrix} Δ_{Gran} (G) & = \frac{1}{| V |} \sum_{v_{i} \in V} \frac{| {SA}_{Q} (v_{i}) |}{| V |} \\ + \frac{1}{| E |} \sum_{e_{m} \in E} \frac{| {SA}_{L} (e_{m}) |}{| E |} \end{matrix}$ is the naive granularity measure of G.

Proposition 5. Let G = (G_Q, G_L) and G^* = (G_{Q
^*}, G_{L
^*}) be two Z-SCRG such that C_P ⪯ $C_{P}^{*}$ and C_T ⪯ $C_{T}^{*}$ then Δ_Gran (G) ≤ Δ_Gran (G^*).

Proof. By Propositions 2 and 3, we get SA_Q (v_i) ⊆ SA_{Q
^*} (v_i) and SA_L (e_m) ⊆ SA_{L
^*} (e_m). $\begin{matrix} \Rightarrow \frac{1}{| V |} \sum_{v_{i} \in V} \frac{| {SA}_{Q} (v_{i}) |}{| V |} & + \frac{1}{| E |} \sum_{e_{m} \in E} \frac{| {SA}_{L} (e_{m}) |}{| E |} \\ \leq \frac{1}{| V |} \sum_{v_{i} \in V} \frac{| {SA}_{Q^{*}} (v_{i}) |}{| V |} \\ + \frac{1}{| E |} \sum_{e_{m} \in E} \frac{| {SA}_{L^{*}} (e_{m}) |}{| E |} . \end{matrix}$ Hence, Δ_Gran (G) ≤ Δ_Gran (G^*) . □

Remark 2. Δ_Gran (G) reaches its maximal value 2 if |SA_Q (v_i) | = |V| and |SA_L (e_m) | = |E| and reaches the minimal value $\frac{1}{| V |} + \frac{1}{| E |}$ if |SA_Q (v_i) | = 1 = |SA_L (e_m) |, for each v_i ∈ V and e_m ∈ E.

Proposition 6. Let G = (G_Q, G_L) be a Z-SCRG then the relationship between the soft adhesion information entropy and the naive granularity measure of G is Δ_Ent (G) =2 - Δ_Gran (G).

Proof. $\begin{matrix} Δ_{Ent} (G) & = \sum_{v_{i} \in V} \frac{1}{| V |} [1 - \frac{| {SA}_{Q} (v_{i}) |}{| V |}] \\ + \sum_{e_{m} \in E} \frac{1}{| E |} [1 - \frac{| {SA}_{L} (e_{m}) |}{| E |}] \\ Δ_{Ent} (G) & = 2 - Δ_{Gran} (G) . \end{matrix}$ □

Example 3. Let G^* = (V, E) be a simple graph as shown in Fig. 2. Let (ρ, B) over V and (σ, B) over E are shown in Tables 5 6 respectively.

Fig. 2

G^* = (V, E).

Table 5

Tabular representation of (ρ, B)

(ρ, B)	v ₁	v ₂	v ₃	v ₄	v ₅	v ₆
b ₁	1	0	1	0	1	0
b ₂	0	1	1	0	0	1
b ₃	0	0	1	1	0	0
b ₄	1	1	0	0	1	1
b ₅	1	0	1	0	1	0

Table 6

Tabular representation of (σ, B)

(σ, B)	e ₁	e ₂	e ₃	e ₄	e ₅	e ₆	e ₇	e ₈	e ₉	e ₁₀
b ₁	1	0	1	0	1	1	1	1	0	0
b ₂	1	0	1	1	1	0	0	0	1	1
b ₃	0	1	0	1	0	1	1	1	0	1
b ₄	0	1	0	0	0	0	1	0	1	0
b ₅	1	0	1	1	1	1	0	1	1	0

Let Q = {V, C_P} and L = {E, C_T} be SVCA and SECA respectively. Let the soft association of each vertex be C_Q (v₁) ={ ρ (b₁) , ρ (b₄) , ρ (b₅) }, C_Q (v₂) ={ ρ (b₂) , ρ (b₄) }, C_Q (v₃) ={ ρ (b₁) , ρ (b₂) , ρ (b₃) , ρ (b₅) }, C_Q (v₄) ={ ρ (b₃) }, C_Q (v₅) ={ ρ (b₁) , ρ (b₄) , ρ (b₅) } and C_Q (v₆) ={ ρ (b₂) , ρ (b₄) }. Then, SA_Q (v₁) ={ v₁, v₅ }, SA_Q (v₂) ={ v₂, v₆ }, SA_Q (v₃) ={ v₃ }, SA_Q (v₄) ={ v₄ }, SA_Q (v₅) ={ v₁, v₅ } and SA_Q (v₆) ={ v₂, v₆ }. Let the soft association of each edge be C_L (e₁) ={ σ (b₁) , σ (b₂) , σ (b₅) }, C_L (e₂) ={ σ (b₃) , σ (b₄) }, C_L (e₃) ={ σ (b₁) , σ (b₂) , σ (b₅) },

C_L (e₄) ={ σ (b₂) , σ (b₃) , σ (b₅) },

C_L (e₅) ={ σ (b₁) , σ (b₂) , σ (b₅) },

C_L (e₆) ={ σ (b₁) , σ (b₃) , σ (b₅) },

C_L (e₇) ={ σ (b₁) , σ (b₃) , σ (b₄) },

C_L (e₈) ={ σ (b₁) , σ (b₃) , σ (b₅) },

C_L (e₉) ={ σ (b₃) , σ (b₄) , σ (b₅) } and C_L (e₁₀) ={ σ (b₂) , σ (b₃) }. Then, SA_L (e₁) ={ e₁, e₃, e₅ }, SA_L (e₂) ={ e₂ }, SA_L (e₃) ={ e₁, e₃, e₅ }, SA_L (e₄) ={ e₄ }, SA_L (e₅) ={ e₁, e₃, e₅ }, SA_L (e₆) ={ e₆, e₈ }, SA_L (e₇) ={ e₇ }, SA_L (e₈) ={ e₆, e₈ }, SA_L (e₉) ={ e₉ }, and SA_L (e₁₀) ={ e₁₀ }.

Thus, $Δ_{Ent} (G) = \frac{347}{225}$ and $Δ_{Gran} (G) = \frac{206}{450}$ .

Definition 29. Let G = (G_Q, G_L) be Z-SCRG. Then $\begin{matrix} Δ_{SA - Ent} (G) & = - \frac{1}{| V |} \sum_{v_{i} \in V} \log_{2} \frac{| {SA}_{Q} (v_{i}) |}{| V |} \\ - \frac{1}{| E |} \sum_{e_{m} \in E} \log_{2} \frac{| {SA}_{L} (e_{m}) |}{| E |} \end{matrix}$ is the soft adhesion elementary entropy of G.

Proposition 7. Suppose G = (G_Q, G_L) and G^* = (G_{Q
^*}, G_{L
^*}) are two Z-SCRG such that C_P ⪯ $C_{P}^{*}$ and C_T ⪯ $C_{T}^{*}$ then Δ_SA-Ent (G^*) ≤ Δ_SA-Ent (G).

Proof. By Propositions 2 and 3, we get SA_Q (v_i) ⊆ SA_{Q
^*} (v_i) and SA_L (e_m) ⊆ SA_{L
^*} (e_m) which implies $- \frac{1}{| V |} \sum_{v_{i} \in V} \log_{2} \frac{| {SA}_{Q} (v_{i}) |}{| V |} \geq - \frac{1}{| V |} \sum_{v_{i} \in V} \log_{2} \frac{| {SA}_{Q^{*}} (v_{i}) |}{| V |}, - \frac{1}{| E |} \sum_{e_{m} \in E} \log_{2} \frac{| {SA}_{L} (e_{m}) |}{| E |} \geq - \frac{1}{| E |} \sum_{e_{m} \in E} \log_{2} \frac{| {SA}_{L^{*}} (e_{m}) |}{| E |}, - \frac{1}{| V |} \sum_{v_{i} \in V} \log_{2} \frac{| {SA}_{Q} (v_{i}) |}{| V |} - \frac{1}{| E |} \sum_{e_{m} \in E} \log_{2} \frac{| {SA}_{L} (e_{m}) |}{| E |} \geq - \frac{1}{| V |} \sum_{v_{i} \in V} \log_{2} \frac{| {SA}_{Q^{*}} (v_{i}) |}{| V |} - \frac{1}{| E |} \sum_{e_{m} \in E} \log_{2} \frac{| {SA}_{L^{*}} (e_{m}) |}{| E |} .$ Hence, Δ_SA-Ent (G^*) ≤ Δ_SA-Ent (G). □

Remark 3. Δ_SA-Ent (G) attains its maximal value log₂|V| + log₂|E| when |SA_Q (v_i) | = |SA_L (e_m) | = 1 and Δ_SA-Ent (G) reaches its minimal value 0 when |SA_Q (v_i) | = |V| and |SA_L (e_m) | = |E|, for each v_i ∈ V and e_m ∈ E.

Definition 30. Let G = (G_Q, G_L) be Z-SCRG. Then $\begin{matrix} Δ_{SA - R - Ent} (G) & = - \frac{1}{| V |} \sum_{v_{i} \in V} \log_{2} \frac{| 1 |}{| {SA}_{Q} (v_{i}) |} \\ - \frac{1}{| E |} \sum_{e_{m} \in E} \log_{2} \frac{| 1 |}{| {SA}_{L} (e_{m}) |} . \end{matrix}$ is the soft adhesion rough entropy of G.

Proposition 8. Suppose G = (G_Q, G_L) and G^* = (G_{Q
^*}, G_{L
^*}) are two Z-SCRG such that C_P ⪯ $C_{P}^{*}$ and C_T ⪯ $C_{T}^{*}$ then Δ_SA-R-Ent (G) ≤ Δ_SA-R-Ent (G^*).

Proof. By Proposition 2 and 3, we get SA_Q (v_i) ⊆ SA_{Q
^*} (v_i) and SA_L (e_m) ⊆ SA_{L
^*} (e_m) which implies $- \frac{1}{| V |} \sum_{v_{i} \in V} \log_{2} \frac{| 1 |}{| {SA}_{Q} (v_{i}) |} \leq - \frac{1}{| V |} \sum_{v_{i} \in V} \log_{2} \frac{| 1 |}{| {SA}_{Q^{*}} (v_{i}) |}, - \frac{1}{| E |} \sum_{e_{m} \in E} \log_{2} \frac{| 1 |}{| {SA}_{L} (e_{m}) |} \leq - \frac{1}{| E |} \sum_{e_{m} \in E} \log_{2} \frac{| 1 |}{| {SA}_{L^{*}} (e_{m}) |}, - \frac{1}{| V |} \sum_{v_{i} \in V} \log_{2} \frac{| 1 |}{| {SA}_{Q} (v_{i}) |} - \frac{1}{| E |} \sum_{e_{m} \in E} \log_{2} \frac{| 1 |}{| {SA}_{L} (e_{m}) |} \leq - \frac{1}{| V |} \sum_{v_{i} \in V} \log_{2} \frac{| 1 |}{| {SA}_{Q^{*}} (v_{i}) |} - \frac{1}{| E |} \sum_{e_{m} \in E} \log_{2} \frac{| 1 |}{| {SA}_{L^{*}} (e_{m}) |} .$ Hence, Δ_SA-R-Ent (G) ≤ Δ_SA-R-Ent (G^*). □

Remark 4. Δ_SA-R-Ent (G) attains its maximal value log₂|V| + log₂|E| when |SA_Q (v_i) | = |V| and |SA_L (e_m) | = |E|. Δ_SA-R-Ent (G) reaches its minimal value 0 when |SA_Q (v_i) | = 1 = |SA_L (e_m) |, for each v_i ∈ V and e_m ∈ E.

Proposition 9. Let G = (G_Q, G_L) be Z-SCRG then Δ_SA-Ent (G) + Δ_SA-R-Ent (G) = log₂|V| + log₂|E|.

Proof. $Δ_{SA - Ent} (G) + Δ_{SA - R - Ent} (G) = [- \frac{1}{| V |} \sum_{v_{i} \in V} \log_{2} \frac{| {SA}_{Q} (v_{i}) |}{| V |} - \frac{1}{| E |} \sum_{e_{m} \in E} \log_{2} \frac{| {SA}_{L} (e_{m}) |}{| E |}] + [- \sum_{v_{i} \in V} \frac{1}{| V |} \log_{2} \frac{| 1 |}{| {SA}_{Q} (v_{i}) |} - \sum_{e_{m} \in E} \frac{1}{| E |} \log_{2} \frac{| 1 |}{| {SA}_{L} (e_{m}) |}] Δ_{SA - Ent} (G) + Δ_{SA - R - Ent} (G) = \log_{2} | V | + \log_{2} | E | .$ □

Proposition 6 states the relationship between the soft adhesion information entropy and the naive granularity measure and Proposition 9 provides the relationship between the soft adhesion elementary entropy and the soft adhesion rough entropy.

5 Relationship between NSCRG and Z-SCRG

In this section, the relationship between NSCRG [21] and Z-SCRG is analyzed. Also, the significance of our model is explored. We define the soft adhesion of vertices and edges using the soft neighborhood of vertices and edges defined in [21] as follows:

Let Q = (V, C_P) be an SVCA and v_i ∈ V. Then the soft adhesion for each v_i is defined as SA_Q (v_i) ={ v_j ∈ V : N_Q (v_i) = N_Q (v_j) } where

N_Q (v_i) =∩ { ρ (b) ∈ C_P : v_i ∈ ρ (b) }. Similarly, we can define the soft adhesion of edges.

Proposition 10. Let $G = (G^{*}, ρ, σ, B)$ be a CSG. Let Q = (V, C_P) and L = (E, C_T) be SVCA and SECA respectively. Then

(1) SA_Q (v_i) ⊆ N_Q (v_i) ,

(2) SA_L (e_m) ⊆ N_L (e_m) .

Proof. Let v_j ∈ SA_Q (v_i) then C_Q (v_i) = C_Q (v_j) which means that v_j ∈ ρ (b) for any ρ (b) ∈ C_Q (v_i). By definition, we can say that v_j ∈ ∩ C_Q (v_i) = N_Q (v_i). Similarly, (2) can be proved. □

Proposition 11. Let $G = (G^{*}, ρ, σ, B)$ be a CSG. Let Q = (V, C_P) and L = (E, C_T) be SVCA and SECA respectively. Then

(1) ${\underline{N}}_{Q} (M) \subseteq {\underline{SC}}_{Q} (M)$ ,

(2) ${\bar{SC}}_{Q} (M) \subseteq {\bar{N}}_{Q} (M)$ for every M ⊆ V.

(3) ${\underline{N}}_{L} (O) \subseteq {\underline{SC}}_{L} (O)$ ,

(4) ${\bar{SC}}_{L} (O) \subseteq {\bar{N}}_{L} (O)$ for every O ⊆ E .

Proof. (1) Let $v_{i} \in {\underline{N}}_{Q} (M)$ then N_Q (v_i) ⊆ M. Since SA_Q (v_i) ⊆ N_Q (v_i) which implies SA_Q (v_i) ⊆ M. Hence $v_{i} \in {\underline{SC}}_{Q} (M)$ and so ${\underline{N}}_{Q} (M) \subseteq {\underline{SC}}_{Q} (M)$ .

(2) Let $v_{i} \in {\bar{SC}}_{Q} (M)$ then SA (v_i) ∩ M ≠ ∅. Since SA_Q (v_i) ⊆ N_Q (v_i) which implies N_Q (v_i) ∩ M ≠ ∅. By definition, $N_{Q} (v_{i}) \in {\bar{N}}_{Q} (M)$ which shows that $v_{i} \in {\bar{N}}_{Q} (M)$ . Thus ${\bar{SC}}_{Q} (M) \subseteq {\bar{N}}_{Q} (M)$ . Similarly, (3) and (4) can be verified. □

From Proposition 10, we can understand the relation between soft adhesion and neighborhood for each vertex and edge. Similarly, Proposition 11 gives the relation between lower and upper approximations of Z-SCRG and NSCRG.

Proposition 12. Let $G = (G^{*}, ρ, σ, B)$ be a CSG. Let Q = (V, C_P) and L = (E, C_T) be SVCA and SECA respectively. Then

(1) ${\underline{SC}}_{Q} (- M) = - {\bar{SC}}_{Q} (M)$ ,

(2) ${\bar{SC}}_{Q} (- M) = - {\underline{SC}}_{Q} (M)$ for every M ⊆ V.

(3) ${\underline{SC}}_{L} (- O) = - {\bar{SC}}_{L} (O)$ ,

(4) ${\bar{SC}}_{L} (- O) = - {\underline{SC}}_{L} (O)$ for every O ⊆ E.

Proof. (1) Let $v_{i} \in {\underline{SC}}_{Q} (- M)$ iff SA_Q (v_i) ⊆ - M iff SA_Q (v_i)∩ M = ∅ iff $v_{i} \notin {\bar{SC}}_{Q} (M)$ iff $v_{i} \in - {\bar{SC}}_{Q} (M)$ . Thus ${\underline{SC}}_{Q} (- M) = - {\bar{SC}}_{Q} (M)$ .

(2) Let $v_{i} \in {\bar{SC}}_{Q} (- M)$ iff SA_Q (v₂)∩ - M ≠ ∅ iff there exits v_j ∈ SA_Q (v_i) such that v_j ∈ - M iff SA_Q (v₂) ⊆ M iff $v_{i} \in - {\underline{SC}}_{Q} (M)$ . Thus ${\bar{SC}}_{Q} (- M) = - {\underline{SC}}_{Q} (M)$ . Similarly, (3) and (4) can be verified. □

Proposition 13. Let $G = (G^{*}, ρ, σ, B)$ be a CSG. Let Q = (V, C_P) and L = (E, C_T) be SVCA and SECA respectively. Then

(1) ${\underline{N}}_{Q} (- M) \neq - {\bar{N}}_{Q} (M)$ ,

(2) ${\bar{N}}_{Q} (- M) \neq - {\underline{N}}_{Q} (M)$ for every M ⊆ V.

(3) ${\underline{N}}_{L} (- O) \neq - {\bar{N}}_{L} (O)$ ,

(4) ${\bar{N}}_{L} (- O) \neq - {\underline{N}}_{L} (O)$ for every O ⊆ E.

Example 4. Let G^* = (V, E) be a simple graph as shown in Fig. 1. Let (ρ, B) be a CSS over V as shown in Table 3. Let Q = (V, C_P) be the SVCA. Then soft neighborhood for each v_i are N_Q (v₁) ={ v₁ }, N_Q (v₂) ={ v₁, v₂, v₅ }, N_Q (v₃) ={ v₃ }, N_Q (v₄) ={ v₁, v₃, v₄ } N_Q (v₅) ={ v₅ }, N_Q (v₆) ={ v₆, v₇ } and N_Q (v₇) ={ v₆, v₇ }.

Let M ={ v₁, v₂, v₅, v₇ } and -M ={ v₃, v₄, v₆ }.

(1) ${\underline{N}}_{Q} (- M) = {v_{3}}$ and $- {\bar{N}}_{Q} (M) = \emptyset$ . Thus ${\underline{N}}_{Q} (- M)$ ≠ $- {\bar{N}}_{Q} (M)$ .

(2) ${\bar{N}}_{Q} (- M) = {v_{1}, v_{3}, v_{4}, v_{6}, v_{7}}$ and $- {\underline{N}}_{Q} (M) = {v_{3}, v_{4}, v_{6}, v_{7}}$ . Thus ${\bar{N}}_{Q} (- M)$ ≠ $- {\underline{N}}_{Q} (M)$ . Similarly, (3) and (4) can be verified.

From Propositions 12 and 13, we can say that the current lower and upper Q-soft vertex covering approximations satisfy more properties of Pawlak’s rough sets when compared with the soft vertex covering lower and upper approximations defined in [21]. This holds true for the current lower and upper L-soft edge covering approximations.

Proposition 14. Let G = (G_Q, G_L) be Z-SCRG and G_n = (G_{Q
_n}, G_{L
_n}) be NSCRG then $\underline{SC} (G_{n}) = \underline{SC} (G)$ .

Proof. To prove $\underline{G_{n}} \subseteq \underline{SC} (G)$ , we have to prove ${\underline{SC}}_{Q} (M_{n}) \subseteq {\underline{SC}}_{Q} (M)$ and ${\underline{SC}}_{L} (O_{n}) \subseteq {\underline{SC}}_{L} (O)$ , where M ⊆ V and O ⊆ E. Let $v_{i} \in {\underline{SC}}_{Q} (M_{n})$ , then v_i ∈ N_Q (v_i). Since SA (v_i) ⊆ N (v_i). Thus, v_i ∈ SA (v_i) such that $v_{i} \in {\underline{SC}}_{Q} (M)$ . Therefore, ${\underline{SC}}_{Q} (M_{n}) \subseteq {\underline{SC}}_{Q} (M)$ . Similarly, we can prove ${\underline{SC}}_{L} (O_{n}) \subseteq {\underline{SC}}_{L} (O)$ . Hence, $\underline{SC} (G_{n}) = \underline{SC} (G)$ . □

From the above proposition, we can observe that granules of information in Z-SCRG (G = (G_Q, G_L)) are finer than NSCRG (G_n = (G_{Q
_n}, G_{L
_n})). Therefore, Z-SCRG is more significant than NSCRG.

6 Application of Z-soft covering based rough graph

In this section, a novel MAGDM model is developed using Z-SCRG. In MAGDM methods, decisions are made only on the basis of information of individuals in the application process. Using this graph-related technique, we consider individuals as vertices and connections between individuals as edges that increase the accuracy of the results. A numerical example is developed to demonstrate the application of Z-SCRG for dealing with real-world problems in the medical field.

6.1 Description and process

Let V ={ v₁, v₂, . . . , v_n } be the set of all objects (patients) and B ={ b₁, b₂, . . . , b_m } be the set of all parameters. Let G^* = (V, E) be a simple graph. Let (ρ, B) and (σ, B) be the soft sets over the vertex set and edge set respectively defined by, $ρ (b_{k}) = {v_{i} \in V : v_{i} have the attributes b_{k}} for k = 1, 2, 3, 4, i = 1, 2, 3, . . ., 10 . σ (b_{k}) = {v_{j} v_{i} \in E : v_{j} may have high risk than v_{i} in relation to the attributes b_{k}} .$ no tag Let D ={ D₁, D₂, D₃ } be the set of expert doctors and N = (τ, D) be the soft sets over V. Let Q = (V, C_P) be an SVCA then $\underline{τ (D_{q})}$ and $\bar{τ (D_{q})}$ are defined by $\underline{τ (D_{q})} = {v_{i} \in M : {SA}_{Q} (v_{i}) \subseteq M}$ and $\bar{τ (D_{q})} = {v_{i} \in M : {SA}_{Q} (v_{i}) \cap M \neq \emptyset}$ , where q = 1,2,3.

Now we calculate the fuzzy sets ς_N, $ς_{\underline{N}}$ and $ς_{\bar{N}}$ defined by, $ς_{N (v_{i})} = \frac{1}{n} \sum_{q = 1}^{n} χ_{τ (D_{q})} (v_{i})$ , $ς_{\underline{N (v_{i})}} = \frac{1}{n} \sum_{q = 1}^{n} χ_{\underline{τ (D_{q})}} (v_{i})$ and $ς_{\bar{N (v_{i})}} = \frac{1}{n} \sum_{q = 1}^{n} χ_{\bar{τ (D_{q})}} (v_{i})$ . where, $χ_{τ (D_{q})} (v_{i}) = {\begin{matrix} 1 & if v_{i} \in τ (D_{q}); \\ 0, & otherwise . \end{matrix}$ $χ_{\underline{τ (D_{q})}} (v_{i}) = {\begin{matrix} 1 & if v_{i} \in \underline{τ (D_{q})}; \\ 0, & otherwise . \end{matrix}$ $χ_{\bar{τ (D_{q})}} (v_{i}) = {\begin{matrix} 1 & if v_{i} \in \bar{τ (D_{q})}; \\ 0, & otherwise . \end{matrix}$

The marginal weight ψ for each v_i is defined by

$ψ (v_{i}) = \frac{1}{| V |} [ψ_{r} (v_{i}) + ψ_{f} (v_{i})]$ for i = 1,2,...,10 where,

$ψ_{r} (v_{i}) = \sum_{i = 1}^{10} χ_{D} (v_{i} v_{j})$ , which measures the high risk factor of v_i than v_j and

$ψ_{f} (v_{i}) = \sum_{i = 1}^{10} χ_{D} (v_{j} v_{i})$ , which measures the high risk factor of v_j than v_i. Let χ_D is an indicator function on D defined by, $χ_{D} (v_{i} v_{j}) = {\begin{matrix} 1 & if v_{i} v_{j} form an edge; \\ 0, & otherwise . \end{matrix}$ Finally, we compute the evaluation function Θ defined by, $Θ (v_{i}) = \frac{1}{3} [ς_{N (v_{i})} + ς_{\underline{N (v_{i})}} + ς_{\bar{N (v_{i})}}] ψ (v_{i}) .$

Patient v_i is at high risk if v_i = max {Θ (v_i)}, i = 1,2,...,10.

6.2 Pseudo code

(1) Input a CSG, $G$ = (G^*, ρ, σ, B).

(2) Compute the upper and lower Q-soft vertex covering approximations for each τ (D_q).

(3) Calculate the fuzzy sets ς_N, $ς_{\underline{N}}$ and $ς_{\bar{N}}$ defined by, $ς_{N (v_{i})} = \frac{1}{n} \sum_{q = 1}^{n} χ_{τ (D_{q})} (v_{i})$ ,

$ς_{\underline{N (v_{i})}} = \frac{1}{n} \sum_{q = 1}^{n} χ_{\underline{τ (D_{q})}} (v_{i})$ ,

$ς_{\bar{N (v_{i})}} = \frac{1}{n} \sum_{q = 1}^{n} χ_{\bar{τ (D_{q})}} (v_{i})$ .

(4) Find the marginal weight ψ for each v_i defined by

$ψ (v_{i}) = \frac{1}{| V |} [ψ_{r} (v_{i}) + ψ_{f} (v_{i})]$ for i = 1,2,...,10.

(5) The Evaluation function Θ is defined as,

$Θ (v_{i}) = \frac{1}{3} [ς_{N (v_{i})} + ς_{\underline{N (v_{i})}} + ς_{\bar{N (v_{i})}}] ψ (v_{i}) .$

The vertex v_i is optimal if v_i = max {Θ (v_i)}, i = 1,2,...,10.

6.3 Example

Suppose ten patients are suffering from chronic kidney disease.. We aim to identify the patient at high risk of kidney failure with the help of the parameters - Blood Glucose level(b₁), Blood Urea level(b₂), Serum Creatinine(b₃) and Haemoglobin(b₄). Our main intention is to assist doctors in finding a patient who needs a kidney transplant. We select 10 patients from the UCI machine learning repository with chronic kidney disease according to the data presented in Table 7. Let V = {v₁, v₂, v₃, v₄, v₅, v₆, v₇, v₈, v₉, v₁₀} be the set of patients (universe) and B = {b₁, b₂, b₃, b₄} be the set of parameters. Patients with blood glucose levels greater than 140, blood urea levels of 90 or greater, serum creatinine greater than 3, and hemoglobin 10 or less than 10 were selected.

Table 7
Tabular representation of parameter values of 10 patients

v ₁ v ₂ v ₃ v ₄ v ₅ v ₆ v ₇ v ₈ v ₉ v ₁₀

b ₁ 117 70 380 157 173 95 264 70 253 163

b ₂ 56 107 60 90 148 163 87 32 142 92

b ₃ 3.8 7.2 2.7 4.1 3.9 7.7 2.7 0.9 4.6 3.3

b ₄ 11.2 9.5 10.8 5.6 7.7 9.8 12.5 10 10.5 9

v ₁	v ₂	v ₃	v ₄	v ₅	v ₆	v ₇	v ₈	v ₉	v ₁₀
b ₁	117	70	380	157	173	95	264	70	253	163
b ₂	56	107	60	90	148	163	87	32	142	92
b ₃	3.8	7.2	2.7	4.1	3.9	7.7	2.7	0.9	4.6	3.3
b ₄	11.2	9.5	10.8	5.6	7.7	9.8	12.5	10	10.5	9

Table 8

Tabular representation of (ρ, B)

v ₁	v ₂	v ₃	v ₄	v ₅	v ₆	v ₇	v ₈	v ₉	v ₁₀
b ₁	0	0	1	1	1	0	1	0	1	1
b ₂	0	1	0	1	1	1	0	0	1	1
b ₃	1	1	0	1	1	1	0	0	1	1
b ₄	0	1	0	1	1	1	0	1	0	1

Let G^* = (V, E) be a simple graph. Let (ρ, B) be a soft set over the vertex set and (σ, B) be a soft set over the edge set defined by, $ρ (b_{k}) = {v_{i} \in V : v_{i} have the attributes b_{k}} fork = 1, 2, 3, 4, i = 1, 2, 3, . . ., 10 . σ (b_{k}) = {v_{j} v_{i} \in E : v_{j} may have high risk than v_{i} in relation to the attributes b_{k}} .$ no tag Let

$\begin{matrix} ρ (b_{1}) & = {v_{3}, v_{4}, v_{5}, v_{7}, v_{9}, v_{10}}, \\ ρ (b_{2}) & = {v_{2}, v_{4}, v_{5}, v_{6}, v_{9}, v_{10}}, \\ ρ (b_{3}) & = {v_{1}, v_{2}, v_{4}, v_{5}, v_{6}, v_{9} . v_{10}}, \\ ρ (b_{4}) & = {v_{2}, v_{4}, v_{5}, v_{6}, v_{8}, v_{10}} . \\ σ (b_{1}) & = {v_{3} v_{4}, v_{3} v_{10}, v_{7} v_{4}, v_{10} v_{4}, v_{7} v_{10}, v_{9} v_{10}, v_{9} v_{5}}, \\ σ (b_{2}) & = {v_{2} v_{4}, v_{2} v_{10}, v_{5} v_{4}, v_{6} v_{10}, v_{6} v_{9}}, \\ σ (b_{3}) & = {v_{2} v_{1}, v_{2} v_{5}, v_{6} v_{2}, v_{6} v_{9}, v_{9} v_{4}, v_{4} v_{10}, v_{2} v_{10}}, \\ σ (b_{4}) & = {v_{4} v_{2}, v_{2} v_{8}, v_{4} v_{6}, v_{6} v_{8}, v_{5} v_{8}, v_{4} v_{10}, v_{5} v_{10}} . \end{matrix}$

where $G$ = (G^*, ρ, σ, B) is a CSG, for all k. Let G_k = (ρ (b_k) , σ (b_k)) be subgraphs of G^* in Fig. 3. The directions from v_i to v_j indicate that v_i has a higher risk than v_j in terms of b_k.

Fig. 3

Subgraphs of G for each parameters.

Let Q = (V, C_P) be SVCA. Then the soft adhesion for each v_i ∈ V are SA_Q (v₁) = { v₁ } , SA_Q (v₂) = { v₂, v₆ } , SA_Q (v₃) = { v₃, v₇ } , SA_Q (v₄) = { v₄, v₅, v₁₀ } , SA_Q (v₅) = { v₄, v₅, v₁₀ } , SA_Q (v₆) = { v₂, v₆ } , SA_Q (v₇) = { v₃, v₇ } ,

SA_Q (v₈) = { v₈ } , SA_Q (v₉) ={ v₉ } and SA_Q (v₁₀) = { v₄, v₅, v₁₀ } .

Let D = {D₁, D₂, D₃} be the set of expert doctors assess patients with the help of parameters. Using the first evaluation values, we construct a soft set N = (τ, D) over V.

Let τ (D₁) = { v₁, v₃, v₅, v₇, v₈, v₁₀ } ,

τ (D₂) = { v₂, v₄, v₅, v₆, v₇, v₁₀ } ,

τ (D₃) ={ v₁, v₂, v₃, v₄, v₇, v₈, v₉ }.

Then $\underline{τ (D_{1})} = {v_{1}, v_{3}, v_{7}, v_{8}}$ ,

$\underline{τ (D_{2})} = {v_{2}, v_{4}, v_{5}, v_{6}, v_{10}}$ ,

$\underline{τ (D_{3})} = {v_{1}, v_{3}, v_{7}, v_{9}}$ .

$\bar{τ (D_{1})} = {v_{1}, v_{3}, v_{4}, v_{5}, v_{7}, v_{8}, v_{10}}$ ,

$\bar{τ (D_{2})} = {v_{2}, v_{3}, v_{4}, v_{5}, v_{6}, v_{7}, v_{10}}$ ,

$\bar{τ (D_{3})} = {v_{1}, v_{2}, v_{3}, v_{4}, v_{5}, v_{6}, v_{7}, v_{8}, v_{9}, v_{10}}$ . The outcomes of the doctors can be formulated into fuzzy sets. Let ς_N, $ς_{\underline{N}}$ and $ς_{\bar{N}}$ be three fuzzy sets representing the optimal measure, the positive optimal measure and the possible optimal measure respectively defined by

ς_N : V ⟶ [0, 1]

$ς_{N} (v_{i}) = \frac{1}{3} \sum_{q = 1}^{3} χ_{τ (D_{q})} (v_{i})$

$ς_{\underline{N}} : V ⟶ [0, 1]$

$ς_{\underline{N}} (v_{i}) = \frac{1}{3} \sum_{q = 1}^{3} χ_{\underline{τ (D_{q})}} (v_{i})$

$ς_{\bar{N}} : V ⟶ [0, 1]$

$ς_{\bar{N}} (v_{i}) = \frac{1}{3} \sum_{q = 1}^{3} χ_{\bar{τ (D_{q})}} (v_{i})$

The values of the three fuzzy sets are shown in Table 9.

Table 9

Tabular representation of the membership of patients

ς _{N₍v_i)}	$ς_{\underline{N_{(} v_{i})}}$	$ς_{\bar{N_{(} v_{i})}}$
v ₁	2/3	2/3	2/3
v ₂	2/3	1/3	2/3
v ₃	2/3	2/3	1
v ₄	2/3	1/3	1
v ₅	2/3	1/3	1
v ₆	1/3	1/3	2/3
v ₇	1	2/3	1
v ₈	2/3	1/3	2/3
v ₉	1/3	1/3	1/3
v ₁₀	2/3	1/3	1

Using MATLAB, the values of three fuzzy sets ς_N, $ς_{\underline{N}}$ and $ς_{\bar{N}}$ which represents the optimal measure, positive optimal measure and possible optimal measure respectively are plotted on the graph shown in Fig. 4 for better understanding.

Fig. 4

Graphical representation of optimal measures.

Now we compute the marginal weight function ψ for each v_i is defined as

$ψ (v_{i}) = \frac{1}{| V |} [ψ_{r} (v_{i}) + ψ_{f} (v_{i})]$ for i = 1,2,...,10.

By calculations, $ψ (v_{1}) = \frac{2}{10}$ , $ψ (v_{2}) = \frac{7}{10}$ , $ψ (v_{3}) = \frac{2}{10}$ , $ψ (v_{4}) = \frac{9}{10}$ , $ψ (v_{5}) = \frac{5}{10}$ , $ψ (v_{6}) = \frac{6}{10}$ , $ψ (v_{7}) = \frac{2}{10}$ , $ψ (v_{8}) = \frac{3}{10}$ , $ψ (v_{9}) = \frac{4}{10}$ , and $ψ (v_{10}) = \frac{8}{10}$ .

The Evaluation function Θ is defined by,

$Θ (v_{i}) = \frac{1}{3} [ς_{N (v_{i})} + ς_{\underline{N (v_{i})}} + ς_{\bar{N (v_{i})}}] ψ (v_{i}) .$

That is, $Θ (v_{1}) = \frac{2}{15}$ , $Θ (v_{2}) = \frac{7}{18}$ , $Θ (v_{3}) = \frac{14}{45}$ , $Θ (v_{4}) = \frac{9}{15}$ , $Θ (v_{5}) = \frac{5}{15}$ , $Θ (v_{6}) = \frac{4}{15}$ , $Θ (v_{7}) = \frac{8}{45}$ , $Θ (v_{8}) = \frac{1}{6}$ , $Θ (v_{9}) = \frac{2}{15}$ , $Θ (v_{10}) = \frac{8}{15}$ .

Using MATLAB, the evaluation function values are plotted on the graph shown in Fig. 5. This shows that the patient v₄ reaches the peak value. Therefore, patient v₄ is at high risk of chronic kidney disease. We conclude that patient v₄ should undergo kidney transplantation.

Fig. 5

Graphical representation of evaluation function.

6.4 Comparative analysis

In this subsection, Z-SCRG is compared with MSR-graphs and NSCRG.

According to the proposition 12 and 13, our approach is more accurate than NSCRG. Additionally, our approach is applied to chronic kidney disease patients to demonstrate the importance of our model in decision-making and to compare our technique with other models. As shown in Table 10, we attain the same optimal solution v₄ with the different models, which shows that our model is effective and feasible.

Table 10
Table for the ranking outcomes

Different models obtain a decision

NSCRG model v₄ ≥ v₁₀ ≥ v₆ ≥ v₅ ≥ v₉ ≥ v₈ ≥ v₇ ≥ v₁ = v₃ ≥ v₂

MSR-graph model v₄ ≥ v₁₀ ≥ v₂ ≥ v₅ ≥ v₃ ≥ v₆ ≥ v₇ ≥ v₈ ≥ v₁ = v₉

Our model v₄ ≥ v₁₀ ≥ v₆ ≥ v₂ ≥ v₅ ≥ v₈ ≥ v₃ = v₇ ≥ v₁ = v₉

Different models	obtain a decision
NSCRG model	v₄ ≥ v₁₀ ≥ v₆ ≥ v₅ ≥ v₉ ≥ v₈ ≥ v₇ ≥ v₁ = v₃ ≥ v₂
MSR-graph model	v₄ ≥ v₁₀ ≥ v₂ ≥ v₅ ≥ v₃ ≥ v₆ ≥ v₇ ≥ v₈ ≥ v₁ = v₉
Our model	v₄ ≥ v₁₀ ≥ v₆ ≥ v₂ ≥ v₅ ≥ v₈ ≥ v₃ = v₇ ≥ v₁ = v₉

7 Conclusion

In this paper, a unique form of graphs called Z-soft covering based rough graphs is proposed and their fundamental properties are explored. Uncertainty measures and their relations are discussed for Z-SCRG. Our results in this paper will provide a strong basis for understanding the concepts of uncertainty measures. The relationship between our model (Z-SCRG) and the existing model (NSCRG) is investigated. Furthermore, Z-SCRG is used to develop a novel MAGDM model to solve the medical diagnosis problem. As a result, we found that patient v₄ is at high risk of chronic kidney disease. Through comparative analysis, Z-SCRG is shown to be effective and significant. One of the recently developing concepts is soft hypergraphs. We plan to extend our study in the following areas: (1) single-valued neutrosophic soft hypergraphs and (2) fuzzy soft competition hypergraphs.

Acknowledgments

The authors take this opportunity to thank the UCI Machine Learning Repository for providing access to the data set. We express our gratitude to them for making this research possible. Special thanks to Dr. P. Soundarapandian, L. Jerlin Rubini and Dr. P. Eswaran for sharing the source.

Conflict of interest

The authors declare that they have no conflict of interest.

References

Abd El-Monsef

M.E.

, Embaby

O.A.

and El-Bably

M.K.

, Comparison between rough set approximations based on different topologies, International Journal of Granular Computing, Rough Sets andIntelligent Systems 3(4) (2014), 292–305.

Akram

and Nawaz

, Operations on soft graphs, Fuzzy Information and Engineering 7(4) (2015), 423–449.

Akram

and Nawaz

, On fuzzy soft graphs, Italian Journal of Pure and Applied Mathematics 34 (2015), 497–514.

Akram

and Shahzadi

, Neutrosophic soft graphs with application, Journal of Intelligent and Fuzzy Systems 32(1) (2017), 841–858.

Akram

, Shahzadi

, Rasool

and Sarwar

, Decision-making methods based on fuzzy soft competition hypergraphs, Complex & Intelligent Systems 8(3) (2022), 2325–2348.

Ali

M.I.

, A note on soft sets, rough soft sets and fuzzy soft sets, Applied Soft Computing 11(4) (2011), 3329–3332.

Atef

, Khalil

A.M.

, Li

S.G.

, Azzam

A.A.

, Atik

and El Fattah

, Comparison of six types of rough approximations based onj-neighborhood space and j-adhesion neighborhood space, Journal of Intelligent and Fuzzy Systems 39(3) (2020), 4515–4531.

Dai

, Huang

, Su

, Tian

and Yang

, Uncertainty measurement for covering rough sets, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 22(02) (2014), 217–233.

El Sayed

, El Safty

M.A.

and El-Bably

M.K.

, Topological approach for decision-making of COVID-19 infection via a nano-topology model, AIMS Mathematics 6(7) (2021), 7872–7894.

10.

El-Bably

M.K.

and Abo-Tabl

E.A.

, A topological reduction for predicting of a lung cancer disease based on generalized rough sets, Journal of Intelligent and Fuzzy Systems (Preprint) (2021), 1–16.

11.

Euler

, Solutio problematis ad geometriam situs pertinentis, Commentarii academiae scientiarum Petropolitanae (1741) 128–140.

12.

Feng

, Liu

, Leoreanu-Fotea

and Jun

Y.B.

, Soft sets and softrough sets, Information Sciences 181(6) (2011), 1125–1137.

13.

Greco

, Matarazzo

and Slowinski

, Rough sets theory for multicriteria decision analysis, European Journal of Operational Research 129(1) (2001), 1–47.

14.

Liang

and Shi

, The information entropy, rough entropy and knowledge granulation in rough set theory, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 12(01) (2004), 37–46.

15.

Maji

P.K.

, Roy

A.R.

and Biswas

, An application of soft sets in adecision making problem, Computers & Mathematics with Applications 44(8-9) (2002), 1077–1083.

16.

Maji

P.K.

, Biswas

and Roy

A.R.

, Soft set theory, Computers & Mathematics with Applications 45(4-5) (2003), 555–562.

17.

Molodtsov

, Soft set theory-first results, Computers & Mathematics with Applications 37(4-5) (1999), 19–31.

18.

Noor

, Irshad

and Javaid

, Soft Rough Graphs, arXiv preprint arXiv:1707.05837 (2017).

19.

Pawlak

, Rough sets, International Journal of Computer &Information Sciences 11(5) (1982), 341–356.

20.

Park

, Shah

, Rehman

, Ali

M.I.

and Shabir

, Soft covering based rough graphs and corresponding decision making, Open Mathematics 17(1) (2019), 423–438.

21.

Rehman

, Shah

, Ali

M.I.

and Park

, Uncertainty measurement for neighborhood based soft covering rough graphs with applications, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 113(3) (2019), 2515–2535.

22.

Shabir

, Ali

M.I.

and Shaheen

, Another approach to soft rough sets, Knowledge-based Systems 40 (2013), 72–80.

23.

Shah

, Rehman

, Shabir

and Ali

M.I.

, Another approach to roughness of soft graphs with applications in decision making, Symmetry 10(5) (2018), 145–.

24.

Shahzadi

and Akram

, Intuitionistic fuzzy soft graphs with applications, Journal of Applied Mathematics and Computing 55(1) (2017), 369–392.

25.

Shahzadi

, Akram

and Davvaz

, Pythagorean fuzzy soft graphswith applications, Journal of Intelligent and Fuzzy Systems 38(4) (2020), 4977–4991.

26.

Yüksel

Ş.

, Ergül

Z.G.

and Tozlu

, Soft covering based rough sets and their application, The Scientific World Journal (2014).

27.

Yüksel

Ş.

, Tozlu

and Dizman

T.H.

, An application of multicriteria group decision making by soft covering based roughsets, Filomat 29(1) (2015), 209–219.

28.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

29.

Zhan

and Wang

, Certain types of soft coverings based roughsets with applications, International Journal of Machine Learning and Cybernetics 10(5) (2019), 1065–1076.

30.

Zhan

, Jiang

and Yao

, Three-way multi-attribute decision-making based on outranking relations, IEEE Transactions on Fuzzy Systems 29(10) (2020), 2844–2858.

31.

Zhan

, Zhang

and Wu

W.Z.

, An investigation on Wu-Leungmulti-scale information systems and multi-expert group decision-making, Expert Systems with Applications 170(2021), 114542.

Uncertainty measure for Z-soft covering based rough graphs with application

Abstract

Keywords

1 Introduction

3 Z-soft covering based rough graphs

6 Application of Z-soft covering based rough graph

6.1 Description and process

6.2 Pseudo code

6.3 Example

Table 7 Tabular representation of parameter values of 10 patients v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 b 1 117 70 380 157 173 95 264 70 253 163 b 2 56 107 60 90 148 163 87 32 142 92 b 3 3.8 7.2 2.7 4.1 3.9 7.7 2.7 0.9 4.6 3.3 b 4 11.2 9.5 10.8 5.6 7.7 9.8 12.5 10 10.5 9

Acknowledgments

Conflict of interest

References