The concepts of graph theory are applied in many areas of computer science including image segmentation, data mining, clustering, image capturing and networking. Fuzzy graph theory is successfully used in many problems, to handle the uncertainty that occurs in graph theory. A single valued neutrosophic graph (SVNG) is an instance of a neutrosophic graph and a generalization of the fuzzy graph, intuitionistic fuzzy graph, and interval-valued intuitionistic fuzzy graph. In this paper, the basic operations on SVNGs such as direct product, Cartesian product, semi-strong product, strong product, lexicographic product, union, ring sum and join are defined. Moreover, the degree of a vertex in SVNGs formed by these operations in terms of the degree of vertices in the given SVNGs in some particular cases are determined. Finally, an application of single valued neutrosophic digraph (SVNDG) in traval time is provided.
Graph representations are generally used for dealing with structural information, in different domains such as operations research, networks, systems analysis, pattern recognition, economics and image interpretation. However, in many situations, some aspects of a graph theoretic problem may be vague or uncertain. For instance, the vehicle travel time or vehicle capacity on a road network may not be known exactly. In such situations, it is natural to deal with the uncertainty applying fuzzy set theory. In 1965, Zadeh [27] originally introduced the concept of fuzzy set. Its characteristic is that a membership degree in [0, 1] is assigned to each element in the set. After the inception of fuzzy set theory, it has become a vigorous area of research in different disciplines including management sciences, medical and life sciences, social sciences, artificial intelligence, signal processing, robotics, expert systems, computer networks, pattern recognition, decision-making, graph theory and automata theory.
Smarandache [21] firstly proposed neutrosophy, a branch of philosophy which discusses the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra. The characteristics of neutrosophic set (NS), a generalization of [3, 29], are described by truth-membership, indeterminacy membership and falsity membership degrees independently. NS as a powerful general formal framework expresses and handles imprecise, indeterminate and inconsistent information, existing in real situations. Intuitionistic fuzzy set (IFS) is a generalization of fuzzy set. Its characteristic is that a membership degree and a non-membership degree are assigned to each element in the set. However IFSs and interval-valued intuitionistic fuzzy sets (IVIFSs) cannot deal with all types of uncertainty, such as indeterminate and inconsistent information, therefore, the concept of NS is more extensive and overcomes the above-mentioned issues. But NSs are difficult to apply in the real applications. To easily apply it to scientific and engineering fields, Wang et al. [23] initiated the concept of a single valued neutrosophic set (SVNS) and provide its various properties. NS, particularly SVNS has attracted significant interest from researchers in recent years. It has been widely applied in various fields, such as information fusion in which data are combined from different sensors [6], control theory [1], image processing [12], medical diagnosis [26], decision making [25], and graph theory [7, 22], etc.
The concept of fuzzy graphs was initiated by Kafmann [13], based on Zadeh’s fuzzy relations. Later, another elaborated definition of fuzzy graph with fuzzy vertex and fuzzy edges was introduced by Rosenfeld [15]. Mordeson and Peng [15] defined some operations on fuzzy graphs and investigated their properties. Later, the degrees of the vertices of the resultant graphs, obtained from two given fuzzy graphs using these operations were determined in [16, 17]. Ghorai and Pal [10, 11] defined the concept of m-polar fuzzy planar graphs. Intuitionistic fuzzy graphs were first introduced by Atanassov [5] and further discussed by Akram [2]. Mishra and Pal [14] initiated the notion of interval-valued intuitionistic fuzzy graphs. Rashmanlou et al. [18, 19] introduced many new concepts, including product of bipolar fuzzy graphs and interval-valued intuitionistic (S, T)-fuzzy graphs. When description of the objects or their relations or both is indeterminate and inconsistent, it cannot be handled by fuzzy, intuitionistic fuzzy and interval valued intuitionistic fuzzy graphs. Therefore some new theories are required. That is why, Smarandache [22] put forward the concept of neutrosophic graphs. Furthermore, Broumi et al. [7–9] discussed several properties of SVNGs and their extensions.
The paper is structured as follows: Section 2 contains a brief background about SVNSs and SVNGs. Section 3 establishes the definitions and properties of direct product, Cartesian product, semi-strong product, strong product, lexicographic product, union, ring sum and join on SVNGs. Section 4 is devoted to the application of SVNDGs in travel time and finally we draw conclusions in Section 5.
Preliminaries
In the following, some basic concepts on SVNSs and SVNGs are reviewed to facilitate nextsections.
A graph is a pair of sets G = (V, E), satisfying E (G) ⊆ V × V. The elements of V (G) and E (G) are the vertices and edges of the graph G, respectively. The standard products of graphs: direct product (tensor product), Cartesian product, semi-strong product, strong product (symmetric composition) and lexicographic product (composition) of two graphs G1 = (V1, E1) and G2 = (V2, E2) will be denoted by G1 × G2, G1 □ G2, G1 • G2, G1 ⊠ G2 and G1 [G2], respectively. Let (x1, x2) , (y1, y2) ∈ V1 × V2, then
A directed graph (or digraph) is nothing but a graph with directed edges.
Definition 2.1. [20, 28] A fuzzy subset η of a set V is a function η : V → [0, 1]. A fuzzy (binary) relation on a set V is a mapping μ : V × V → [0, 1] such that μ (x, y) ≤ min {η (x) , η (y)} for all x, y ∈ V. A fuzzy relation μ is symmetric if μ (x, y) = μ (y, x) for all x, y ∈ V. A fuzzy graph is a pair , where η is a fuzzy subset of a set V and μ is a (symmetric) fuzzy relation on η .
Definition 2.2. [3] An IFS X in V is an object having the form
where the functions μX : V → [0, 1] and νX : V → [0, 1] define the degree of membership and degree of non-membership of the element x ∈ V, respectively, such that 0 ≤ μX (x) + νX (x) ≤1 for all x ∈ V .
For each IFS X in V, πX (x) =1 - μX (x) - νX (x) is called a hesitancy degree of x in X. If πX (x) =0 for all x ∈ V, then IFS reduces to Zadeh’s fuzzy set.
Definition 2.3. Let V be a universal set. A NS X in V is an object having the following form
which is characterized by a truth-membership function TX, an indeterminacy-membership function IX and a falsity-membership function FX, where
There is no restriction on the sum of TX (x) , IX (x) and FX (x) , therefore 0- ≤ sup TX (x) + sup IX (x) + sup FX (x) ≤3+ . The functions TX (x) , IX (x) and FX (x) are real standard or nonstandard subsets of ] 0-, 1+ [.
Definition 2.4. [23] Let V be a universal set. A SVNS X in V is an object having the form
which is characterized by a truth-membership function TX, an indeterminacy-membership function IX and a falsity-membership function FX, where
For convenience, γ = 〈T, I, F〉 is called a single valued neutrosophic number, where T, I, F ∈ [0, 1], 0 ≤ T + I + F ≤ 3 . To measure the degree of suitability, Zhang et al. [30] presented a score function s of γ as s (γ) = T + 1 - I + 1 - F .
Definition 2.5. [24] A single valued neutrosophic relation in V, denoted by
where TR : V × V → [0, 1], IR : V × V → [0, 1] and FR : V × V → [0, 1] represent the truth-membership function, indeterminacy membership function and falsity-membership function of R, respectively.
Definition 2.6. [7] A SVNG of a (crisp) graph G = (V, E) is defined to be a pair , where
the functions TX : V → [0, 1], IX : V → [0, 1] and FX : V → [0, 1] represent the degree of truth-membership, indeterminacy-membership and falsity membership of the element x ∈ V, respectively. There is no restriction on the sum of TX (x) , IX (x) and FX (x), therefore 0 ≤ TX (x) + IX (x) + FX (x) ≤3 for all x ∈ V,
the functions TY : E ⊆ V × V → [0, 1] , IY : E ⊆ V × V → [0, 1] and FY : E ⊆ V × V → [0, 1] are defined by
There is no restriction on the sum of TY (xy) , IY (xy) and FY (xy), therefore 0 ≤ TY (xy) + IY (xy) + FY (xy) ≤3 for all xy ∈ E . Here X is the single valued neutrosophic vertex set of and Y is the single valued neutrosophic edge set of .
Definition 2.7. The degree of a vertex x ∈ V in a SVNG is defined as where dT (x) = ∑x,y≠x∈VTY (xy) , dI (x) = ∑x,y≠x∈VIY (xy) and dF (x) = ∑x,y≠x∈VFY (xy) .
Operations on SVNGs and their degree
In this section, the basic operations on graphs such as direct product, Cartesian product, semi-strong product, strong product, lexicographic product, union, ring sum and join are defined under single valued neutrosophic environment and their properties are investigated.
Definition 3.1. The direct product of two SVNGs and of the graphs G1 = (V1, E1) and G2 = (V2, E2), respectively, is defined as follows:
for all (x1, x2) ∈ V1 × V2,
for all x1y1 ∈ E1, forall x2y2 ∈ E2 .
Proposition 3.2.If and are the SVNGs, then is the SVNG.
Proof. Consider x1y1 ∈ E1, x2y2 ∈ E2 . Then
Definition 3.3. Let and be two SVNGs. For any vertex (x1, x2) V1 × V2,
Theorem 3.4.Let and be two SVNGs. If TY2 ≥ TY1, IY2 ≤ IY1, FY2 ≤ FY1, then and if TY1 ≥ TY2, IY1 ≤ IY2, FY1 ≤ FY2, then for all (x1, x2) ∈ V1 × V2 .
Proof. By definition of vertex degree of , we must have
Hence Similarly, it is easy to show that, if TY1 ≥ TY2, IY1 ≤ IY2, FY1 ≤ FY2, then □
Example 3.5. Consider two SVNGs and , where , , and . SVNGs and their direct product are shown in Fig. 1.
Since TY2 ≥ TY1, IY2 ≤ IY1, FY2 ≤ FY1, so, by Theorem 3.4, we have Therefore, Similarly, it is easy to find the degree of all vertices in .
Definition 3.6. The Cartesian product of two SVNGs and of the graphs G1 = (V1, E1) and G2 = (V2, E2), respectively, is defined as follows:
for all (x1, x2) ∈ V1 × V2,
for all x ∈ V1, forall x2y2 ∈ E2,
for all z ∈ V2, forall x1y1 ∈ E1 .
Proposition 3.7.If and are the SVNGs, then is the SVNG.
Proof. Consider x ∈ V1, x2y2 ∈ E2 . Then
Similarly, for z ∈ V2, x1y1 ∈ E1, we have
Definition 3.8. Let and be two SVNGs. For any vertex (x1, x2) ∈ V1 × V2,
Theorem 3.9.Let and be two SVNGs. If TX1 ≥ TY2, IX1 ≤ IY2, FX1 ≤ FY2 and TX2 ≥ TY1, IX2 ≤ IY1, FX2 ≤ FY1. Then for all (x1, x2) ∈ V1 × V2 .
Proof. By definition of vertex degree of , we must have
Similarly, it is easy to show that and Hence □
Example 3.10. Consider two SVNGs and as in Example 3.5, where TX1 ≥ TY2, IX1 ≤ IY2, FX1 ≤ FY2 and TX2 ≥ TY1, IX2 ≤ IY1, FX2 ≤ FY1. Their Cartesian product is shown in Fig. 2.
Then by Theorem 3.9, we have , and . Therefore, . Similarly, we can find the degree of all the vertices in
Definition 3.11. The semi-strong product of two SVNGs and of the graphs G1 = (V1, E1) and G2 = (V2, E2), respectively, is defined as follows:
for all (x1, x2) ∈ V1 × V2,
for all x ∈ V1, forall x2y2 ∈ E2,
for all x1y1 ∈ E1, forall x2y2 ∈ E2 .
Proposition 3.12.If and are the SVNGs, then is the SVNG.
Proof. The proof follows from proof of Propositions 3.2 and 3.7. □
Definition 3.13. Let and be two SVNGs. For any vertex (x1, x2) ∈ V1 × V2,
Theorem 3.14.Let and be two SVNGs. If TX1 ≥ TY2, IX1 ≤ IY2, FX1 ≤ FY2, TY1 ≤ TY2, IY1 ≥ IY2, FY1 ≥ FY2. Then for all (x1, x2) V1 × V2 .
Proof. By definition of vertex degree of , we must have
Analogously, it is easy to show that and Hence □
Example 3.15. Consider two SVNGs and as given in Example 3.5, where TX1 ≥ TY2, IX1 ≤ IY2, FX1 ≤ FY2, TY1 ≤ TY2, IY1 ≥ IY2, FY1 ≥ FY2, and their semi-strong product is shown in Fig. 3.
So, by Theorem 3.14, we have , and . Therefore, . Similarly, we can find the degree of all the vertices in
Definition 3.16. The strong product of two SVNGs and of the graphs G1 = (V1, E1) and G2 = (V2, E2), respectively, is defined as follows:
for all (x1, x2) ∈ V1 × V2,
for all x ∈ V1, forall x2y2 ∈ E2,
for all z ∈ V2, forall x1y1 ∈ E1 .
for all x1y1 ∈ E1, forall x2y2 ∈ E2 .
Proposition 3.17.If and are the SVNGs, then is the SVNG.
Proof. The proof follows from proof of Propositions 3.2 and 3.7. □
Definition 3.18. Let and be two SVNGs. For any vertex (x1, x2) ∈ V1 × V2,
Theorem 3.19.Let and be two SVNGs. If TX1 ≥ TY2, IX1 ≤ IY2, FX1 ≤ FY2, TX2 ≥ TY1, IX2 ≤ IY1, FX2 ≤ FY1, TY1 ≤ TY2, IY1 ≤ IY2, FY1 ≥ FY2. Then for all (x1, x2) ∈ V1 × V2 .
Proof. By definition of vertex degree of , we must have
Analogously, it is easy to show that and Hence □
Example 3.20. Consider two SVNGs and as in Example 3.5, where TX1 ≥ TY2, IX1 ≤ IY2, FX1 ≤ FY2, TX2 ≥ TY1, IX2 ≤ IY1, FX2 ≤ FY1, TY1 ≤ TY2, IY1 ≥ IY2, FY1 ≥ FY2 and their strong product is shown in Fig. 4.
Then by Theorem 3.19, we have
Therefore, Similarly, we can find the degree of all the vertices in
Definition 3.21. The lexicographic product of SVNGs of G1 = (V1, E1) and of G2 = (V2, E2) is defined as follows:
for all (x1, x2) ∈ V1 × V2,
for all x ∈ V1, forall x2y2 ∈ E2,
for all z ∈ V2, forall x1y1 ∈ E1,
for all x1y1 ∈ E1, x2 ≠ y2 .
Proposition 3.22.If and are the SVNGs, then is the SVNG.
Proof. From the proof of Proposition 3.7, it follows that
Now consider x1y1 ∈ E1, x2 ≠ y2. Then
Similarly, (IY1 ∘ IY2) ((x1, x2) (y1, y2)) = max {(IX1 ∘ IX2) (x1, x2) , (IX1 ∘ IX2) (y1, y2)} and (FY1 ∘ FY2) ((x1, x2) (y1, y2)) = max {(FX1 ∘ FX2) (x1, x2) , (FX1 ∘ FX2) (y1, y2)} . □
Definition 3.23. Let and be two SVNGs. For any vertex (x1, x2) ∈ V1 × V2,
Theorem 3.24.Let and be two SVNGs. If TX1 ≥ TY2, IX1 ≤ IY2, FX1 ≤ FY2 and TX2 ≥ TY1, IX2 ≤ IY1, FX2 ≤ FY1. Then for all (x1, x2) ∈ V1 × V2 .
Proof. For any vertex (x1, x2) ∈ V1 × V2,
Analogously, we can show that and . Hence . □
Definition 3.25. The union of two SVNGs and of the graphs G1 = (V1, E1) and G2 = (V2, E2), respectively, is defined as follows:
Proposition 3.26.If and are the SVNGs, then is the SVNG.
Proof. Consider xy ∈ E1 \ E2 . Then there are three different cases, (i) x, y ∈ V1 \ V2, (ii) x ∈ V1 \ V2, y ∈ V1 ∩ V2 and (iii) x, y ∈ V1 ∩ V2.
Suppose x, y ∈ V1 \ V2
Suppose x ∈ V1 \ V2, y ∈ V1 ∩ V2 . Then
Suppose x ∈ V1 ∩ V2, y ∈ V1 ∩ V2 . Then
Also, it is easy to find (IY1 ∪ IY2) (xy) ≥ max {(IX1 ∪ IX2) (x) , (IX1 ∪ IX2) (y)} and (FY1 ∪ FY2) (xy) ≥ max {(FX1 ∪ FX2) (x) , (FX1 ∪ FX2) (y)} in thethree possible cases. Similarly, if xy ∈ E2 \ E1,then (TY1 ∪ TY2) (xy) ≤ min {(TX1 ∪ TX2) (x) , (TX1 ∪ TX2) (y)} , (IY1 ∪ IY2) (xy) ≥ max {(IX1 ∪ IX2) (x) , (IX1 ∪ IX2) (y)} , (FY1 ∪ FY2) (xy) ≥ max {(FX1 ∪ FX2) (x) , (FX1 ∪ FX2) (y)} .
Let xy ∈ E1 ∩ E2 . Then
The converse of above result holds if V1 ∩ V2 = ∅ . □
Theorem 3.27.The union of and is a SVNG of G1 ∪ G2 if and only if and are SVNGs of G1 and G2, respectively, where V1∩ V2 = ∅.
Proof. Assume that is a SVNG. Let xy ∈ E1, then xy ∉ E2 and x, y ∈ V1 \ V2 . Thus
Thus is a SVNG of G1. Similarly, it is easy to show that is a SVNG of G2. □
Definition 3.28. Let and be two SVNGs. For any vertex x ∈ V1 ∪ V2, there are three cases to consider.
Case 1. Either x ∈ V1 \ V2 or x ∈ V2 \ V1. Then no edge incident at x lies in E1 ∩ E2. So, for x ∈ V1 \ V2
For x ∈ V2 \ V1,
Case 2.x ∈ V1 ∩ V2 but no edge incident at x lies in E1 ∩ E2. Then any edge incident at x is either in E1 \ E2 or in E2 \ E1.
Similarly, and
Case 3.x ∈ V1 ∩ V2 and some edges incident at x are in E1 ∩ E2.
Similarly, and .
Definition 3.29. The ring sum of two SVNGs and of the graphs G1 = (V1, E1) and G2 = (V2, E2), respectively, is defined as follows:
Proposition 3.30.If and are the SVNGs, then is the SVNG.
Definition 3.31. The join of two SVNGs and of the graphs G1 = (V1, E1) and G2 = (V2, E2), respectively, is defined as follows:
where E′ is the set of all edges joining the vertices of V1 and V2, V1∩ V2 = ∅.
Theorem 3.32.The join of and is a SVNG of G1 + G2 if and only if and are SVNGs of G1 and G2, respectively, where V1∩ V2 = ∅.
Proof. Suppose that is a SVNG. Then from Theorem 3.27, and are SVNGs.
Conversely, assume that and are SVNGs of G1 and G2, respectively. Consider xy ∈ E1 ∪ E2 . Then the required result follows from Proposition 3.26. Let xy ∈ É. Then
Similarly, we can show (IY1 + IY2) (xy) = max {(IX1 + IX2) (x) , (IX1 + IX2) (y)} and (FY1 + FY2) (xy) = max {(FX1 + FX2) (x) , (FX1 + FX2) (y)} . □
Definition 3.33. Let and be two SVNGs. For any vertex x ∈ V1 + V2,
For any x ∈ V1,
Similarly, and .
For any x ∈ V2,
Similarly, and
Single valued neutrosophic digraph in travel time
In modern age, planning efficient routes is essential for industry and business, with applications as varied as product distribution, laying new fiber optic lines for broadband internet, and suggesting new friends within social network websites such as Facebook. When we visit a website like Google Maps and looking for directions from one city to another city in USA, we are usually asking for a shortest path between the two cities. These computer applications use representations of the road maps as graphs, with estimated travel times as edge weights. The travel time is a function of traffic density on the road or the length of the road. The traffic density is a fuzzy, while the length of a road is a crisp quantity. In a road network, crossings are represented by vertices, roads by edges and traffic density on the road is usually calculated between adjacent crossings. These factors can be represented as a SVNS. Any model of a road network can be represented as a SVNDG where is a SVNS of crossings (vertices) at which the traffic density is calculated and connectivity conditions as truth-membership degree , indeterminacy membership degree and falsity membership degree ,
and is a SVNS of roads (edges) between crossings, whose truth-membership degree , indeterminacy membership degree and falsity membership degree can be calculated as:
The SVNDG of the travel time is given in Fig. 5. The single valued neutrosophic out neighbourhoods are given in Table 1.
The final weights on edges can be calculated by finding the score function of single valued neutrosophic edges as The final weighted digraph given in Fig. 6, which can be used for finding the shortest/optimal path between two locations (vertices) by any of the known methods, like Djkastra and A star. Weighted relations are given in Table 2.
Algorithm given below generates the weighted digraph, WR, for given SVNDG and uses it to calculate the optimal path from a source vertex.
Algorithm
void single valued neutrosophic shortest path(){
SVNS of crossings;
number of crossings=count
Empty SVNS of roads;
for (int c = 0 ; c < numberofcrossings ; c ++) {
for (int
if ( is adjacent to
}
}
}
SVNS of edges;
R = Single valued neutrosophic relation;
WR =Weighted relation;
no of edges;
for (int i = 0;i<no of edges; i ++) {
;
c =Adjacent from Node of ;
Adjacent to Node of ;
;
}
print WR;
Calculate optimal path using WR;
}
Conclusions
Single valued neutrosophic models are more flexible and practical than fuzzy, interval-valued fuzzy, intuitionistic fuzzy and interval-valued intuitionistic fuzzy models. SVNGs can be used in computer technology, networking, communication, economics, genetics, linguistics, sociology etc, when the concept of indeterminacy is present. So, in this paper, we have defined the basic operations on SVNGs such as direct product, Cartesian product, semi-strong product, strong product, lexicographic product, union, ring sum and join, and investigated some of their properties. Moreover, the degree of a vertex in SVNGs formed by these operations in terms of the degree of vertices in the given SVNGs in some particular cases are determined. They will be helpful especially when the graphs are very large. We have also provided an application of SVNDG in travel time.
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