On Antipodal Single Valued Neutrosophic Graph

2017·0.1 MB·English

by  J. Malarvizhi, G. Divya;

Annals of Pure and Applied Mathematics
Vol. 15, No. 2, 2017, 235-242 Annals of
ISSN: 2279-087X (P), 2279-0888(online)
Published on 11 December 2017
www.researchmathsci.org
DOI: http://dx.doi.org/10.22457/apam.v15n2a10

On Antipodal Single Valued Neutrosophic Graph

J. Malarvizhi1 and G.Divya2

PG and Research Department of Mathematics
K.N.Govt. Arts College (Autonomous) for Women
Thanjavur – 7, Tamilnadu, India.
1Email: [email protected]; 2Email: [email protected]
Received 10 October 2017; accepted 9 December 2017
Abstract. In this paper we define Antipodal single valued neutrosophic graph and some
Isomorphic properties of Antipodal single valued neutrosophic graph are discussed.
Keywords: Single valued neutosophic graph, antipodal single valued neutrosophic graph
AMS Mathematics Subject Classification (2010): 05C72

1. Introduction
Graph theory has been highly successful in certain academic fields, including natural
sciences and engineering. Graph theoretic models can sometimes provide a useful
structure upon which analytical techniques can be used. It is often convenient to depict
the relationships between pairs of elements of a system by means of a graph or a digraph.
The vertices of the graph represent the system elements and its edges or arcs represent the
relationships between the elements. This approach is especially useful for transportation,
scheduling, sequencing, allocation, assignment, and other problems which can be
modelled as networks. Such a graph theoretic model is often useful as an aid in
communicating. Zadeh introduced the concept of fuzzy set. Attanassov introduced the
intuitionistic fuzzy sets which is a generalization of fuzzy sets. Fuzzy set theory and
intuitionistic fuzzy sets theory are useful models for dealing with uncertainty and
incomplete information. But they may not be sufficient in modeling of indeterminate and
inconsistent information encountered in real world. In order to cope with this issue,
neutrosophic set theory was proposed by Smarandache as a generalization of fuzzy sets
and intuitionistic fuzzy sets. Neutrosophic set proposed by Smarandache is a powerful
tool to deal with incomplete, indeterminate and inconsistent information in real world. It
is a generalization of the theory of fuzzy set, intuitionistic fuzzy sets, interval-valued
fuzzy sets and interval-valued intuitionistic fuzzy sets, then the neutrosophic set is
characterized by a truth-membership degree (T), an indeterminacy-membership degree (I)
and a falsity-membership degree (F) independently, which are within the real standard or
nonstandard unit interval ]−0, 1+[. There are some difficulties in modeling of some
problems in Engineering and Sciences. To overcome these difficulties, in 2010, concept
of single-valued neutrosophic sets and its operations were defined by Wang et al. as a
generalization of intuitionistic fuzzy sets.
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J. Malarvizhi and G.Divya
In crisp graph theory the concept of antipodal graph of a given graph G was
introduced by Smith. In fuzzy graph theory the concept of antipodal fuzzy graph of a
given fuzzy graph was introduced [1].
In this paper we define Antipodal single valued neutrosophic graph and some
Isomorphic properties of Antipodal single valued neutrosophic graph are discussed.

2. Preliminaries
Definition 2.1. [5] Let G: (σ ,µ ) be a fuzzy graph with the underlying set V.
A(G) : (σ A(G), µ A(G)) is defined as follows:
The node set of G is taken as the node set of A(G) also. Two nodes in A(G) are
made as neighbors if the µ - distance between them is diam(G).i.e.,
σ (u) = σ (u) for all u∈V and if δ (u,v) = diam(G) then
A(G)
µ (u,v) = µ (u,v) if u and v are neighbors in G
A(G)
= σ (u) ∧ σ (v) if u and v are not neighbors in G
µ (u,v) = 0 otherwise.
A(G)
This pair A(G) is a fuzzy graph, because σ (u)=σ (u) for all u∈V implies σ is a
A(G) A(G)
fuzzy subset on V and by the definition of µ , it is a fuzzy relation on σ , such that,
A(G) A(G)
µ (u,v) ≤ σ (u) ∧ σ (v) = σ (u) ∧ σ (v) for all u,v in V. This fuzzy graph A(G) is
A(G) A(G) A(G)
termed as Antipodal fuzzy graph of G.

Definition 2.2. [8] Let X be a space of points (objects). A neutrosophic set A in X is
characterized by a truth-membership function T (x), an indeterminacy-membership
A
function I (x) and a falsity-membership function F (x). The functions T (x), I (x), and
A A A A
F (x) are real standard or non-standard subsets of ]0−, 1+[ . That is, T (x) : X → ]0−, 1+[,
A A
I (x) : X → ]0−, 1+[ and F (x) : X → ]0−, 1+[ and 0−≤ T (x) + I (x) + F (x) ≤ 3+.
A A A A A

Definition 2.3. [7] A Single-Valued Neutrosophic graph(SVN graph)is a pair G = (A,B)
of the crisp graph G* = (V, E)(i.e., with underlying set V), where A : V → [0, 1] is
single-valued neutrosophic set in V and B : V × V → [0, 1] is single-valued neutrosophic
relation on V such that
T (xy) ≤ min{T (x), T (y)},
B A A
I (xy) ≤ min{I (x), I (y)},
B A A
F (xy) ≤ max{F (x), F (y)}
B A A
for all x, y ∈ V . A is called single-valued neutrosophic vertex set of G and B is called
single-valued neutrosophic edge set of G, respectively.

Example 1.
V1(0.2, (0.2, 0.2,0.5) V2(0.4,

3) 7
2,0. 2,0.
0.1, 0. (0.,30.)
(

V3(0.6, (0.4, 0.3,0.6 ) V4(0.5, 0.4,0.8)
Figure 1: Single valued neutrosophic graph
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On Antipodal Single Valued Neutrosophic Graph
Definition 2.4. A partial single valued neutrosophic subgraph of a single valued
neutrosophic graph G= (A,B) is a single valued neutrosophic graph H= (A(cid:10),B(cid:10)) such
that
i) A(cid:10) ⊆ A where T(cid:13)(cid:14)(v(cid:16))≤T(cid:13)(v(cid:16)), I(cid:13)(cid:14)(v(cid:16))≤I(v(cid:16)), F(cid:13)(cid:14)(v(cid:16))≤ F(cid:13)(v(cid:16)) for all v(cid:16) ∈ A(cid:10)
ii) B(cid:10) ⊆B where T(cid:21)(cid:14)(cid:22)v(cid:16),v(cid:23)(cid:24)≤T(cid:21)(v(cid:16),v(cid:23)), I(cid:21)(cid:14)(cid:22)v(cid:16),v(cid:23)(cid:24) ≤I(cid:21)(v(cid:16),v(cid:23)), F(cid:21)(cid:14)(cid:22)v(cid:16),v(cid:23)(cid:24)≤F(cid:21)(v(cid:16),v(cid:23))
for every (v,v)∈ B(cid:10).
(cid:16) (cid:23)

Definition 2.5. A single valued neutrosophic subgraph of a single valued neutrosophic
graph G=(A,B) is a single valued neutrosophic graph H =(A(cid:10),B(cid:10)) such that
i) A(cid:10) ⊆ A where T(cid:13)(cid:14)(v(cid:16))= T(cid:13)(v(cid:16)), I(cid:13)(cid:14)(v(cid:16))=I(v(cid:16)), F(cid:13)(cid:14)(v(cid:16))= F(cid:13)(v(cid:16)) for all v(cid:16) ∈A(cid:10)
ii) B(cid:10) ⊆B where T(cid:21)(cid:14)(cid:22)v(cid:16),v(cid:23)(cid:24)= T(cid:21)(v(cid:16),v(cid:23)), I(cid:21)(cid:14)(cid:22)v(cid:16),v(cid:23)(cid:24) =I(cid:21)(v(cid:16),v(cid:23)), F(cid:21)(cid:14)(cid:22)v(cid:16),v(cid:23)(cid:24)=F(cid:21)(v(cid:16),v(cid:23))
for every (v,v)∈ B(cid:10).
(cid:16) (cid:23)

Definition 2.6. The two vertices are said to be adjacent in a single valued neutrosophic
graph G= (A, B) if T (x,y) = min{T (x), T (y)},
B A A
I (x,y) = min{I (x), I (y)},
B A A
F (x,y) = max{F (x), F (y)}.
B A A
In this case, x and y are said to be neighbors and (x , y) is incident on x and y .

Definition 2.7. [7] A single-valued neutrosophic graph G = (A,B) is called completeif
the following conditions are satisfied:
T (x,y) = min{T (x), T (y)},
B A A
I (x,y) = min{I (x), I (y)},
B A A
F (x,y) = max{F (x), F (y)}
B A A
for all x, y ∈ V .

Definition 2.8. [4] A single valued neutrosophic path P in a single valued neutrosophic
graph G=(A, B) is a sequence of distinct vertices such that
v ,v ,v ,… v T (v ,v )>
(cid:25) (cid:26) (cid:27) (cid:29) (cid:21) (cid:16)(cid:30)(cid:26) (cid:16)
, and for . Here is called the length of
0 I (v ,v)>0 F (v ,v )>0 0≤i≤1 n≥1
(cid:21) (cid:16)(cid:30)(cid:26) (cid:16) (cid:21) (cid:16)(cid:30)(cid:26) (cid:16)
the path P. The consecutive pairs are called edges of the path.
(v ,v)
(cid:16)(cid:30)(cid:26) (cid:16)
A single node or vertex may also be considered as a path. In this case the path
v
(cid:16)
is of the length 0.
We call P a cycle if and .
v = v n ≥3
(cid:25) (cid:29)

Definition 2.9. [4] A single valued neutrosophic graph G= (A, B) is said to be connected
if every pair of vertices has at least one single valued neutrosophic path between them,
otherwise it is disconnected.

Definition 2.10. Let G be a connected single valued neutrosophic graph.
The T – distance , is the smallest T – length of any u – v path
δ(u,v)
P : u=v(cid:25),v(cid:26),v(cid:27),… v(cid:29) =v in G is ℓ)(P)=∑(cid:29)(cid:16)1(cid:26)),(-.(cid:26)/0,-.), where vi∈ V,
i.e.,
δ (u,v)= min(ℓ (P)).
) )
The I – distance , is the smallest I – length of any u – v path
δ(u,v)
P : u=v(cid:25),v(cid:26),v(cid:27),… v(cid:29) =v in G is ℓ4(P)=∑(cid:29)(cid:16)1(cid:26)4,(-.(cid:26)/0,-.), where vi∈ V,
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J. Malarvizhi and G.Divya
i.e.,
δ (u,v)=min(ℓ (P)).
4 4
The F – distance , is the smallest F – length of any u – v path
δ(u,v)
P : u=v(cid:25),v(cid:26),v(cid:27),… v(cid:29) =v in G is ℓ5(P) =∑(cid:29)(cid:16)1(cid:26)5,(-.(cid:26)/0,-.), where vi∈ V,
i.e.,
δ (u,v)= min(ℓ (P)).
5 5
The distance is defined as .
δ(u,v) δ(u,v)= (cid:22)δ (u,v),δ (u,v),δ (u,v)(cid:24)
) 4 5
For each v V, the T – eccentricity of v is denoted by and is defined as
i∈ i e)(v(cid:16))
.
e (v)=max9δ (cid:22)v ,v(cid:24)│v ∈V,v ≠ v=
) (cid:16) ) (cid:16) (cid:23) (cid:23) (cid:16) (cid:23)
For each v V, the I – eccentricity of v is denoted by and is defined as
i∈ i e4(v(cid:16))
.
e (v)= max9δ (cid:22)v ,v(cid:24)│v ∈V,v ≠ v=
4 (cid:16) 4 (cid:16) (cid:23) (cid:23) (cid:16) (cid:23)
For each v V, the F – eccentricity of v is denoted by and is defined as
i∈ i e5(v(cid:16))
.
e (v)= max9δ (cid:22)v,v(cid:24)│ v ∈ V,v ≠v=
5 (cid:16) 5 (cid:16) (cid:23) (cid:23) (cid:16) (cid:23)
For each v V, the eccentricity of v is denoted by and is defined as
i∈ i e(v(cid:16))
.
e(v)= (cid:22)e (v),e (v),e (v)(cid:24)
(cid:16) ) (cid:16) 4 (cid:16) 5 (cid:16)
The T – radius of G is denoted by and is defined as .
r (G) r (G)= min9e (v)│v ∈V=
) ) ) (cid:16) (cid:16)
The I – radius of G is denoted by and is defined as .
r (G) r (G)=min9e (v)│v ∈ V=
4 4 4 (cid:16) (cid:16)
The F – radius of G is denoted by and is defined as .
r (G) r (G)= min9e (v)│v ∈ V=
5 5 5 (cid:16) (cid:16)
The radius of G is denoted by r(G) and is defined by r(G) = .
(r (G),r (G),r (G))
) 4 5
The T – diameter of G is denoted by and is defined as
d (G) d (G)=
) )
. The I – diameter of G is denoted by and is defined as
max9e (v)│v ∈ V= d(G)
) (cid:16) (cid:16)
. The F – diameter of G is denoted by and is defined
d (G)=max9e (v)│v ∈V= d (G)
4 4 (cid:16) (cid:16) 5
as . The diameter of G is denoted by d(G) and is defined by
d (G)=max9e (v)│v ∈ V=
5 5 (cid:16) (cid:16)
d(G) = .
(d (G),d (G),d(G))
) 4
A connected single valued neutrosophic graph is a self centred graph, if every
vertex of G is a central vertex, that is
r (G)= e (v),r (G)=e (v) and r (G)=
) ) (cid:16) 4 4 (cid:16) 5
.
e (v)
5 (cid:16)

Example 2. An SVN graph with r(G) = (5.5, 5, 1.7) , d(G) = (10, 10, 3.3)

V1(0 .2, 0.3,0.7) (0.2, 0 .2,0.5) V2(0.4, 0.3, 0.6)

1, 0.1,0.3) (0.1, 0.2, 0.6) 3,0.2,0.7)
0. 0.
( (

V(0.6, 0.9,0.4) (0.4, 0.3,0.6) V(0.5, 0.4,0.8)
3 4
Figure 2:

Definition 2.11. Let G and G(cid:10) be single valued neutrosophic graphs with underlying sets
V and V(cid:10) respectively. A homomorphism of single valued neutrosophic graphs, h∶G→
G(cid:10) is a map h∶ V→ V(cid:10) which satisfies
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On Antipodal Single Valued Neutrosophic Graph
, , for all
T(cid:13)(u)≤T(cid:13)(cid:14)(h(u)) I(cid:13)(u)≤I(cid:13)(cid:14)(h(u)) F(cid:13)(u)≤F(cid:13)(cid:14)(h(u)) u∈V
, ,
T(cid:21)(u,v)≤ T(cid:13)(cid:14)(h(u),h(v))I(cid:21)(u,v)≤I(cid:21)(cid:14)(h(u),h(v))F(cid:21)(u,v)≤F(cid:21)(cid:14)(h(u),h(v))
for all .
u,v∈V

Definition 2.12. Let G and G(cid:10) be single valued neutrosophic graphs with underlying sets
V and V(cid:10) respectively. An isomorphism of single valued neutrosophic graphs, h∶G→G(cid:10)
is a bijective map h∶V→V(cid:10) which satisfies
, , for all
T(cid:13)(u)= T(cid:13)(cid:14)(h(u)) I(cid:13)(u)= I(cid:13)(cid:14)(h(u)) F(cid:13)(u)= F(cid:13)(cid:14)(h(u)) u∈V
, , for all
T(cid:21)(u,v)= T(cid:21)(cid:14)(h(u),h(v))I(cid:21)(u,v)= I(cid:21)(cid:14)(h(u),h(v))F(cid:21)(u,v)= F(cid:21)(cid:14)(h(u),h(v))
.
u,v∈ V
Then G is said to be isomorphic to G(cid:10).

Definition 2.13. A weak isomorphism of single valued neutrosophic graphs, h∶G→ G(cid:10)
is a map h∶ V→V(cid:10) which is a bijective homomorphism that satisfies
, , for all
T(cid:13)(u)= T(cid:13)(cid:14)(h(u)) I(cid:13)(u)= I(cid:13)(cid:14)(h(u)) F(cid:13)(u)= F(cid:13)(cid:14)(h(u)) u∈V

Definition 2.14. A co-weak isomorphism of single valued neutrosophic graphs,
h ∶G→
G(cid:10) is a map h∶ V→ V(cid:10) which is a bijective homomorphism that satisfies T (u,v)=
(cid:21)
, , for all .
T(cid:21)(cid:14)(h(u),h(v)) I(cid:21)(u,v)= I(cid:21)(cid:14)(h(u),h(v)) F(cid:21)(u,v)= F(cid:21)(cid:14)(h(u),h(v)) u,v∈ V

3. Main results
Definition 3.1. Let G = (A,B) be a single valued neutrosophic graph. An antipodal single
valued neutrosophic graph A(G) = (P, Q) is a single valued neutrosophic graph of G =
(A,B) in which:
(a) An single valued neutrosophic vertex set of G is taken as single valued neutrosophic
vertex set of A(G), that is,
T (x) = T (x),
P A
I (x) = I (x),
P A
F (x) = F (x) for all x ∈ V ,
P A
(b) If (x, y) = d(G), then x and y are made as neighbours in A(G) such that
δ

T (x,y)= T (x,y) if x and y are neighbors in G
C (cid:21)

T (x,y)= min(cid:22)T (x),T (y)(cid:24) if x and y are not neighbors in G
C (cid:13) (cid:13)

I (x,y)= I (x,y) if x and y are neighbors in G
C (cid:21)

I (x,y)=min(cid:22)I (x),I (y)(cid:24) if x and y are not neighbors in G
C (cid:13) (cid:13)

F (x,y)= F (x,y) if x and y are neighbors in G
C (cid:21)

F (x,y)= max(cid:22)F (x),F (y)(cid:24) if x and y are not neighbors in G
C (cid:13) (cid:13)
That is, two vertices in A(G) are made as neighborhood if the T,I,F –distance between
them is diameter of G.

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J. Malarvizhi and G.Divya
Example 3.2.

V1(0.2, 0.3,0.7) (0.2, 0.2,0.5) V2(0.4, 0.3,0.6) V1(0.2, 0.3,0.7)
V(0.4, 0.3,0.6)
2

1, 0.1,0.3) (0.1, 0.2, 0.6) 3,0.2,0.7) 2, 0.3,0.7)
0. 0. 0.
( ( (
V(0.5, 0.4,0.8)
4
V(0.6, 0.9,0.4) (0.4, 0.3,0.6) V(0.5, 0.4,0.8) V(0.6, 0.9,0.4)
3 4 3
(a) SVN graph (b) Antipodal SVN graph
Figure 3:

Theorem 3.3. Let G = (A,B) be a complete single valued neutrosophic graph where (T ,
A
I , F )is constant function then G is isomorphic to A(G) = (P, Q).
A A
Proof: Given that G = (A,B) be a complete single valued neutrosophic graph with (T ,
A
I , F ) = (k , k , k ), where k , k , and k are constants, which implies that (v, v) = (l ,
A A 1 2 3 1 2 3 K i j 1
l , l ) ∀v, v∈ V . Therefore, eccentricity e(v) = (l , l , l ), ∀vi ∈ V , which implies that
2 3 i j i 1 2 3
d(G) = (l , l , l ). Hence (v, v) = (l , l , l ) = d(G), ∀v, v∈ V . Hence those vertices
1 2 3 K i j 1 2 3 i j
which are neighbous in G are also neighbors in A(G) such that
(a) A single valued neutrosophic vertex set of G is taken as single valued neutrosophic
vertex set of A(G), that is,
T (v) = T (v),
P i A i
I (v) = I (v),
P i A i
F (v) = F (v)
P i A i
for all v∈ V ,
i
(b)
T (cid:22)vv(cid:24)= T (cid:22)vv(cid:24) since v and v are neighbors in G
C (cid:16) (cid:23) (cid:21) (cid:16) (cid:23) (cid:16) (cid:23)

I (cid:22)vv(cid:24)= I (cid:22)vv(cid:24) since v and v are neighbors in G
C (cid:16) (cid:23) (cid:21) (cid:16) (cid:23) (cid:16) (cid:23)

F (cid:22)vv(cid:24)= F (cid:22)vv(cid:24) since v and v are neighbors in G
C (cid:16) (cid:23) (cid:21) (cid:16) (cid:23) (cid:16) (cid:23)
Taking the mapping from G → A(G) as i : A → P, G A(G).
≅

Theorem 3.4. If G and G are isomorphic to each other, then A(G ) and A(G ) are also
1 2 1 2
isomorphic.
Proof: As G and G are isomorphic, the isomorphism h, between them pre-serves the
1 2
edge weights, so the T,I,F - length and T,I,F- distance will also be pre-served. Hence, if
the vertex v has the maximum T – eccentricity, maximum I – eccentricity and minimum
i
F - eccentricity, in G , then h(v) has the maximum T – eccentricity, maximum I –
1 i
eccentricity and minimum F - eccentricity, in G . So, G and G will have the same
2 1 2
diameter. If the T,I,F - distance between v and v is (k , k , k ) in G , then h(v) and h(v)
i j 1 2 3 1 i j
will also have their T,I,F - distance as (k , k , k ) in G . The same mapping h itself is a
1 2 3 2
bijection between A(G ) and A(G ) satisfying the isomorphism condition.
1 2
(i)
T (v)=T (v)= T (cid:22)h(v)(cid:24)= T (cid:22)h(v)(cid:24) ∀v ∈ G
N0 (cid:16) (cid:13)0 (cid:16) (cid:13)O (cid:16) NO (cid:16) (cid:16) (cid:26)
(ii)
I (v)= I (v)= I (cid:22)h(v)(cid:24)= I (cid:22)h(v)(cid:24) ∀v ∈G
N0 (cid:16) (cid:13)0 (cid:16) (cid:13)O (cid:16) NO (cid:16) (cid:16) (cid:26)
(iii)
F (v)=F (v )=F (cid:22)h(v)(cid:24)=F (cid:22)h(v)(cid:24) ∀v ∈G
N0 (cid:16) (cid:13)0 (cid:16) (cid:13)O (cid:16) NO (cid:16) (cid:16) (cid:26)
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On Antipodal Single Valued Neutrosophic Graph
(iv)
T (cid:22)v,v(cid:24)= T (cid:22)v,v(cid:24) if v and vare neighbors in G
C0 (cid:16) (cid:23) (cid:21)0 (cid:16) (cid:23) (cid:16) (cid:23) (cid:26)

T (cid:22)v,v(cid:24)=minQT (cid:22)v),T (v(cid:24)R if v and v are not neighbors in G
C0 (cid:16) (cid:23) N0 (cid:16) N0 (cid:23) (cid:16) (cid:23) (cid:26)
(v)
I (cid:22)v,v(cid:24)= I (cid:22)v,v(cid:24) if v and v are neighbors in G
C0 (cid:16) (cid:23) (cid:21)0 (cid:16) (cid:23) (cid:16) (cid:23) (cid:26)

I (cid:22)v,v(cid:24)= minQI (cid:22)v),I (v(cid:24)R if v and v are not neighbors in G
C0 (cid:16) (cid:23) N0 (cid:16) N0 (cid:23) (cid:16) (cid:23) (cid:26)
(vi)
F (cid:22)v,v(cid:24)= F (cid:22)v,v(cid:24) if v and v are neighbors in G
C0 (cid:16) (cid:23) (cid:21)0 (cid:16) (cid:23) (cid:16) (cid:23) (cid:26)

F (cid:22)v,v(cid:24)=maxQF (cid:22)v),F (v(cid:24)R if v and v are not neighbors in G
C0 (cid:16) (cid:23) N0 (cid:16) N0 (cid:23) (cid:16) (cid:23) (cid:26)
As is an isomorphism,
h∶ G → G
(cid:26) (cid:27)
T (cid:22)v,v(cid:24)
C0 (cid:16) (cid:23)
= T Qh(v),h(cid:22)v(cid:24)R if v and vare neighbors in G T (cid:22)v,v(cid:24)
(cid:21)O (cid:16) (cid:23) (cid:16) (cid:23) (cid:26) C0 (cid:16) (cid:23)
= minQT (cid:22)h(v)),T (h(v (cid:24)R if v and v are not neighbors in G I (cid:22)v,v(cid:24)
(cid:21)O (cid:16) (cid:21)O (cid:23)) (cid:16) (cid:23) (cid:26) C0 (cid:16) (cid:23)
= I Qh(v),h(cid:22)v(cid:24)R if v and vare neighbors in G I (cid:22)v,v(cid:24)
(cid:21)O (cid:16) (cid:23) (cid:16) (cid:23) (cid:26) C0 (cid:16) (cid:23)

= minQI (cid:22)h(v)),I (h(v (cid:24)R if v and v are not neighbors in G
(cid:21)O (cid:16) (cid:21)O (cid:23)) (cid:16) (cid:23) (cid:26)

F (cid:22)v,v(cid:24)= F Qh(v),h(cid:22)v(cid:24)R if v and vare neighbors in G
C0 (cid:16) (cid:23) (cid:21)O (cid:16) (cid:23) (cid:16) (cid:23) (cid:26)
Hence and
T (cid:22)v,v(cid:24)= T Qh(v) ,h(cid:22)v(cid:24)R ,I (cid:22)v,v(cid:24)= I Qh(v),h(cid:22)v(cid:24)R
C0 (cid:16) (cid:23) CO (cid:16) (cid:23) C0 (cid:16) (cid:23) CO (cid:16) (cid:23)

F (cid:22)v,v(cid:24)= maxQF (cid:22)h(v)),F (h(v (cid:24)R if v and v are not neighbors in G
C0 (cid:16) (cid:23) (cid:21)O (cid:16) (cid:21)O (cid:23)) (cid:16) (cid:23) (cid:26)
. So, the same h is an isomorphism between A(G ) and
FC0(cid:22)v(cid:16),v(cid:23)(cid:24)= FCOQh(v)(cid:16),h(cid:22)v(cid:23)(cid:24)R 1
A(G ).
2

Theorem 3.5. If G =(A , B ) and G =(A , B ) are complete single valued neutrosophic
1 1 1 2 2 2
graph such that G is co-weak isomorphic to G then A(G ) is co-weak isomorphic to
1 2 1
A(G ).
2
Proof: As G is co-weak isomorphic to G , there exists a bijection h : G → G satisfying,
1 2 1 2
.
T (v )≤ T (cid:22)h(v)(cid:24), T (cid:22)v,v(cid:24)= T Qh(v),h(cid:22)v(cid:24)R, ∀ v ,v ∈ V
(cid:13)0 (cid:16) (cid:13)O (cid:16) (cid:21)0 (cid:16) (cid:23) (cid:21)O (cid:16) (cid:23) (cid:16) (cid:23) (cid:26)
If G has n vertices, arrange the vertices of G in such a way that
1 1 T(cid:13)0(v(cid:26))≤T(cid:13)0(v(cid:27))≤
. As G and G are complete, co-weak isomorphic single valued
T(cid:13)0(vS)≤⋯≤T(cid:13)0(v(cid:29)) 1 2
neutrosophic graph . By Theorem 3.4 and
T (cid:22)v ,v(cid:24)= T Qh(v),h(cid:22)v(cid:24)R, ∀ v,v ∈ V
(cid:21)0 (cid:16) (cid:23) (cid:21)O (cid:16) (cid:23) (cid:16) (cid:23) (cid:26)
the definition of antipodal single valued neutrosophic graph, we have A(G) contains all
i
the vertices of G, where i = 1, 2. That is,
T (x)=T (x),I (x)=I (x) and F (x)=
N0 (cid:13)0 N0 (cid:13)0 N0
for all and . So, the same
F (x) x∈ V T (cid:22)v,v(cid:24)= T Qh(v),h(cid:22)v(cid:24)R, ∀v,v ∈V
(cid:13)0 C0 (cid:16) (cid:23) CO (cid:16) (cid:23) (cid:16) (cid:23) (cid:26)
bijection h is a co-weak isomorphism between A(G ) and A(G ).
1 2

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