Abstract
Neutrosophic cubic graph (NCG) belonging to FG family has good capabilities when facing problems that cannot be expressed by FGs. When an element membership is not clear, neutrality is a good option that can be well supported by a NCG. Hence, in this paper, some types of edge irregular neutrosophic cubic graphs (EI-NCGs) such as neighborly edge totally irregular (NETI), strongly edge irregular (SEI), and strongly edge totally irregular (SETI) are introduced. A comparative study between NEI-NCGs and NETI-NCGs is done. Finally, an application of neutrosophic cubic digraph to find the most effective person in a school has been presented.
1. Introduction
The fuzzy set theory was introduced by Zadeh [1]. It focuses on the membership degree of an object in a particular set. Kaufmann [2] represented FGs based on Zadeh’s fuzzy relation [3, 4]. Rosenfeld [5] described the structure of FGs obtaining analogs of several graph theoretical concepts. Bhattacharya [6] gave some remarks on FGs. Several concepts on FGs were introduced by Mordeson et al. [7]. The existence of a single degree for a true membership could not resolve the ambiguity on uncertain issues, so the need for a degree of membership was felt. Afterward, to overcome the existing ambiguities, Atanassov [8] defined an extension of fuzzy set by introducing non-membership function and defined intuitionistic fuzzy set (IFS). But after a while, Atanassov and Gargov [9] developed IFS and presented interval-valued intuitionistic fuzzy set (IVIFS). Hongmei and Lianhua [10] defined interval-valued fuzzy graph and studied its properties. Zhang et al. [11] introduced bipolar fuzzy sets and relations. Smarandache [12–14] gave the idea of neutrosophic sets. Kandasamy [15] defined neutrosophic graphs. Akram et al. [16–19] studied new results in NGs. Jun et al. [20] introduced cubic set. For more details about cubic sets and their applications in different research areas, we refer the readers to [21–23]. Rashid et al. [24] investigated cubic graphs. Jun et al. [25, 26] gave the idea of neutrosophic cubic set and defined different operations on it. Gulistan et al. [27, 28] presented complex bipolar fuzzy sets, NCGs, and some binary operations on it. Karunambigai et al. [29] discussed edge regular-IFG. Gani and Radha [30] studied the concept of regular fuzzy graphs and defined degree of a vertex in FGs. Gani et al. [31] investigated the concept of IFGs, NI-FGs, and HI-FGs in 2008. Nandhini [32] described the concept of SI-FG and studied its properties. Maheswari and Sekar defined the concepts of edge irregular-FGs and edge totally irregular-FGs [33]. Also, they analyzed some properties of NEI-FGs, NETI-FGs, SEI-FGs, and SETI-FGs [34, 35]. Rao et al. [36–38] studied dominating set, equitable dominating set, valid degree, isolated vertex, and some properties of VGs with novel application. Kou et al. [39] investigated g-eccentric node and vague detour g-boundary nodes in VGs. Shi et al. [40, 41] introduced total dominating set, perfect dominating set, and global dominating set in product vague graphs. Rashmanlou et al. [42] presented some properties of cubic graphs. Amanathulla et al. [43] studied on distance two surjective labeling of paths and interval graphs. Bhattacharya and Pal [44] gave the fuzzy covering problem of fuzzy graphs and its application. Borzooei et al. [45, 46] defined inverse fuzzy graphs and new results of domination in vague graphs. Kalaiarasi et al. [47] presented regular and irregular m-polar fuzzy graphs. Ramprasad et al. [48] investigated some properties of highly irregular, edge regular, and totally edge regular m-polar fuzzy graphs. Poulik and Ghorai [49] defined certain indices of graphs under bipolar fuzzy environment. Ullah et al. [50] introduced new results on bipolar-valued hesitant fuzzy sets. Jan et al. [51] presented some root level modifications in interval valued fuzzy graphs. Broumi et al. [52] introduced a novel system and method for telephone network planning based on neutrosophic graph. Muhiuddin et al. [53, 54] presented reinforcement number of a graph and new results in cubic graphs. Talebi et al. [55–57] presented some properties of irregularity and edge irregularity on intuitionistic fuzzy graphs and single valued neutrosophic graphs.
NCGs have many applications in psychology and medical sciences and can play a significant role in solving the vague and complex problems that exist around our lives. With the help of this fuzzy graph, the most effective person in an organization can be determined according to the amount of its performance in a specific period. Therefore, in this paper, some types of EI-NCGs such as neighborly edge totally irregular (NETI)-NCGs, strongly edge irregular (SEI)-NCGs, and strongly edge totally irregular (SETI)-NCGs are introduced. Also, we have given some interesting results about EI-NCGs, and several examples are investigated. Finally, an application of neutrosophic cubic digraph to find the most effective person in a school has been presented.
2. Preliminaries
Definition 1. A graph is a mathematical model consisting of a set of nodes and a set of edges , where each is an unordered pair of distinct nodes.
Definition 2 (see [5]). A FG is a non-empty set together with a pair of functions and so that , .
All the basic notations are shown in Table 1.
3. New Concepts of Edge Irregular-NCGs
Definition 3. Let be a graph. By NCG of , we mean a pair where is the NCS representation of and is the NCS representation of so that(i).(ii).(iii).
Definition 4. Let be a NCG on . Then, the degree of a node is defined as where , . , . and .
Definition 5. Let be a NCG on . The TD of a node is defined by where , . , . and .
Definition 6. Let be a NCG on . Then:(i) is irregular, if there is a node that is neighbor to nodes with VDs.(ii) is TI, if there is a node which is neighbor to nodes with various TDs.
Definition 7. Let be a CNCG. Then, is called a(i)NI-NCG if each pair of neighbor nodes has VDs.(ii)NTI-NCG if each pair of neighbor nodes has various TDs.(iii)SI-NCG if each pair of nodes has VDs.(iv)STI-NCG if each pair of nodes has various TDs.(v)HI-NCG if each node in is neighbor to the nodes having VDs.(vi)HTI-NCG if each node in is neighbor to the nodes having various TDs.
Definition 8. Let be a NCG. The degree of an edge is defined as where , . , . and .
Definition 9. Let be a NCG. The TD of an edge is defined as where . . . . . .
Definition 10. Let be a CNCG on . Then, is called a(1)NEI-NCG if each pair of AEs has VDs.(2)NETI-NCG if each pair of AEs has various TDs.
Example 1. Consider a graph which is both NEI-NCG and NETI-NCG.
Consider where and are defined asFrom Figure 1,Clearly, is a NEI-NCG.So, is a NETI-NCG.
Therefore, is both NEI-NCG and NETI-NCG.
Example 2. NEI-NCG need not to be NETI-NCG.
Let be a NCG and be a star that includes four nodes where
and are defined asFrom Figure 2,.
Here, . Hence, is a NEI-NCG. But is not a NETI-NCG, since all edges have same TDs.
Example 3. NETI-NCG does not need to be NEI-NCG. The following shows this subject.
Let be a NCG so that is a path that consists of four nodes where
and are defined asFrom Figure 3,Here, . Hence, is not a NEI-NCG. But is a NETI-NCG, since and .
Theorem 1. Let be a CNCG on and be a CF. Then, is a NEI-NCG, iff is a NETI-NCG.
Proof. Assume that is a CF, and let , in , where is constant.
Let and be pair of AEs in . Then,Therefore, adjacent edges have various degrees if and only if they have various total
degrees. So, is a NEI-NCG iff is a NETI-NCG.
Remark 1. Let be a CNCG on . If is both NEI-NCG and NETI-NCG, then does not need to be a CF.
Example 4. Let be a NCG and be a path that consists of four nodes where and are defined asFrom Figure 4,Here, and . Hence, is a NEI-NCG. Also, and . Hence, is a NETI-NCG. But is not CF.
Theorem 2. Let be a CNCG on and be a CF. If is a SI-NCG, then is a NEI-NCG.
Proof. Let be a CNCG. Assume that is a CF, and let , in , where is constant.
Let and be any two AEs in and be a SI-NCG. Then, each pair of nodes in
has VDs, and henceTherefore, each pair of AEs has VDs. Hence, is a NEI-NCG.
Theorem 3. Let be a CNCG on and be a CF. If is a SI-NCG, then is a NETI-NCG.
Remark 2. Converse of Theorems 3 is not generally true.
Example 5. Let be a NCG so that is a path on four nodes where and are defined asFrom Figure 5,Here, is not a SI-NCG. . . . .It is noted that and . Also, and . Hence, is both NEI-NCG and NETI-NCG. But is not a SI-NCG.
Theorem 4. Let be a CNCG and be a CF. Then, is a HI-NCG if and only if is a NEI-NCG.
Proof. Let be a CNCG. Assume that is a CF, and let , in , in which is constant.
Let and be any two AEs in . Then,