Special irregular neutrosophic graphs

Abstract

Recently, the neutrosophic graph has been introduced as an extension of fuzzy graphs and intuitionistic fuzzy graphs, which offers more compatibility and flexibility than these two types in modeling and structuring many actual issues. In this article, using neutrosophic highly strong arc, the new notions of (totally) special irregular, highly special irregular, strongly special irregular, neighborly special irregular and special arc-irregular of neutrosophic graphs are stated. Finally, one of their utilizations relevant to offering a fixed optimization model in decision making in diverse conditions is presented. In fact,we present a decision-making problem in real-world applied example which discusses the factors influencing a companys efficiency. The presented model is, in fact, a factor-based model wherein the impact score of each factor is divided into two types of direct and indirect influences, in which the concept of neutrosophic special dominating set plays a significant role.

Keywords

Neutrosophic graph special irregular neurosophic graph special homomorphism special isomorphism

1 Introduction

The idea of neutrosophic sets (NSs) was offered by Smarandache [22] as an extension of the fuzzy sets [27], intuitionistic fuzzy sets [8], interval-valued fuzzy sets [26] and interval-valued intuitionistic fuzzy sets [4] theories. The neutrosophic sets are characterized by a truth-membership mapping T, an indeterminacy-membership mapping I and a falsity membership mapping F independently, which are within the absolute standard or nonstandard unit interval ] ^-0, 1⁺ [. In some scientific studies, fuzzy sets do not have the necessary yield to display and resolve mental obscurity and Neutrosophic sets show more flexibility and capability in this field. Smarandache [23, 24] defined two principal classes of neutrosophic graphs. In other perusal, Satham Hussain, Jahir Hussain and Smarandache [21] offered the concept of domination in neutrosophic soft graphs. Banitalebi and Borzooei [8, 9] introduced notions of neutrosophic special domination and neutrosophic special n-domination in neutrosophic graphs.

Characteristics of Neutrosophic graphs urged us to examine different meanings regarding their domination sets and in the following, expand the factor-based decision modeling technique Banitalebi et. al. [5–7 , 11], this time in Neutrosophic cognitive maps. In this modeling method, optimization and modeling solutions are proposed using the concept of dominating sets, and by reducing the dominating set size, the effective weight of the graph of factors on the control goal increases.

The aim of this article is to introduces the notions of (totally) special irregular, highly special irregular, strongly special irregular, neighborly special irregular and special arc-irregular of neutrosophic graphs. Finally, a fixed optimization model in decision making in diverse conditions will be offered.

This paper is organized as follows: Section 2 contains some of the preliminaries. Section 3 describes T_hs-degree, I_hs-degree, F_hs-degree and etc of nodes and arcs in a neutrosophic graph and Special notions such as special irregular, highly special irregular, strongly special irregular and etc are introduced. Also, we investigate some relevant results in this section. Section 4 is dedicated to applications and it is devoted to providing applications. Finally, we closed this paper by a conclusion.

2 Preliminaries

Definition 2.1. [22] If $𝕍$ is a space of nodes with universal elements in $𝕍$ marked by r, then the neutrosophic set $ℍ$ is an object having the form $ℍ = {〈 r : T_{ℍ} (r), I_{ℍ} (r), F_{ℍ} (r) >, r \in 𝕍 〉},$ where the functions $T, I, F : 𝕍 \to]^{-} 0, 1^{+} [$ describe respectively, the truth-membership mapping, the indeterminacy-membership mapping and the falsity-membership mapping of the element $r \in 𝕍$ to the set $ℍ .$

Definition 2.2. [12] A Neutrosophic Graph (or shortly NG) on crisp graph $G^{*} = (𝕍, 𝔼)$ is indicated with $G = (𝕄, ℕ)$ , where $𝕄 = (T_{𝕄}, I_{𝕄}, F_{𝕄})$ such that $T_{𝕄}, I_{𝕄}, F_{𝕄} : 𝕍 \to [0, 1]$ with the condition $0 \leq T_{𝕄} (r) + I_{𝕄} (r) + F_{𝕄} (r) \leq 3,$ for all $r \in 𝕍$ and $ℕ = (T_{ℕ}, I_{ℕ}, F_{ℕ})$ where $T_{ℕ}, I_{ℕ}, F_{ℕ} : 𝔼 \to [0, 1]$ with conditions $T_{ℕ} (rs) \leq T_{𝕄} (r) \land T_{𝕄} (s),$ $I_{ℕ} (rs) \geq I_{𝕄} (r) \lor I_{𝕄} (s),$ $F_{ℕ} (rs) \geq F_{𝕄} (r) \lor F_{𝕄} (s),$ and $0 \leq T_{ℕ} (rs) + I_{ℕ} (rs) + F_{ℕ} (rs) \leq 3,$ for all $rs \in 𝔼 .$

Definition 2.3. [8] Let $G = (𝕄, ℕ)$ be a NG on G^*. Then, (i) the neutrosophic order of G is interpreted with, $| 𝕍 | = \sum_{r_{i} \in 𝕍} (\frac{3 + T_{𝕄} (r_{i}) - (I_{𝕄} (r_{i}) + F_{𝕄} (r_{i}))}{2}),$ (ii) the neutrosophic size of G is interpreted with, $| 𝔼 | = \sum_{r_{i} r_{j} \in 𝔼} (\frac{3 + T_{ℕ} (r_{i} r_{j}) - (I_{ℕ} (r_{i} r_{j}) + F_{ℕ} (r_{i} r_{j}))}{2}),$

(iii) the neutrosophic cardinality of G is interpreted with, $| G | = | 𝕍 | + | 𝔼 |,$

(iv) for any $A \subset 𝕍,$ the neutrosophic node cardinality of A is indicated with O (A) and interpreted with, $O (A) = \sum_{r_{i} \in A} (\frac{3 + T_{𝕄} (r_{i}) - (I_{𝕄} (r_{i}) + F_{𝕄} (r_{i}))}{2}),$ (v) for any $R \subset 𝔼,$ the neutrosophic arc cardinality of R is indicated with $𝕊 (R)$ and interpreted with, $𝕊 (R) = \sum_{r_{i} r_{j} \in R} (\frac{3 + T_{ℕ} (r_{i} r_{j}) - (I_{ℕ} (r_{i} r_{j}) + F_{ℕ} (r_{i} r_{j}))}{2}) .$

Definition 2.4. [8] Let $G = (𝕄, ℕ)$ be a NG on G^*. Then, (i) an arc e = pq in G is named a neutrosophic highly strong arc (NHStA), if $\begin{matrix} T_{ℕ} (pq) > T_{ℕ}^{\infty} (pq), I_{ℕ} (pq) < I_{ℕ}^{\infty} (pq), \\ F_{ℕ} (pq) < F_{ℕ}^{\infty} (pq) . \end{matrix}$ (ii) The neutrosophic highly strong neighborhood of $p \in 𝕍$ is indicated with N_hs (p) and interpreted as follows: $N_{hs} (p) = {q \in 𝕍 ∣ pq is a highly strong arc in G} .$

(iii) A node $p \in 𝕍$ of a NG G is named a neutrosophic slightly isolated node (NSlIN) if N_hs (p) = ∅ .

Definition 2.5. [14] Let $G = (𝕄, ℕ)$ be a NG on G^* and $r, s \in 𝕍$ . Then, (i) T-strength of connectedness between r and s is $T_{ℕ}^{\infty} (rs) = sup {T_{ℕ}^{t} (rs) | t = 1, 2, \dots, p},$ and $\begin{matrix} T_{ℕ}^{t} (rs) = min {T_{ℕ} ({rw}_{1}), T_{ℕ} (w_{1} w_{2}), \dots, \\ T_{ℕ} (w_{t - 1} s) | r, w_{1}, \dots, w_{t - 1}, s \in 𝕍, \\ t = 1, 2, \dots, p} . \end{matrix}$ (ii) I-strength of connectedness between r and s is $I_{ℕ}^{\infty} (rs) = inf {I_{ℕ}^{t} (rs) | t = 1, 2, \dots, p},$ and $\begin{matrix} I_{ℕ}^{t} (rs) = max {I_{ℕ} ({rw}_{1}), I_{ℕ} (w_{1} w_{2}), \dots, \\ I_{ℕ} (w_{t - 1} v) | r, w_{1}, \dots, w_{t - 1}, s \in 𝕍, \\ t = 1, 2, \dots, p} . \end{matrix}$ (iii) F-strength of connectedness between r and s is $F_{ℕ}^{\infty} (rs) = inf {F_{ℕ}^{t} (rs) | t = 1, 2, \dots, p},$ and $\begin{matrix} F_{ℕ}^{t} (rs) = max {F_{ℕ} ({rw}_{1}), F_{ℕ} (w_{1} w_{2}), \dots, \\ F_{ℕ} (w_{t - 1} s) | r, w_{1}, \dots, w_{t - 1}, s \in 𝕍, \\ t = 1, 2, \dots, p} . \end{matrix}$

Definition 2.6. [14] Let $G = (𝕄, ℕ)$ be a NG on G^*. An arc $rs \in 𝔼$ is named a neutrosophic strong arc if $\begin{matrix} T_{ℕ} (rs) \geq T_{ℕ}^{\infty} (rs), I_{ℕ} (rs) \leq I_{ℕ}^{\infty} (rs) and \\ F_{ℕ} (rs) \leq F_{ℕ}^{\infty} (rs) . \end{matrix}$

Definition 2.7. [8] Let G be a NG on G^* and $p, q \in 𝕍$ . Then: (i) p specially dominate q in G, if there is a NHStA between p and q . (ii) $R \subset 𝕍$ is named a neutrosophic special dominating set (NSpDS) in G, if for any $q \in 𝕍 \ R$ , there is p ∈ R such that p specially dominates q. (iii) A NSpDS R in G is named a minimal NSpDS if no proper subset of R is a NSpDS. (iv) Minimum neutrosophic node cardinality amid all minimal NSpDSs of G is named lower neutrosophic special domination number of G and is indicated with $(nsdn)_{𝕍} (G)$ . (v) Maximum neutrosophic node cardinality amid all minimal NSpDSs of G is named upper neutrosophic special domination number of G and is indicated with $(NSDN)_{𝕍} (G)$ . (vi) The neutrosophic special domination number of G is indicated with NSΔN (G) and interpreted with $NS Δ N (G) = \frac{(nsdn)_{𝕍} (G) + (NSDN)_{𝕍} (G)}{2} .$

Definition 2.8. [8] Let G be a NG on G^*. Then: (i) Two nodes $p, q \in 𝕍$ are named neutrosophic slightly independent if there is no NHStA between them. (ii) $R \subset 𝕍$ is named a neutrosophic slightly independent set (NSlIS) in G if for any p, q ∈ R, $T_{ℕ} (pq) \leq T_{ℕ}^{\infty} (pq)$ , $I_{ℕ} (pq) \geq I_{ℕ}^{\infty} (pq)$ and $F_{ℕ} (pq) \geq F_{ℕ}^{\infty} (pq) .$

(iii) A NSlIS R in G is named a maximal NSlIS if for any node $q \in 𝕍 \ R$ , the set R ∪ {q} is not NSlIS. (iv) Minimum neutrosophic node cardinality amid all maximal NSlISs is named a lower neutrosophic slightly independent number of G and is indicated with $(ni)_{𝕍} (G)$ . (v) Maximum neutrosophic node cardinality amid all maximal NSlISs is named an upper neutrosophic slightly independent number of G and is indicated with $(NI)_{𝕍} (G)$ . (vi) The neutrosophic slightly independent number of G is indicated with NI (G) and interpreted as follows, $NI (G) = \frac{(ni)_{𝕍} (G) + (NI)_{𝕍} (G)}{2} .$

Definition 2.9. [8] Let G be a NG on G^* .

(i) The neutrosophic special cobondage set (NSpCS) is the set C of additional NHStAs to G, that reduces the neutrosophic special domination number, i.e. $NS Δ N (G_{C}) < NS Δ N (G) .$ (ii) A NSpCS Q of G is named a minimal NSpCS if no proper subset of Q is a NSpCS.

(iii) Minimum neutrosophic arc cardinality amid all minimal NSpCSs of G is named lower neutrosophic special cobondage number of G and indicated with $(nsbn)_{𝔼} (G)$ .

(iv) Maximum neutrosophic arc cardinality amid all minimal NSpCSs of G is named upper neutrosophic special cobondage number of G and indicated with $(NSBN)_{𝔼} (G)$ .

Definition 2.10. [9] Let G be a NG on G^* .

(i) $D_{m} \subset 𝕍$ is named a neutrosophic special m-dominating set $\tilde{Sp m - D}$ of G if for each node $s \in 𝕍 \ D_{m}$ we get s specially dominate with at least m nodes in D_m.

(ii) An $\tilde{Sp m - D}$ D_m of G is named a minimal $\tilde{Sp m - D}$ if no proper subset of D_m is a $\tilde{Sp m - D}$ of G.

(iii) Minimum neutrosophic node cardinality amid all minimal $\tilde{Sp m - D}$ s of G is named a lower neutrosophic special m-domination number of G and indicated with ${\overset{ˇ}{N}}_{𝔸_{𝕍}}^{n} (𝔾_{Ne}) .$

(iv) Maximum neutrosophic node cardinality amid all minimal $\tilde{Sp m - D}$ s of G is named an upper neutrosophic special m-domination number of G and indicated with ${\hat{N}}_{𝔸_{𝕍}}^{n} (𝔾_{Ne}) .$

(v) The neutrosophic special m-domination number of G is indicated with $(\bar{N})^{n} (𝔾_{Ne})$ and interpreted with: $(\bar{N})^{n} (𝔾_{Ne}) = \frac{{\overset{ˇ}{N}}_{𝔸_{𝕍}}^{n} (𝔾_{Ne}) + {\hat{N}}_{𝔸_{𝕍}}^{n} (𝔾_{Ne})}{2} .$

Notation 1. Henceforth, in this article, we presume $G = (𝕄, ℕ)$ be a neutrosophic graph on crisp graph $G^{*} = (𝕍, 𝔼)$ .

3 Special irregular neutrosophic graph

In this part, first, we describe T_hs-degree, I_hs-degree, F_hs-degree, hs-degree and ths-degree of nodes and arcs in a neutrosophic graph. Then we offer the notions of special irregular, highly special irregular, strongly special irregular, neighborly special irregular, totally special irregular and special arc-irregular neutrosophic graphs and we investigate some relevant results.

Definition 3.1. Let G be a NG on G^*. Then,

(i) T_hs-degree of a node p is indicated with d_{T
_hs} (p) and interpreted with, $d_{T_{hs}} (p) = \sum_{q \in N_{hs} (p)} T_{ℕ} (pq) .$ (ii) I_hs-degree of a node p is indicated with d_{I
_hs} (p) and interpreted with, $d_{I_{hs}} (p) = \sum_{q \in N_{hs} (p)} I_{ℕ} (pq) .$ (iii) F_hs-degree of a node p is indicated with d_{F
_hs} (p) and interpreted with, $d_{F_{hs}} (p) = \sum_{q \in N_{hs} (p)} F_{ℕ} (pq) .$ (iv) hs-degree of a node p is indicated with d_hs (p) and interpreted with, $d_{hs} (p) = [d_{T_{hs}} (p), d_{I_{hs}} (p), d_{F_{hs}} (p)] .$ (v) ths-degree of a node p is indicated with d_ths (p) and interpreted with, $\begin{matrix} d_{ths} (p) = [d_{T_{hs}} (p) + T_{𝕄} (p), d_{I_{hs}} (p) + I_{𝕄} (p), \\ d_{F_{hs}} (p) + F_{𝕄} (p)] . \end{matrix}$

Definition 3.2. Let G be a NG on G^*. Then,

(i) T_hs-degree of an arc $c_{ij} = v_{i} v_{j} \in 𝔼$ is indicated with d_{T
_hs} (c_ij) and interpreted with, $d_{T_{hs}} (c_{ij}) = {\begin{matrix} d_{T_{hs}} (v_{i}) + d_{T_{hs}} (v_{j}) - 2 T_{ℕ} (c_{ij}), & v_{j} \in N_{hs} (v_{i}), \\ d_{T_{hs}} (v_{i}) + d_{T_{hs}} (v_{j}), & otherwise . \end{matrix}$ (ii) I_hs-degree of an arc $c_{ij} = v_{i} v_{j} \in 𝔼$ is indicated with d_{I
_hs} (c_ij) and interpreted with, $d_{I_{hs}} (c_{ij}) = {\begin{matrix} d_{I_{hs}} (v_{i}) + d_{I_{hs}} (v_{j}) - 2 I_{ℕ} (c_{ij}), & v_{j} \in N_{hs} (v_{i}), \\ d_{I_{hs}} (v_{i}) + d_{I_{hs}} (v_{j}), & otherwise . \end{matrix}$ (iii) F_hs-degree of an arc $c_{ij} = v_{i} v_{j} \in 𝔼$ is indicated with d_{F
_hs} (c_ij) and interpreted with, $d_{F_{hs}} (c_{ij}) = {\begin{matrix} d_{F_{hs}} (v_{i}) + d_{F_{hs}} (v_{j}) - 2 F_{ℕ} (c_{ij}), & v_{j} \in N_{hs} (v_{i}), \\ d_{F_{hs}} (v_{i}) + d_{F_{hs}} (v_{j}), & otherwise . \end{matrix}$ (vi) hs-degree of an arc $c_{ij} \in 𝔼$ is indicated with d_hs (c_ij) and interpreted with, $d_{hs} (c_{ij}) = [d_{T_{hs}} (c_{ij}), d_{I_{hs}} (c_{ij}), d_{F_{hs}} (c_{ij})] .$

Theorem 3.3. Let that G^* is cycle and G is a connected NG on G^*. Then $\sum_{v_{i} \in 𝕍} d_{hs} (v_{i}) = \sum_{c_{ij} \in 𝔼} d_{hs} (c_{ij}) .$

Proof. Consider G^* be a cycle as v₁v₂v₃ . . . v_nv₁ . Then, we investigate the following cases: Case 1. If G does not have any NHStA, then all nodes of G have the hs-degrees equal to zero. Then, the proof is clear. Case 2. If G has a NHStA c, then c is a unique. Presume that c = c_jj+1 is an unique NHStA of G. Then, we have $\begin{matrix} \sum_{c_{ij} \in 𝔼} d_{hs} (c_{ij}) & = & (\sum_{c_{ij} \in 𝔼} d_{T_{hs}} (c_{ij}), \sum_{c_{ij} \in 𝔼} d_{I_{hs}} (c_{ij}), \sum_{c_{ij} \in 𝔼} d_{F_{hs}} (c_{ij})) \\ = & (2 T_{ℕ} (c_{jj + 1}), 2 I_{ℕ} (c_{jj + 1}), 2 F_{ℕ} (c_{jj + 1})) \\ = & (d_{T_{hs}} (v_{i}) + d_{T_{hs}} (v_{i + 1}), d_{I_{hs}} (v_{i}) + d_{I_{hs}} (v_{i + 1}), \\ d_{I_{hs}} (v_{i}) + d_{I_{hs}} (v_{i + 1})) \\ = & (\sum_{i = 1}^{n} d_{T_{hs}} (v_{i}), \sum_{i = 1}^{n} d_{I_{hs}} (v_{i}), \sum_{i = 1}^{n} d_{F_{hs}} (v_{i})) \\ = & \sum_{v_{i} \in 𝕍} d_{hs} (v_{i}) . \end{matrix}$ □

Definition 3.4. Let G be a connected NG on G^*. Then:

(i) G is named a special irregular neutrosophic graph (SpING) if there exists a node of G that is adjoining to nodes with distinguished hs-degrees.

(ii) G is named a highly SpING if every node of G is adjoining to nodes with distinguished hs-degrees.

(iii) G is named a strongly SpING if every pair of nodes of G have distinguished hs-degrees.

(iv) G is named a neighborly SpING if every pair of adjoining nodes of G have distinguished hs-degrees.

(v) G is called a totally SpING if there exists a node of G that is adjoining to nodes with distinguished ths-degrees.

(vi) G is named a special arc-irregular neutrosophic graph (Sp arc-ING) if there exists an arc of G that is adjoining to arcs with distinguished hs-degrees.

Example 3.5. Let G be a NG on G^* as Fig. 1.

Fig. 1

G a NG on G^*.

It is clear that, u₁u₂, u₁u₅ and u₂u₃ are NHStAs and $\begin{matrix} d_{T_{hs}} (u_{1}) = 0.4, d_{T_{hs}} (u_{2}) = 0.4, d_{T_{hs}} (u_{3}) = 0.2, \\ d_{T_{hs}} (u_{4}) = 0, d_{T_{hs}} (u_{5}) = 0.2, \end{matrix}$ $\begin{matrix} d_{I_{hs}} (u_{1}) = 0.6, d_{I_{hs}} (u_{2}) = 0.7, d_{I_{hs}} (u_{3}) = 0.4, \\ d_{I_{hs}} (u_{4}) = 0, d_{I_{hs}} (u_{5}) = 0.3, \end{matrix}$ $\begin{matrix} d_{F_{hs}} (u_{1}) = 1.3, d_{F_{hs}} (u_{2}) = 1.2, d_{F_{hs}} (u_{3}) = 0.5, \\ d_{F_{hs}} (u_{4}) = 0, d_{F_{hs}} (u_{5}) = 0.6 . \end{matrix}$ Then, $\begin{matrix} d_{hs} (u_{1}) = [0.4, 0.6, 1.3], d_{hs} (u_{2}) = [0.4, 0.7, 1.2], \\ d_{hs} (u_{3}) = [0.2, 0.4, 0.5], \end{matrix}$ $d_{hs} (u_{4}) = [0, 0, 0], d_{hs} (u_{5}) = [0.2, 0.3, 0.6] .$ Also, $\begin{matrix} d_{ths} (u_{1}) = [0.6, 0.9, 1.8], d_{ths} (u_{2}) = [0.7, 0.9, 1.6], \\ d_{ths} (u_{3}) = [0.4, 0.7, 0.9], \end{matrix}$ $d_{ths} (u_{4}) = [0.1, 0.3, 0.5], d_{ths} (u_{5}) = [0.5, 0.6, 1] .$ Therefore, G is strongly SpING and totally SpING.

Example 3.6. Let G be a NG on G^* as shown in Fig. 2.

Fig. 2

NG G.

Clear that, u₁u₃ and u₃u₄ are NHStAs. Obviously, $\begin{matrix} d_{hs} (u_{1}) = [0.7, 0.3, 0.3], d_{hs} (u_{2}) = [0, 0, 0], \\ d_{hs} (u_{3}) = [1.4, 0.6, 0.7], \\ d_{hs} (u_{4}) = [0.7, 0.3, 0.4] . \end{matrix}$ Hence, $\begin{matrix} d_{hs} (u_{1} u_{2}) = [0.7, 0.3, 0.3], d_{hs} (u_{1} u_{3}) \\ = [1.4, 0.6, 0.7], d_{hs} (u_{2} u_{3}) = [1.4, 0.6, 0.7], \\ d_{hs} (u_{3} u_{4}) = [1.4, 0.6, 0.7] . \end{matrix}$ Therefore, G is a Sp arc-ING.

Theorem 3.7. Let G be a complete NG on G^* and the set of all nodes of G be indicated with $𝕍 = {v_{1}, v_{2}, v_{3}, . . ., v_{n}}$ such that $T_{𝕄} (v_{1}) < T_{𝕄} (v_{2}) < T_{𝕄} (v_{3}) < . . . < T_{𝕄} (v_{n})$ , $I_{𝕄} (v_{1}) > I_{𝕄} (v_{2}) > I_{𝕄} (v_{3}) > . . . > I_{𝕄} (v_{n})$ and $F_{𝕄} (v_{1}) > F_{𝕄} (v_{2}) > F_{𝕄} (v_{3}) > . . . > F_{𝕄} (v_{n})$ . Then G is a SpING.

Proof. Let G be a complete NG on G^*. Then for any $v_{i}, v_{j} \in 𝕍$ , $T_{ℕ} (v_{i} v_{j}) = T_{𝕄} (v_{i}) \land T_{𝕄} (v_{j}),$ $I_{ℕ} (v_{i} v_{j}) = I_{𝕄} (v_{i}) \lor I_{𝕄} (v_{j}),$ $F_{ℕ} (v_{i} v_{j}) = F_{𝕄} (v_{i}) \lor F_{𝕄} (v_{j}) .$ Hence, e = v_n-1v_n is a unique NHStA in G. Therefore, G is a SpING. □

Note 1. If T is a strictly decreasing chain and I and F are two strictly increasing chain on node-set $𝕍$ of G, then Theorem 3.7 is true.

Theorem 3.8. Set G be a connected NG and $𝕄 = (T_{𝕄}, I_{𝕄}, F_{𝕄})$ be a constant mapping. Then the following expressions are tantamount: (i) G is a SpING. (ii) G is a totally SpING.

Proof. Let that for each $p \in 𝕍$ , $𝕄 (p) = (T_{𝕄} (p), I_{𝕄} (p), F_{𝕄} (p)) = (c_{1}, c_{2}, c_{3}),$ where c₁, c₂ and c₃ are constant values. Hence, for any $p, q \in 𝕍,$ we have $d_{hs} (q) \neq d_{hs} (p) \Leftrightarrow d_{ths} (q) \neq d_{ths} (p) .$ □

Theorem 3.9. Let G be a connected NG on path graph G^*. Then $𝕄 = (T_{𝕄}, I_{𝕄}, F_{𝕄})$ is a constant mapping iff the subsequent expressions are tantamount: (i) G is a SpING. (ii) G is a totally SpING.

Proof. (⇒) From Theorem 3.8, it is clear. (⇐) Clearly, each arc of G is a NHStA. Let that $𝕄 = (T_{𝕄}, I_{𝕄}, F_{𝕄})$ is variable mapping with the contrary. If $ℕ = (T_{ℕ}, I_{ℕ}, F_{ℕ})$ is a constant mapping, then G is a totally SpING, but G is not a SpING; that is a contradiction. Therefore, the proof is complete. □

Theorem 3.10. Set G be a connected NG on Petersen graph G^*. Then the subsequent expressions are tantamount. (i) G is a SpING. (ii) G is a Sp arc-ING. (iii) G has a unique NHStA.

Proof. (i ⇔ iii) Set G is a SpING. If G does not have any NHStA, then all nodes of G have the hs-degrees equal to zero. That is a contradiction. Hence, G has a NHStA. Otherhand, if there are $e_{1}, e_{2} \in 𝔼$ where e₁ and e₂ are NHStAs, then e₁ and e₂ belong to a cycle with five nodes of G. That is a contradiction. Therefore, G has a unique NHStA. Conversely, if G has a unique NHStA, clearly, there is a node of G that is adjoining to nodes with hs-degrees 0 and 1. (ii ⇔ iii) Let that G is a Sp arc-ING. If G does not have any NHStA, then all arcs of G have the hs-degrees equal to zero, that is a contradiction. Hence, G has a NHStA. Otherhand, if there are $e_{1}, e_{2} \in 𝔼$ where e₁ and e₂ are NHStAs, then e₁ and e₂ belongs to a cycle with 5 nodes of G. That is a contradiction. Therefore, G has a unique NHStA. Conversely, if G has a unique NHStA, clearly, there is an arc of G adjoining to arcs with hs-degrees 0 and 1. □

Theorem 3.11. Set G be a connected NG on Petersen graph G^*. Then G is never a strongly SpING, a highly SpING and a neighborly SpING.

Proof. If G does not have any NHStA, then all arcs and nodes of G have the hs-degrees equal to zero. If G has a NHStA c, then c is a unique NHStA in G. Hence, the proof is clear. □

Theorem 3.12. Let G be a connected NG on path graph G^* with n ≥ 4 nodes. If alternate arcs take the same membership values, then G is a SpING

Proof. Obviously, each arc of G is a NHStA. Assume that alternate arcs take the same membership values. Then, $T_{ℕ} (e_{k}) = {\begin{matrix} α_{1}, & if k is odd, \\ α_{2}, & if k is even, \end{matrix}$ and $I_{ℕ} (e_{k}) = {\begin{matrix} h_{1}, & if k is odd, \\ h_{2}, & if k is even, \end{matrix}$ and $F_{ℕ} (e_{k}) = {\begin{matrix} λ_{1}, & if k is odd, \\ λ_{2}, & if k is even . \end{matrix}$ Thus, $\begin{matrix} d_{hs} (v_{1}) = [α_{1}, h_{1}, λ_{1}], \\ d_{hs} (v_{j}) = [α_{1} + α_{2}, h_{1} + h_{2}, λ_{1} + λ_{2}], \end{matrix}$ where k = 2, 3, . . . , n - 1 . Hence there is a node v₂ that is adjoining to v₁, v₃ with distinguished hs-degrees. Therefore, G is a SpING. □

Corollary 3.13. Set G be a connected NG on path graph G^* with n ≥ 4 nodes. If alternate arcs take the same membership values, then G is a Sp arc-ING.

Definition 3.14. A special homomorphism h of SpINGs G₁ and G₂ is a mapping $h : 𝕍_{1} \to 𝕍_{2}$ that satisfies the subsequent situations:

(i) h (t) = t′ where t adjoining to vertices with distinguished hs-degrees in G₁ and t′ adjoining to vertices with distinguished hs-degrees in G₂, (ii) for any $t \in 𝕍_{1}$ , $T_{𝕄_{1}} (t) \leq T_{𝕄_{2}} (h (t)), I_{𝕄_{1}} (t) \geq I_{𝕄_{2}} (h (t)), F_{𝕄_{1}} (t) \geq F_{𝕄_{2}} (h (t))$ , (iii) for any $pq \in 𝔼_{1}$ , $T_{ℕ_{1}} (pq) \leq T_{ℕ_{2}} (h (p) h (q))$ , $I_{ℕ_{1}} (pq) \geq I_{ℕ_{2}} (h (p) h (q)), F_{ℕ_{1}} (pq) \geq F_{ℕ_{2}} (h (p) h (q))$ .

Definition 3.15. Set h be a bijective special homomorphism of SpINGs G₁ and G₂. Then: (i) h is named a weak special isomorphism of SpINGs G₁ and G₂ if for any $t \in 𝕍_{1}$ , $\begin{matrix} T_{𝕄_{1}} (t) = T_{𝕄_{2}} (h (t)), I_{𝕄_{1}} (t) = I_{𝕄_{2}} (h (t)), \\ F_{𝕄_{1}} (t) = F_{𝕄_{2}} (h (t)), \end{matrix}$ (ii) h is named a co-weak special isomorphism of SpINGs G₁ and G₂ if for any $pq \in 𝔼_{1}$ , $\begin{matrix} T_{ℕ_{1}} (pq) = T_{ℕ_{2}} (h (p) h (q)), I_{ℕ_{1}} (pq) \\ = I_{ℕ_{2}} (h (p) h (q)), F_{ℕ_{1}} (pq) = F_{ℕ_{2}} (h (p) h (q)), \end{matrix}$ (iii) h is named a special isomorphism of SpINGs G₁ and G₂ if it is a weak special isomorphism and a co-weak special isomorphism.

Notation. If h is a special isomorphism of SpINGs G₁ and G₂, then we say G₁ is a special isomorphic neutrosophic graph of G₂ and denote it with G₁ ≈ G₂ .

Example 3.16. Investigate SpINGs G₁ and G₂ in Fig. 3. If h is a mapping h : V₁ → V₂ where $h (u_{1}) = a, h (u_{2}) = b, h (u_{3}) = c, h (u_{4}) = d,$ then with routine computations, clearly h is a special homomorphism of SpINGs G₁ and G₂.

Fig. 3

SpINGs G₁ and G₂.

Example 3.17. Investigate SpINGs G₁ and G₂ in Fig. 4. If h : V₁ → V₂ is a special homomorphism of SpINGs G₁ and G₂ where $\begin{matrix} h (u_{1}) = a, h (u_{2}) = e, h (u_{3}) = d, \\ h (u_{4}) = c, h (u_{5}) = b, \end{matrix}$ then with routine computations, obviously h is a weak special isomorphism of SpINGs G₁ and G₂.

Fig. 4

SpINGs G₁ and G₂.

Example 3.18. Investigate SpINGs G₁ and G₂ in Fig. 5. If h : V₁ → V₂ is a special homomorphism of SpINGs G₁ and G₂ where $\begin{matrix} h (u_{1}) = e, h (u_{2}) = d, h (u_{3}) = b, h (u_{4}) = c, \\ h (u_{5}) = f, h (u_{6}) = a, \end{matrix}$ then with routine computations, clearly h is a co-weak special homomorphism of SpINGs G₁ and G₂.

Fig. 5

SpINGs G₁ and G₂.

Note 2. (i) If G₁ ≈ G₂, then G₁ and G₂ have the same neutrosophic order and neutrosophic size. (ii) If G₁ and G₂ are weak special isomorphic, then G₁ and G₂ have the same neutrosophic order. (ii) If G₁ and G₂ are co-weak special isomorphic, then G₁ and G₂ have the same neutrosophic size.

Theorem 3.19. If G₁ is a co-weak special isomorphism of SpING of G₂, then

(i) $(nsdn)_{𝕍_{1}} (G_{1}) \leq (nsdn)_{𝕍_{2}} (G_{2})$ and $(NSDN)_{𝕍_{1}} (G_{1}) \leq (NSDN)_{𝕍_{2}} (G_{2}),$ and so NSΔN (G₁) ≤ NSΔN (G₂) .

(ii) $(nsdn)_{𝕍_{1}}^{n} (G_{1}) \leq (nsdn)_{𝕍_{2}}^{n} (G_{2})$ and $(NSDN)_{𝕍_{1}}^{n} (G_{1}) \leq (NSDN)_{𝕍_{2}}^{n} (G_{2}),$ and so (NSΔN) ⁿ (G₁) ≤ (NSΔN) ⁿ (G₂) .

Proof. (i) Assume that $h : 𝕍_{1} \to 𝕍_{2}$ which satisfies the next situations: is a co-weak special isomorphism of G₁ and G₂. If D₁ is a minimal NSpDS of G₁, then with Definition 3.15, it is clear that G₁ and G₂ have the same set of NHStAs and we get h (D₁) is a minimal NSpDS of G₂ such that O (D₁) ≤ O (h (D₁)). Therefore, (nsdn) _{V
₁} (G₁) ≤ (nsdn) _{V
₂} (G₂), (NSDN) _{V
₁} (G₁) ≤ (NSDN) _{V
₂} (G₂) and so NSΔN ((G₁) ≤ NSΔN ((G₂) .

(ii) Assume that $h : 𝕍_{1} \to 𝕍_{2}$ which satisfies the next situations: is a co-weak special isomorphism of G₁ and G₂. If D₁ is a minimal NSp n-DS of G₁, then with Definition 3.15, clearly G₁ and G₂ have the same set of NHStAs and we get h (D₁) is a minimal NSp n-DS of G₂ where O (D₁) ≤ O (h (D₁)) . Therefore, $(nsdn)_{𝕍_{1}}^{n} (G_{1}) \leq (nsdn)_{𝕍_{2}}^{n} (G_{2})$ , $(NSDN)_{𝕍_{1}}^{n} (G_{1}) \leq (NSDN)_{𝕍_{2}}^{n} (G_{2})$ and so (NSΔN) ⁿ (G₁) ≤ (NSΔN) ⁿ (G₂) . □

Corollary 3.20. If G₁ ≈ G₂, then

(i) $(nsdn)_{𝕍_{1}} (G_{1}) \leq (nsdn)_{𝕍_{2}} (G_{2})$ and $(NSDN)_{𝕍_{1}} (G_{1}) \leq (NSDN)_{𝕍_{2}} (G_{2})$ and so NSΔN (G₁) ≤ NSΔN (G₂) .

(ii) $(nsdn)_{𝕍_{1}}^{n} (G_{1}) \leq (nsdn)_{𝕍_{2}}^{n} (G_{2})$ and $(NSDN)_{𝕍_{1}}^{n} (G_{1}) \leq (NSDN)_{𝕍_{2}}^{n} (G_{2})$ and so (NSΔN) ⁿ (G₁) ≤ (NSΔN) ⁿ (G₂) .

Theorem 3.21. If G₁ is a co-weak special isomorphism of SpING of G₂, then $\begin{matrix} (nsbn)_{𝔼_{1}} (G_{1}) = (nsbn)_{𝔼_{2}} (G_{2}), \\ (NSBN)_{𝔼_{1}} (G_{1}) = (NSBN)_{𝔼_{2}} (G_{2}) . \end{matrix}$

Proof. Assume that $h : 𝕍_{1} \to 𝕍_{2}$ , which satisfies the subsequent situations: is a co-weak special isomorphism of G₁ and G₂. If e = uv is an additional NHStA in $G_{1}^{*}$ , then with Definition 3.15, we get e′ = h (u) h (v) , with coordinates $T_{ℕ_{1}} (e) = T_{ℕ_{2}} (e^{'}),$ $I_{ℕ_{1}} (e) = I_{ℕ_{2}} (e^{'})$ and $F_{ℕ_{1}} (e) = F_{ℕ_{2}} (e^{'}),$ is an additional NHStA in $G_{2}^{*} .$ Therefore, $(nsbn)_{𝔼_{1}} (G_{1}) = (nsbn)_{𝔼_{2}} (G_{2})$ and $(NSBN)_{𝔼_{1}} (G_{1}) = (NSBN)_{𝔼_{2}} (G_{2}) .$ □

Corollary 3.22. If G₁ ≈ G₂, then $\begin{matrix} (nsbn)_{𝔼_{1}} (G_{1}) = (nsbn)_{𝔼_{2}} (G_{2}), \\ (NSBN)_{𝔼_{1}} (G_{1}) = (NSBN)_{𝔼_{2}} (G_{2}) . \end{matrix}$

Theorem 3.23. If G₁ is a co-weak special isomorphism of SpING of G₂, then NI (G₁) ≤ NI (G₂) .

Proof. Assume that $h : 𝕍_{1} \to 𝕍_{2}$ , which satisfies the subsequent situation: is a co-weak special isomorphism of G₁ and G₂. If S₁ is a maximal NSlIS of G₁, then with Definition 3.15, it is clear that G₁ and G₂ have the same set of NHStAs and we get h (S₁) is a maximal NSlIS of G₂ where O (S₁) ≤ O (h (S₁)) . Therefore, $(ni)_{𝕍_{1}} (G_{1}) \leq (ni)_{𝕍_{2}} (G_{2})$ and $(NI)_{𝕍_{1}} (G_{1}) \leq (NI)_{𝕍_{2}} (G_{2})$ and so NI (G₁) ≤ NI (G₂) . □

Corollary 3.24. If G₁ ≈ G₂, then NI (G₁) ≤ NI (G₂).

4 Application

Today, NG models are beneficial for modeling many real phenomena in diverse sciences and engineering fields. In this article, we present the idea of special isomorphic images in NG theory. The special isomorphic images of a neutrosophic network can be usages to dissolve many actual issues.

Table 1

a b c d k NSΔN (G) $(nsbn)_{𝔼} (G)$

G (0.2,0.5,0.4)   (0.2,0.5,0.5) (0.2,0.5,0.4) (0.2,0.5,0.4) (0.2,0.5,0.5) 2.825 1.1

G ₁ (0.2,0.4,0.4)   (0.2,0.5,0.5) (0.3,0.5,0.4) (0.2,0.5,0.4) (0.2,0.5,0.5) 2.875 1.1

G ₂ (0.2,0.5,0.4)   (0.2,0.5,0.5) (0.2,0.4,0.4) (0.3,0.5,0.4) (0.2,0.5,0.5) 2.875 1.1

G ₃ (0.3,0.4,0.5)   (0.2,0.5,0.5) (0.2,0.5,0.5) (0.2,0.5,0.5) (0.2,0.5,0.5) 2.8 1.1

G ₄ (0.2,0.5,0.4)   (0.2,0.5,0.5) (0.3,0.4,0.5) (0.2,0.5,0.4) (0.2,0.5,0.5) 2.85 1.1

G ₅ (0.2,0.5,0.4)   (0.2,0.5,0.4) (0.2,0.5,0.4) (0.2,0.4,0.4) (0.3,0.4,0.4) 2.95 1.1

4.1 Application of special isomorphic images in presenting a fixed optimization model in decision making in diverse situations

NG models are considered as efficacious, beneficial and widely used models in variant fields because they show more pliability than variant types of fuzzy graph models in dealing with actual issues. Monitoring activities and making decisions at different levels of the company with favorable efficiency criteria and prearranged efficiency ideals in the ashy situations between certitude and incertitude, performs a significant role in elevating the efficiency level and effectuality rate of a company. The set of influencing factors the efficiency of a company in the ashy situations between certitude and incertitude can be considered as NG. We describe the T-strength, I-strength and F-strength values in each node and arc (path) as follows. For any x, y ∈ V and xy ∈ E, we have: T_K (r): The heaviness of the direct efficacy of factor r on the efficiency of the company in ashy situations. I_K (r): The heaviness of the inefficacy of factor r on the efficiency of the company in ashy situations. F_K (r): The heaviness of the indirect efficacy of factor r on the efficiency of the company in ashy situations. T_L (rs): The heaviness of direct influence rs on the efficiency of the company in ashy situations. I_L (rs): The heaviness of the inefficacy rs on the efficiency of the company in ashy situations. F_L (rs): The heaviness of indirect influence rs on the efficiency of the company in ashy situations. Hereon, the next relationships beseem logical: $\begin{matrix} T_{L} (rs) \leq T_{K} (r) \land T_{K} (s), I_{L} (rs) \geq I_{K} (r) \lor I_{K} (s), \\ F_{L} (rs) \geq F_{K} (r) \lor F_{K} (s) . \end{matrix}$

The relevance between r and s is efficient when the rs is a NHStA. Thus, the NSpDS of this graph contains factors that other factors are specially dominated with at least one of the elements (factors) of this set. Actually, the NSpDS creates a chance for officials and chiefs of the company to concentrate on the factors of the NSpDS and align decisions with these factors in lieu of paying attention and monitoring many influencing factors in ashy situations. This aids company chiefs and officials make the best decisions with the uttermost sureness in the shortest possible time.

As we saw in [8, 9], with strengthening the relevance between the influencing factors the efficiency of the company in the neutrosophic special dominating set or in the inverse neutrosophic special dominating set, the optimal normal heaviness of the factors graph in achieving the favorable efficiency can be elevated, which leads to increased exactitude and certitude in the decision-making process and the diminution of the neutrosophic node cardinality in NSpDS or inverse NSpDS. What is certain is that the heaviness of the impact of each of the influencing factors the efficiency of the company in different political, social, cultural, economic and etc situations varies. Nevertheless, officials and chiefs of a company often look to achieve a constant optimization model in different situations. This possibility seems to have been provided with special isomorphism images.

For example, in Fig. 6, the G neutrosophic graph and co-weak special isomorphisms are examined as factors graphs. It is noteworthy that the number of NStHAs, as well as the number of nodes of NSpDSs, are all constant and only the neutrosophic node cardinality of each node has changed according to political, social, cultural, economic, etc. situations.

Fig. 6

NG G and co-weak special isomorphic images of G.

5 Conclusion

The theory of NGs has many applications in new science and technology. Since neutrosophic models show more accuracy, flexibility and compatibility than fuzzy and vague models, in this paper, special irregular and special arc-irregular neutrosophic graphs and some of their variants are presented and examined. Also discussed, some special situations in which irregularities are matched together are discussed. Finally, by using special isomorphic images of a special irregular neutrosophic graph, a model for optimizing the neutrosophic special domination number parameters were provided. In contrast, unlike models presented, the number of nodes of NSpDS as well as the neutrosophic special cobondage number parameter remains constant.

References

Akram

, Feng

, Sawar

and Jun

Y.B.

, Certain types of vaguegraphs, University Politehnica of Bucharest Scientific BulletinSeries A 76(1) (2014), 141–154.

Akram

, Chen

W.J.

and Shum

K.P.

, Some properties of vague graphs, Southeast Asian Bulletin of Mathematics 37(3) (2013), 307–324.

Atanassov

, Intuitionistic fuzzy sets, Fuzzy Sets andSystems 20 (1986), 87–96.

Atanassov

and Gargov

, Interval valued intuitionistic fuzzysets, Fuzzy Sets and Systems 31 (1989), 343–349.

Banitalebi

, Irregular vague graphs, J. algebr. hyperstrucreslog. algebr 2 (2021), 73–90.

Banitalebi

and Borzooei

R.A.

, Domination of vague graphs by usingof strong arcs, Journal of Mathematical Extension 16 (2020).

Banitalebi

, Borzooei

R.A.

and Mohamadzadeh

, 2-Domination invague graphs, Algebraic Structures and their Applications 8(2) (2021), 203–222.

Banitalebi

and Borzooei

R.A.

, Neutrosophic special dominating setin neutrosophic graphs, Neutrosophic Sets and Systems 45(1) (2021), 3.

Banitalebi

and Borzooei

R.A.

, (Inverse) Neutrosophic specialn-domination in neutrosophic graphs with application in decisionmaking, Mahani Mathematical Research 13(1) (2024), 127–142.

10.

Banitalebi

, Borzooei

R.A.

Domination in Pythagorean fuzzy graphs, Granular Computing (2023), 1–8.

11.

Banitalebi

, Ahn

S.S.

, Jun

Y.B.

, Borzooei

R.A.

Normal m-domination and inverse m-domination in Pythagorean fuzzy graphs with application in decision making, Journal of Intelligent & Fuzzy Systems (Preprint) (2022), 1–10.

12.

Broumi

, Smarandache

, Talea

, Bakali

Single valued neutrosophic graphs: degree, order and size. In 2016 IEEE international conference on fuzzy systems (FUZZIEEE) (2016) (pp. 2444–2451). IEEE.

13.

Cockayne

E.J.

and Hedetniem

, Towards a theory of domination ingraphs, Networks 7(3) (1977), 247–261.

14.

Fei

, Study on neutrosophic graph with application in wirelessnetwork, CAAI Transactions on Intelligence Technology 5(4) (2020), 301–307.

15.

Gau

W.L.

and Buehrer

D.J.

, Vague sets, IEEE Transactions on Systems, Man, and Cybernetics 23(2) (1993), 610–614.

16.

Kulli

V.R.

and Janakiram

, The cobondage number of a graph, Discussiones Mathematicae Graph Theory 16(2) (1996), 111–117.

17.

Nagoor Gani

and Chandrasekaran

V.T.

, Domination in fuzzy graph, Advances in Fuzzy Sets and Systems 1(1) (2006), 17–26.

18.

Nagoor Gani

and Prasanna Devi

, 2-domination in fuzzy graphs, International Journal of Fuzzy Mathematical Archive 9(1) (2015), 119–124.

19.

Nagoor Gani

and Prasanna Devi

, Reduction of dominationparameter in fuzzy graphs, Global Journal of Pure and AppliedMathematics 13(7) (2017), 3307–3315.

20.

Rosenfeld

Fuzzy graphs. In Fuzzy sets and their applications to cognitive and decision processes (1975) (pp. 77–95). Academic Press.

21.

Hussain

S.S.

, Hussain

and Smarandache

, Domination number inneutrosophic soft graphs, Neutrosophic Sets and Systems 28 (2019), 228–244.

22.

Smarandache

Neutrosophic set-a generalization of the intuitionistic fuzzy set. In 2006 IEEE international conference on granular computing (2006) (pp. 38–42). IEEE.

23.

Smarandache

, Refined literal indeterminacy and the multiplicationlaw of sub-indeterminacies, Neutrosophic Sets and Systems 9(1) (2015), 58–63.

24.

Smarandache

Symbolic Neutrosophic Theory, Brussels, Europanova (2015), 195.

25.

Somasundaram

and Somasundaram

, Domination in fuzzy graphs-I, Pattern Recognition Letters 19(9) (1998), 787–791.

26.

Turksen

, Interval valued fuzzy sets based on normal forms, Fuzzy Sets and Systems 20(2) (1986), 191–210.

27.

Zadeh

L.A.

, Fuzzy sets, Inform Control 8 (1965), 338–353.

	a	b	c	d	k	NSΔN (G)	$(nsbn)_{𝔼} (G)$
G	(0.2,0.5,0.4)	(0.2,0.5,0.5)	(0.2,0.5,0.4)	(0.2,0.5,0.4)	(0.2,0.5,0.5)	2.825	1.1
G ₁	(0.2,0.4,0.4)	(0.2,0.5,0.5)	(0.3,0.5,0.4)	(0.2,0.5,0.4)	(0.2,0.5,0.5)	2.875	1.1
G ₂	(0.2,0.5,0.4)	(0.2,0.5,0.5)	(0.2,0.4,0.4)	(0.3,0.5,0.4)	(0.2,0.5,0.5)	2.875	1.1
G ₃	(0.3,0.4,0.5)	(0.2,0.5,0.5)	(0.2,0.5,0.5)	(0.2,0.5,0.5)	(0.2,0.5,0.5)	2.8	1.1
G ₄	(0.2,0.5,0.4)	(0.2,0.5,0.5)	(0.3,0.4,0.5)	(0.2,0.5,0.4)	(0.2,0.5,0.5)	2.85	1.1
G ₅	(0.2,0.5,0.4)	(0.2,0.5,0.4)	(0.2,0.5,0.4)	(0.2,0.4,0.4)	(0.3,0.4,0.4)	2.95	1.1