Some Results on the Graph Theory for Complex Neutrosophic Sets

Fuzzy graph theory plays an important role in the study of the symmetry and asymmetry properties of fuzzy graphs. With this in mind, in this paper, we introduce new neutrosophic graphs called complex neutrosophic graphs of type 1 (abbr. CNG1). We then present a matrix representation for it and study some properties of this new concept. The concept of CNG1 is an extension of the generalized fuzzy graphs of type 1 (GFG1) and generalized single-valued neutrosophic graphs of type 1 (GSVNG1). The utility of the CNG1 introduced here are applied to a multi-attribute decision making problem related to Internet server selection.


Introduction
Smarandache [1] introduced a new theory called neutrosophic theory, which is basically a branch of philosophy that focuses on the origin, nature, and scope of neutralities and their interactions with different ideational spectra.On the basis of neutrosophy, Smarandache defined the concept of a neutrosophic set (NS) which is characterized by a degree of truth membership T, a degree of indeterminacy membership I, and a degree of falsity membership F. The concept of neutrosophic set theory generalizes the concept of classical sets, fuzzy sets by Zadeh [2], intuitionistic fuzzy sets by Atanassov [3], and interval-valued fuzzy sets by Turksen [4].In fact, this mathematical tool is apt for handling problems related to imprecision, indeterminacy, and inconsistency of data.The indeterminacy component present in NSs is independent of the truth and falsity membership values.To make it more convenient to apply NSs to real-life scientific and engineering problems, Smarandache [1] proposed the single-valued neutrosophic set (SVNS) as a subclass of neutrosophic sets.Later on, Wang et al. [5] presented the set-theoretic operators and studied some of the properties of SVNSs.The NS model and its generalizations have been successfully applied in many diverse areas, and these can be found in [6].
Graphs are among the most powerful and convenient tools to represent information involving the relationship between objects and concepts.In crisp graphs, two vertices are either related or not related to one another so, mathematically, the degree of relationship is either 0 or 1.In fuzzy graphs on the other hand, the degree of relationship takes on values from the interval [0, 1].Subsequently, Shannon and Atanassov [7] defined the concept of intuitionistic fuzzy graphs (IFGs) using five types of Cartesian products.The concept fuzzy graphs and their extensions have a common property that each edge must have a membership value of less than, or equal to, the minimum membership of the nodes it connects.
In the event that the description of the object or their relations or both is indeterminate and inconsistent, it cannot be handled by fuzzy, intuitionistic fuzzy, bipolar fuzzy, vague, or intervalvalued fuzzy graphs.For this reason, Smarandache [8] proposed the concept of neutrosophic graphs based on the indeterminacy (I) membership values to deal with such situations.Smarandache [9,10] then gave another definition for neutrosophic graph theory using the neutrosophic truth-values (T, I, F) and constructed three structures of neutrosophic graphs: neutrosophic edge graphs, neutrosophic vertex graphs and neutrosophic vertex-edge graphs.Subsequently, Smarandache [11] proposed new versions of these neutrosophic graphs, such as the neutrosophic off graph, neutrosophic bipolar graph, neutrosophic tripolar graph, and neutrosophic multipolar graph.Presently, works on neutrosophic vertex-edge graphs and neutrosophic edge graphs are progressing rapidly.Broumi et al. [12] combined the SVNS model and graph theory to introduce certain types of SVNS graphs (SVNG), such as strong SVNG, constant SVNG, and complete SVNG, and proceeded to investigate some of the properties of these graphs with proofs and examples.Broumi et al. [13] then introduced the concept of neighborhood degree of a vertex and closed neighborhood degree of a vertex in SVNG as a generalization of the neighborhood degree of a vertex and closed neighborhood degree of a vertex found in fuzzy graphs and intuitionistic fuzzy graphs.In addition, Broumi et al. [14] proved a necessary and sufficient condition for a SVNG to be an isolated SVNG.
Recently, Smarandache [15] initiated the idea of the removal of the edge degree restriction for fuzzy graphs, intuitionistic fuzzy graphs and SVNGs.Samanta et al. [16] proposed a new concept called generalized fuzzy graphs (GFG) and defined two types of GFG.Here the authors also studied some of the major properties of GFGs, such as the completeness and regularity of GFGs, and verified the results.In [16], the authors claim that fuzzy graphs and their extensions are limited to the representations of only certain systems, such as social networks.Broumi et al. [17] then discussed the removal of the edge degree restriction of SVNGs and presented a new class of SVNG, called generalized SVNG of type 1, which is an extension of generalized fuzzy graphs of type 1 proposed in [16].Since the introduction of complex fuzzy sets (CFSs) by Ramot et al. in [18], several new extensions of CFSs have been proposed in literature [19][20][21][22][23][24][25].The latest model related to CFS is the complex neutrosophic set (CNS) model which is a combination of CFSs [18] and complex intuitionistic fuzzy sets [21] proposed by Ali and Smarandache [26].The CNS model is defined by three complex-valued membership functions which represent the truth, indeterminate, and falsity components.Therefore, a complexvalued truth membership function is a combination of the traditional truth membership function with the addition of the phase term.Similar to fuzzy graphs, complex fuzzy graphs (CFG) introduced by Thirunavukarasu et al. [27] have a common property that each edge must have a membership value of less than or equal to the minimum membership of the nodes it connects.
In this paper, we extend the research works mentioned above, and introduce the novel concept of type 1 complex neutrosophic graphs (CNG1) and a matrix representation of CNG1.To the best of our knowledge, there is no research on CNGs in the literature at present.We also present an investigation pertaining to the symmetric properties of CNG1 in this paper.In the study of fuzzy graphs, a symmetric fuzzy graph refers to a graph structure with one edge (i.e., two arrows on opposite directions) or no edges, whereas an asymmetric fuzzy graph refers to a graph structure with no arcs or only one arc present between any two vertices.Motivated by this, we have dedicated an entire section in this paper (Section 7) to study the symmetric properties of our proposed CNG1.
The remainder of this paper is organized as follows: in Section 2, we review some basic concepts about NSs, SVNSs, CNSs, and generalized SVNGs of type 1; in Section 3, the formal definition of CNG1 is introduced and supported with illustrative examples; in Section 4 a representation matrix of CNG1 is introduced; some advanced theoretical results pertaining to our CNG1 is presented in Section 5, followed by an investigation on the shortest CNG1 in Section 6; the symmetric properties of ordinary simple CNG1 is presented in Section 7; and Section 8 outlines the conclusion of this paper and suggests directions for future research.This is followed by the acknowledgments and the list of references.

Preliminaries
In this section, we present brief overviews of NSs, SVNSs, SVNGs, and generalize fuzzy graphs that are relevant to the present work.We refer the readers to [1,5,17,18,27] for further information related to these concepts.
The key feature that distinguishes the NS from the fuzzy and intuitionistic fuzzy set (IFS) models is the presence of the indeterminacy membership function.In the NS model the indeterminacy membership function is independent from the truth and falsity membership functions, and we are able to tell the exact value of the indeterminacy function.In the fuzzy set model this indeterminacy function is non-existent, whereas in the IFS model, the value of the indeterminacy membership function is dependent on the values of the truth and falsity membership functions.This is evident from the structure of the NS model in which  +  +  ≤ 3, whereas it is  +  ≤ 1 and  = 1 −  −  in the IFS model.This structure is reflective of many real-life situations, such as in sports (wining, losing, draw), voting (for, against, abstain from voting), and decision-making (affirmative, negative, undecided), in which the proportions of one outcome is independent of the others.The NS model is able to model these situations more accurately compared to fuzzy sets and IFSs as it is able to determine the degree of indeterminacy from the truth and falsity membership function more accurately, whereas this distinction cannot be done when modelling information using the fuzzy sets and IFSs.Moreover, the NS model has special structures called neutrosophic oversets and neutrosophic undersets that were introduced by Smarandache in [11], in which the values of the membership functions can exceed 1 or be below 0, in order to cater to special situations.This makes the NS more flexible compared to fuzzy sets and IFSs, and gives it the ability to cater to a wider range of applications.The flexibility of this model and its ability to clearly distinguish between the truth, falsity, and indeterminacy membership functions served as the main motivation to study a branch of graph theory of NSs in this paper.We refer the readers to [28,29] for more information on the degree of dependence and independence of neutrosophic sets, and [11] for further information on the concepts of neutrosophic oversets and undersets.
Definition 1 [1].Let  be a space of points and let  ∈ .A neutrosophic set  ∈  is characterized by a truth membership function , an indeterminacy membership function , and a falsity membership function .The values of , ,  are real standard or nonstandard subsets of ] − 0, 1 + [, and , , :  →] − 0, 1 + [.A neutrosophic set can therefore be represented as: Since , ,  ∈ [0, 1], the only restriction on the sum of , ,  is as given below: Although theoretically the NS model is able to handle values from the real standard or nonstandard subsets of ] − 0, 1 + [, it is often unnecessary or computationally impractical to use values from this non-standard range when dealing with real-life applications.Most problems in engineering, and computer science deal with values from the interval [0, 1] instead of the interval ] − 0, 1 + [, and this led to the introduction of the single-valued neutrosophic set (SVNS) model in [5].The SVNS model is a special case of the general NS model in which the range of admissible values are from the standard interval of [0, 1], thereby making it more practical to be used to deal with most real-life problems.The formal definition of the SVNS model is given in Definition 2.
The union and the intersection of the phase terms of the complex truth, falsity and indeterminacy membership functions can be calculated from, but not limited to, any one of the following operations: Definition 5 [17].Let  Here we shall restate the concept of complex fuzzy graph of type 1.Moreover, for all the remaining parts of this paper, we shall denote the set { ∈ ℂ: || ≤ 1} as  1 .
Definition 6 [27].Let the following statements hold: Then: (i) the structure  ̇= 〈, , ̇〉 is said to be a complex fuzzy graph of type 1 (abbr.CFG1).(ii) For each  ∈ ,  is said to be a vertex of  ̇.The entire set  is thus called the vertex set of  ̇. (iii) For each ,  ∈ , (, ) is said to be a directed edge of  ̇.In particular, (, ) is said to be a loop of  ̇.

Complex Neutrosophic Graphs of Type 1
By using the concept of complex neutrosophic sets [26], the concept of complex fuzzy graph of type 1 [27], and the concept of generalized single valued neutrosophic graph of type 1 [17], we define the concept of complex neutrosophic graph of type 1 as follows: Then the structure  = 〈, , 〉 is said to be a complex neutrosophic graph of type 1 (abbr.CNG1).
Remark 2.  depends on   ,   ,   , and  depends on   ,   ,   .Hence there are seven mutually independent parameters in total that make up a CNG1: V,   ,   ,   ,   ,   ,   .Furthermore, in analogy to a GSVNG1: (i) For each  ∈ ,  is said to be a vertex of .The entire set  is thus called the vertex set of .
As mentioned, CNG1 is generalized from both GSVNG1 and CFG1.As a result, we have   ,   and   being functions themselves.This further implies that   (, ),   (, ) and   (, ) can only be single values from O 1 .In particular,   (, ),   (, ), and   (, ) can only be single values.
As a result, each vertex  in a CNG1 possess a single, undirected loop, whether void or not.And each of the two distinct vertices ,  in a CNG1 possess two directed edges, resulting from (, ) and (, ), whether void or not.
Recall that in classical graph theory, we often deal with ordinary (or undirected) graphs, and also simple graphs.To further relate our CNG1 with it, we now proceed with the following definition.We now discuss a real life scenario that can only be represented by a CNG1.

The Scenario
Note: All the locations mentioned are fictional Suppose there is a residential area in Malaysia with four families: , , , .All of them have Internet access.In other words, they are Internet clients, which will access the Internet servers from around the world (including those servers located within Malaysia) depending on which website they are visiting.
If they access the internet on their own, the outcomes can be summarized as given in the Table 1 and Figure 1.
Table 1.The outcomes of individuals, for Scenario 3.1.

Activities
Some members will seek excitement (e.g., playing online games) Happens on 80% of the day, and those will be connecting towards 0° (because that server is located in China ⋇) Happens on 70% of the day, and those will be connecting towards 30° Happens on 90% of the day, and those will be connecting towards 120° Happens on 80% of the day, and those will be connecting towards 250° Some members will want to surf around (e.g., online shopping) Happens on 50% of the day, and those will be connecting towards 130° (because that server is located in Australia ⋇) Happens on 60% of the day, and those will be connecting towards 180° Happens on 20% of the day, and those will be connecting towards 340° Happens on 40% of the day, and those will be connecting towards 200° Some members will need to relax (e.g., listening to music) Happens on 20% of the day, and those will be connecting towards 220° (because that server is located in Sumatra ⋇, Indonesia) Happens on 30% of the day, and those will be connecting towards 200° Happens on 50% of the day, and those will be connecting towards 40° Happens on 10% of the day, and those will be connecting towards 110° (⋇) as illustrated in Figure 1.Thus, each pair of family mentioned (e.g., {, }) may invite one another for a visit, accessing the Internet as one team.In particular: (i) When {, } or {, } access the internet together, they will simply search for "a place of common interest".This is regardless of who initiates the invitation.(ii)  and  rarely meet.Thus, each time they do, everyone (especially the children) will be so excited that they would like to try something fresh, so all will seek excitement and connect towards to a local broadcasting server at 240° to watch soccer matches (that server will take care of which country to connect to) for the entire day.This is also regardless of who initiates the visitation.(iii) The size and the wealth of  far surpasses .Thus, it would always be  who invites  to their house, never the other way, and during the entire visit, members of  will completely behave like members of  and, therefore, will visit the same websites as .
Denote the first term of the ordered pair (, ) as the family who initiates the invitation, and the second term as family who receives the invitation and visit the other family.The outcomes of the seven possible teams (, ), (, ) , (, ) , (, ), (, ) , (, ) , (, ) are, thus, summarized by Table 2.
Table 2.The outcomes of teams in pairs, for the scenario.

Some members will seek excitement
Happens on 80% of the day, and those will be connecting towards 15° Happens on the entire day, all will be connecting towards 240° Happens on 80% of the day, and those will be connecting towards 305° Happens on 80% of the day, and those will be connecting towards 250°

Some members will want to surf around
Happens on 60% of the day, and those will be connecting towards 155°

Does not happen
Happens on 50% of the day, and those will be connecting towards 165 o Happens on 40% of the day, and those will be connecting towards 200°

Some members will need to relax
Happens on 30% of the day, and those will be connecting towards 210°

Does not happen
Happens on 50% of the day, and those will be connecting towards 40° Happens on 10% of the day, and those will be connecting towards 110° On the other hand, {, } and {, } are mutual strangers.So  and  will visit each other.The same goes to  and .

Representation of the Scenario with CNG1
We now follow all the steps from (a) to (e) in Definition 7, to represent the scenario with a particular CNG1.
One of the way of representing the entire 〈 0 ,  0 ,  0 〉 is by using a diagram that is analogous with graphs as in classical graph theory, as shown in the Figure 2. In other words, only the non-void edges (whether directed or ordinary) and vertices are to be drawn in such a diagram.Hence, we have shown how a CGN1 can be constructed from a data set with four homes.The same concept mentioned can certainly be used on a larger dataset, such as one with thousands of locations and thousands of homes, which will result in a more complicated diagram being generated.However, one will definitely require computer algebraic systems, such as SAGE, to process the data and to display the data in diagram form.
Additionally, recall that, in classical graph theory, a graph can be represented by an adjacency matrix, for which the entries are either a positive integer (connected) or 0 (not connected).
This motivates us to represent CNG1 using a matrix as well, in a similar manner.Nonetheless, instead of a single value that is either 0 or 1, we have three values to deal with:   ,   ,   , with each of them capable of being anywhere in  1 .Moreover, each of the vertices themselves also contain   ,   ,   , which must be taken into account as well.

Two Methods of Representation
In this section, we discuss the representation of CNG1 in two ways, which are both analogous to the one encountered in classical literature.
In other words, each element of the matrix M is itself an ordered set of three elements, instead of just a number of either 0 or 1 in the classical literature.
Remark 3. Since  can only possess undirected loops, we decided not to multiply the main diagonal elements of  by 2, as seen in adjacency matrices for graphs classical literature (2 for undirected, 1 for directed, 0 for void).
Meanwhile, also recall that each of the vertices in  contains   ,   ,   , which must be taken into account as well.
Thus, we form another matrix  as shown: ), where   = (  ) for all .
To accomplish one of our methods of representing the entire ξ we, therefore, augment the matrix  with M, forming the adjacency matrix of CNG1, [K|M], as shown: where  , = (  ,   ), and   = (  ), for all , .
Although [K|M] is an n × (n + 1) matrix and therefore not a square, this representation will save us another separate ordered set to represent the   ,   ,   values of the vertices themselves.
Sometimes it is more convenient to separately deal with each of the three kinds of membership values for both edges and vertices.As a result, here we provide another method of representing the entire : using three n × (n + 1) matrices ).
, all the entries of   are either 1 or 0, then  is reduced to a graph in classical literature.Furthermore, if that   is symmetrical and with main diagonal elements being zero, then  is further reduced to a simple ordinary graph in the classical literature.

Illustrative Example
For the sake of brevity, we now give representation for our example for the scenario in 3.1 by the latter method using three matrices: As in Section 3, we have shown how a matrix representation of a CNG1 with || = 4 can be constructed.Likewise, the same concept mentioned can certainly be used on a larger CNG1 but, again, one will definitely require computer algebraic systems, such as SAGE to process the data and to display such a matrix representation.

Some Theoretical Results on Ordinary CNG1
We now discuss some theoretical results that follows from the definition of ordinary CNG1, as well as its representation with adjacency matrix.Since we are concerned about ordinary CNG1, all the edges that we will be referring to are ordinary edges.Definition 11.Let  = 〈, , 〉 be an ordinary CNG1.Let  = { 1 ,  2 , … ,   } be the vertex set of .Then, for each , the resultant degree of   , denoted as (  ), is defined to be the ordered set (  (  ),   (  ),   (  )), for which: +   (  ,   ).
Remark 8.In analogy to classical graph theory, each undirected loop has both its ends connected to the same vertex, so is counted twice.

Remark 9.
Each of the values of   (  ),   (  ), and   (  ) need not be an integer as in a classical graph.Definition 12. Let  = 〈, , 〉 be an ordinary CNG1.Let  = { 1 ,  2 , … ,   } be the vertex set of .Then, the resultant amount of edges of , denoted as   , is defined to be the ordered set ( For each vertex   , just because (  ) equals 0, that does not mean that all the edges connect to   are void.It could be two distinct edges {  ,  1 } and {  ,  2 } with   (  ,  1 ) = −  (  ,  2 ),   (  ,  1 ) = −  (  ,  2 ) and   (  ,  1 ) = −  (  ,  2 ) (i.e., equal in magnitude, but opposite in phase).The same goes to the value of  ξ .This differs from the classical theory of graphs and, therefore, it motivates us to look at a CNG1 in yet another approach.We, thus, further define the following: Definition 13.Let  = 〈, , 〉 be an ordinary CNG1.Let  = { 1 ,  2 , … ,   } be the vertex set of .Then, for each i, the absolute degree of   , denoted as ||(  ) , is defined to be the ordered set (||  (  ), ||  (  ), ||  (  )), for which: On the other hand, sometimes we are particularly concerned about the number of non-void edges in an ordinary CNG1.In other words, we just want to know how many edges {  ,   } with: (  ,   ) ≠ (0,0,0).
Instead of a mere visual interpretation, we must however form a precise definition as follows: +   (  ,   ).
Proof.Since ξ is ordinary, (  ,   ) = (  ,   ) for all i and r.The lemma thus follows.□ Lemma 2. Let  = 〈, , 〉 to be an ordinary CNG1.If  is simple.then, for each i: + |  (  ,   )| by Definition 13, it is always a non-negative real number.Thus, in order that ∑ ||  (  )  =1 = 0, there can be only one possibility: all ||  (  ) must be zero.□ Remark 14.A similar statement does not hold for the resultant degree.
We now proceed with two of our theorems which both serve as generalizations of the wellknown theorem in classical literature: "For an ordinary graph, the sum of the degree of all its vertices is always twice the number of its edges."Theorem 1.Let  = 〈, , 〉 be an ordinary CNG1.Then ∑ (  )  =1 = 2  .
This motivates to consider what is the least possible values of   and   , for the special case of an ordinary  being simple, with () = (0,0,0) and () = (1,0,0) for all of its vertices .

The Shortest CNG1 of Certain Conditions
We now proceed with the following definitions.Furthermore,  is said to be trivial if the entire  consist of one single vertex  with () = (1,0,0).
On the other hand,  is said to be gross if it is not net.

Lemma 7.
Let  = 〈, , 〉 be a non-trivial net CNG1.Then each vertex must have least two non-void edges adjacent to it.
Let  be an end-point of some of those non-void edges.From Lemma 7, we conclude that at least two non-void edges must be adjacent to .
Since  is simple, it now follows that those 2 non-void edges must be {, } and {, }, with , ,  being 3 mutually distinct vertices of .
If   = 3: There can only be one more non-void edges besides {, } and {, }.By Lemma 7:  must be an end-point of another non-void edge besides {, }; and c must also be an end-point of another non-void edge besides {, }.
Suppose   < 4. Since ξ is simple, the maximum possible number of edges (whether it is void or not) is 3 + Theorem 5.The smallest non-trivial net CNG1 must be of the structure in Figure 5: Consider the scenario where   = 4 and   = 4 (i.e., the least possible number).
If the edge {, } is non-void, then we would have formed Figure 4, as mentioned in the proof of Theorem 3 That leaves us with only one vertex  and only one extra non-void edge being adjacent to .This is a contradiction.
There is now only one choice left: both the edges {, } and {, } must be non-void.This gives rise to the following structure in Proof.Every single non-void edge is connected to two vertices.Thus, if we count the total number of adjacent non-void edges for each vertex, and then summing the results for all the vertices together, the result will be 2  (note: this paragraph is analogous to classical graph theory).By Lemma 7, each vertex must have at least two non-void edges connect to it.We now have 2  ≥ 2  , so   ≥   follows.□ Theorem 6.Let  = 〈, , 〉 be a non-trivial net CNG1 with both   and   being odd numbers.Then   ≥ 7 and   ≥ 5.
Since both   and   are odd, it follows that   ≥ 5 and   ≥ 5.So in addition to , , , there exist another 2 vertices , .
Consider the scenario where   = 5 and   = 5 (i.e., the least possible number).Case 1. Suppose the edge {, } is non-void.Then we would have formed Figure 4, as mentioned in the proof of Theorem 3.
That leaves us with two vertices  and , and two extra non-void edge, which both of them must be adjacent to .Even if {, } is non-void, the other non-void edge adjacent to  cannot possibly be {, } itself.Therefore, we have, at most, one non-void edge being adjacent to .This is a contradiction.
Case 2. Without loss of generality, suppose the edges {, } and {, } are non-void.Then we would have formed Figure 6, as mentioned in the proof of Theorem 5.
That leaves us with only one vertex  and only one extra edge being adjacent to , which is, again, a contradiction.
Since both   and   are odd, either one of the following must hold: Furthermore, by Lemma 8,   ≥   .Hence (c) will not occur, which implies that   ≥ 7 and  ξ ≥ 5.This completes the proof.□ Theorem 7. The smallest non-trivial net CNG1 , with both   and   being odd numbers, must be of the structure as shown in Figure 8: Proof.Let  = 〈, , 〉 be a non-trivial net CNG1 with both   and   being odd numbers.Then   ≥ 7 and   ≥ 5.
Consider the scenario where   = 7 and   = 5 (i.e., the least possible number).Since ξ is an ordinary CNG1, each vertex must have 5 edges adjacent to it (whether void or not).Since ξ is simple, one of the five edges for each vertex, which is a loop, must be void.As a result, we now conclude that each vertex must have at most 4 non-void edges adjacent to it.
On the other hand, by Lemma 7, each vertex must have at least two non-void edges adjacent to it.Since every single non-void edge is adjacent to two vertices.Thus, if we count the total number of adjacent non-void edges for each vertex, and then summing the results for all the vertices together, the result will be 7 × 2 = 14 (note: this paragraph is analogous to classical graph theory).

Without loss of generality:
Let  be that one vertex which is an end-point to only 2 non-void edges {, } and {, }.(i.e., {, }, {, } are void) (Figure 9) Then, each one among , , ,  must be an end-point of three non-void edges.
Since {, }, {, } are void, both  and  cannot be that vertex which is an end-point to four non-void edges.
By symmetry, fix  to be that vertex which is an end-point to four non-void edges.Then: {, }, {, }, {, }, {, } are all non-void.( 14) From ( 13) and ( 14), we have now arrived at the following structure (Figure 12): Suppose {, } is void.Then exactly one out of {, } and {, } must be non-void.Similarly, exactly one out of {, } and {, } must be non-void.By symmetry and the rules of graph isomorphism, fix {, } to be non-void, then  would have been an end-point of three non-void edges: {, }, {, }, {, }.So {, } must be void and, therefore, {, } is non-void.We, thus, obtain the following structure (Figure 13): Suppose {, } is non-void.Then we now arrived at the following structure (Figure 14): By symmetry, fix  to be a vertex which is an end-point of three non-void edges.Then exactly one edge out of {, } and {, } must be non-void.By the rules of graph isomorphism, we can fix {, } to be non-void.Again we obtain the following structure (Figure 15): Likewise, since {, } and {, } are the only two non-void edges adjacent to , we now have (, ) = − (Figure 16).
In all the following passages of this paper, let ℐ:  →  be the identity mapping from  to itself.Like classical graph theory, whenever  ≡  , ℐ is an isomorphism from  to  itself in accordance with Definition 19.It is, therefore, motivational to investigate if there are other nonidentity bijections from  to itself, which is also an isomorphism from  to  itself.Additionally, recall that, in classical graph theory, an isomorphism from a graph to itself will be called an automorphism on that graph.Thus, we proceed with the following definition: Definition 20.Let  be a non-void set.Let  = 〈, , 〉 be an ordinary CNG1's.Let :  →  be a bijection such that: (a) () = (()) for all  ∈ .(b) (, ) = ((), ()) for all ,  ∈ .
Then  is said to be an automorphism of .

Remark 16.
As  is ordinary, (, ) = (, ) follows for all ,  ∈ .Then: (i)  1 is an isomorphism from  1 to the following ordinary CNG1 (Figure 21).which is not equal to  1 in accordance with Definition 18.  1 is therefore not an automorphism of  1 .
Remark 25.An ordinary simple CNG1 with all non-void vertices having different membership value is classified as weak point-wise symmetric as well.
Definition 27.Let  = 〈, , 〉 be an ordinary simple CNG1. is said to be asymmetric if it is both edgewise asymmetric and point-wise asymmetric.
Based on the definition, we now state such symmetric properties on the smallest non-trivial net CNG1, as mentioned in Theorem 5, as well as the smallest non-trivial net CNG1 with both  ξ and  ξ being odd numbers, as mentioned in Theorem 7.Only the trivial automorphism ℐ can be formed.As ℐ() ≠ ,  ̃ is point-wise asymmetric.Moreover, as ℐ() ≠ ,  ̃ is edge asymmetric.
We end this section by giving a conjecture, which shall be dealt with in our future work: Conjecture 1.The smallest non-trivial net CNG1 , with both   and   being odd numbers, and is both SPS and wES, must be of the structure as shown in Figure 26:

Conclusions
In this article, we presented a new concept of the neutrosophic graph called complex neutrosophic graphs of type 1 (CNG1), and also proceeded to present a matrix representation of it.
The strength of CNG1 lies in the presence of both magnitude and direction for the parameters involved, as has been illustrated in Section 3. As the parameters have directions, even when the resultant degree of a vertex is zero, the edges to that vertex need not necessarily be void.Thus the concept of CNG1 may also be used in engineering, such as in metal frameworks, for example in the construction of power lines, so that even when the beams are under tension, the resultant force at a point (possibly being a cornerstone) joining all those beams are zero.
The concept of CNG1 can also be applied to the case of bipolar complex neutrosophic graphs (BCNG1).We have plans to expand on this interesting concept in the near future, and plan to study the concept of completeness, regularity, and CNGs of type 2.
As we can see in Section 6, when the choices of   and   becomes more restrictive, the smallest non-trivial net CNG1  increases in complexity.This makes us wonder what will be the smallest non-trivial net CNG1  in the case when both   and   are not divisible by all primes up to 5 (7, 11, etc.), as well as whether their symmetric properties, as outlined in Section 7.However, the proof of such cases will become much more tedious and, therefore, we would have to utilize computer programs, such as MATLAB and SAGE, in order to find those non-trivial net CNG1 .Therefore our future research in this area involves plans to deal with those non-trivial net CNG1 .We are motivated by the interest to know if there exist some general patterns or general theorems governing such smallest non-trivial net CNG1 as   and   become more restrictive.

Figure 1 .
Figure 1.The illustration of the servers' relative positions using a public domain map, for Scenario 3.1.

Figure 6 :Lemma 8 .
Without loss of generality, let (, ) = .Then both (, ) = − and (, ) = − must leaving us with (, ) =  as the only valid option.We are therefore left with the only way of assigning  as shown by the theorem.□ Let  = 〈, , 〉 be a non-trivial net CNG1.Then   ≥   .

Figure 6 .
Figure 6.The only choices left because eace vertex must have at least 2 adjacent non void edges.Mentioned in Theorem 5, 6.

Figure 7 .
Figure 7.The only choice left for the case of 5 non-void edges connecting to 5 vertices.Mentioned in Theorem 6.

Corollary 1 .Definition 18 .Definition 19 .
The smallest non-trivial net CNG1 , with both   and   not divisible by 2 or 3, must also be of the structure as shown in Figure: 7. Symmetric Properties of Ordinary Simple CNG1 Let  and  be two non-void sets.Let  = 〈, , 〉 and  = 〈, , 〉 be two ordinary CNG1s.If  = ,  =  and  = , then  and  are said to be equal, and shall be denoted by  ≡ .Let  and  be a non-void set.Let  = 〈, , 〉 and  = 〈, , 〉 be two ordinary CNG1s.If there exist a bijection :  →  such that: (a) () = (()) for all  ∈ .(b) (, ) = ((), ()) for all ,  ∈ .Then:(i) Such  is said to be an isomorphism from  to , and we shall denote such case by [] ≡ .(ii)  and  are said to be isomorphic, and shall be denoted by  ≅ .

Figure 21 .
Figure 21.This is not an automorphism of  1 as mentioned in Example 2.
1.A complex neutrosophic set  defined on a universe of discourse , which is characterized by a truth membership function   (), an indeterminacy-membership function   (), ̆ depends on  ̆,  ̆,  ̆ and  ̆ depends on  ̆ , ̆ , ̆.Hence there are seven mutually independent parameters in total that make up a CNG1: ,  ̆,  ̆,  ̆,  ̆, ̆, ̆.(i)For each  ∈ ,  is said to be a vertex of  ̆.The entire set  is thus called the vertex set of  ̆. (ii) For each ,  ∈ , (, ) is said to be a directed edge of  ̆.In particular, (, ) is said to be a loop of  ̆.

Table 3 .
(),where k represents any of the 3 functions on V0   ,   ,   , for the scenario.Also mentioned in Section 4.2.In accordance with the scenario, define the three functions   ,   ,   , as illustrated in Tables 4-6.

Table 4 .
The outcomes of   (, ), for the scenario.Also mentioned in Section 4.2.

Table 5 .
The outcomes of   (, ), for the scenario.Also mentioned in Section 4.2.

Table 6 .
The outcomes of   (, ), for the scenario.Also mentioned in Section 4.2.