Single Valued Neutrosophic Detour Distance

2020·1 MB·English

by  Y. Srinivasa Rao, Ch. Shashi Kuma, T. Siva Nageswara Rao, V. Venkateswara Rao

Journal of Critical Reviews

ISSN- 2394-5125 Vol 7, Issue 8, 2020

SINGLE VALUED NEUTROSOPHIC DETOUR DISTANCE

Y. Srinivasa Rao1, Ch. Shashi Kumar2, T. Siva Nageswara Rao3, V. Venkateswara Rao4

1Research Scholar, Department of Mathematics, Acharya Nagarjuna University, Guntur (Dt.), A.P, India
2Department of Basic Science & Humanities , Vignan’s Institute of Technology and Science, Deshmukhi, T.S, India
3,4Division of Mathematics, Vignan’s Foundation for Science Technology & Research (Deemed to be University),
Vadlamudi, Guntur (Dt.), A.P, India
E-mail: [email protected], [email protected], [email protected],
[email protected]

Received: 02.05.2020 Revised: 01.06.2020 Accepted: 25.06.2020

Abstract
In the present article, we deduce a characterization of SVN detour eccentric vertex. The concepts of SVN graph are examined. Also we
obtained some definitions SVN on a vertex like SVN detour eccentric vertex, SVN detour radius, SVN detour diameter, SVN detour
centered and SVN detour periphery. We derive some important results based on these SVN detour radius, diameter, center and
periphery.

Keywords --- Detour distance, SVN detour eccentric, SVN detour distance

© 2020 by Advance Scientific Research. This is an open-access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/)
DOI: http://dx.doi.org/10.31838/jcr.07.08.173

INTRODUCTION W w, w , w
Example. Let
The Neutrosophic sets launch by Smarandache [10, 11] are a 1 2 3
great exact implement for the situation uncertainty in the real Aw,0.3,0.2,0.7,w,0.5,0.3,0.1,w,0.8,0.05,0.4
world. These uncertainty idea comes from the theories of fuzzy 1 2 3
sets [5], intuitionistic fuzzy sets [2, 4] and interval valued
W
intuitionistic fuzzy sets [3]. The representation of the is a SVN set in
neutrosophic sets are truth, indeterminacy and falsity value.
These T, I, F values belongs to standard or nonstandard unit Explanation 2.2 SVN relation on W
interval denoted by ]−0, 1+[ [6,9].
W
Let be a non-empty set. Then we call mapping
The idea of subclass of the NS and SVNS introduced by Wang et AW,T,I,F
,
al. [12]. The idea of SVNS initiation by intuitionistic fuzzy sets [1
,7], in this the functions Truth, Indeterminacy, Falsity are not Fw ,w :WW 0,10,1
is a SVN
dependent and these values are present within [0, 1] [8]. 1 2
Neutrosophic theory is widely expands in all fields especially W
relation on such that
authors discoursed about topology with respect to neutrosophic
[13]. T w,w 0,1,I w,w 0,1,F w,w 0,1
.
A 1 2 A 1 2 A 1 2
Graph theory has at this time turn into a most important branch
of mathematics. It is the division of combinatory. The Graph is a Explanation 2.3.
extensively important to analyze combinatorial complication in Let AT ,I ,F  and BT ,I ,F  be a SVN
dissimilar areas in mathematics, optimization and computer A A A B B B
science. Mainly significant object is well-known. The uncertainty graphs on a set W . If B is a SVN relation on A, then
on the subject of vertice and edges or both representation to be a
fuzzy graph. T w ,w minT w ,T w 
B 1 2 A 1 A 2
In this manuscript, discus about SVN graphs and neutrosophic
ddeetfionuer SdVisNta encccee bnetrtiwcieteyn, rtawdoiu vse, rdtiicaemse otef rt,h cee gnrtaepr ha bnads epde roipnh tehriys IBw1,w2maxIAw1,IAw2
with respect to detour distance. Also find some important results

o n these topics. F w ,w maxF w ,F w  for all
B 1 2 A 1 A 2
PRELIMINARIES
Explanation 2.1 SVN sets w ,w W
1 2
A SVN set is explained as the membership functions represented

W
as a triplet set in is denoted by Explanation 2.4.
w,T,I,F :wW, these functions are mapping The symmetric property defined on SVN relationBon W is
explained by
from W to [0,1]. Where T denote truth membership, I
denote indeterminate value and F denote false value of W.
Journal of critical reviews 810
SINGLE VALUED NEUTROSOPHIC DETOUR DISTANCE

TB w1,w2TB w2,w1 , A path P:xw0,w1,w2,........wk1,wk  y in G is
I w ,w I w ,w  sequence of distinct vertices such that
B 1 2 B 2 1 , T w ,w0, I w ,w0, F w ,w0,

F w ,w F w ,w  B i1 i B i1 i B i1 i

B 1 2 B 2 1 i1,2,..........,k k
and length of the path is ,where x is
Explanation 2.5. SVN graph called initial vertex and y is called terminal vertex in the path.
G V,E
The new graph in SVN is denoted by is a pair
Explanation 2.7.
GA,B, where AT ,I ,F  is a SVN in V A SVN graph GA,B of G V,E is called strong
A A A
andBTB,IB,FB  is SVN in V2 defined as STVN grwaph, wif minT w ,T w 

TB w1,w2minTA w1,TA w2 IBwi,wjmaxIAwi,IAwj

I w ,w maxI w ,I w  B i j A i A j
B 1 2 A 1 A 2 F w ,w maxF w ,F w 
F w ,w maxF w ,F w  for all B i j A i A j ,for all
B 1 2 A 1 A 2
 
w ,w E
w ,w V i j
1 2
P:r w ,w,w ,........,w ,w s
SVNG of an edge denoted by ww V2 If 0 1 2 k1 k be a path of
1 2
length k between r and s then
Explanation 2.6. T r,s,I r,s,F r,sk
Let GA,B be a SVNG and x,yV B B B is defined as
supT r,w T w ,w .....T w ,s,
 B 1 B 1 2 B k1
T r,s,I r,s,F r,sk  infI r,w I w ,w .....I w ,s,
B B B B 1 B 1 2 B k1

infF r,w F w ,w .....F w ,s
 B 1 B 1 2 B k1

T r,s,I r,s,F r,sis said to be the strength of connectedness between two vertices x and y in G, where
B B B
T r,s,I r,s,F r,s supT r,s, infI r,s, infF r,s
B B B  kN B kN B kN B 
 
If T r,sT r,s,I r,sI r,s,F r,sF r,s then the arc rs
B B B B B B
G
in is called a strong arc.

A path xy is strong path if all arcs on the path are strong. implies thatba . Hence b is a SVN detour eccentric node
S.N.D

of G.
SVN DETOUR DISTANCE

Explanation 3.1
Conversely, let each vertex of G is a SVN detour eccentric vertex.
SVN detour distance is defined as the length of , x − y strong path
If possible, let G be not SVN detour self centered SVN graph. Then
between x and y if there is no other strong path longer than P
between x and y and we denote this by S.N.D(x, y). Any x−y strong radS.N.D (G) diamS.N.D (G) and a node rGsuch that eS.N.D
path whose length is S.N.D(x, y) is called a x−y SVN detour path.
(r) = diamS.N.D (G). Also let prS.N.D. Let U be a r − p SVN
SVN DETOUR PERIPHERY (PerS.N.D (G)) AND SVN DETOUR detour in G. So there must have a node q on U for which the node
ECCENTRIC SUB GRAPH (EccS.N.D (G)) q is not a SVN detour eccentric node of U. Also q cannot be a SVN
Theorem 4.1 detour eccentric node of every other node. Again if q be a SVN
A SVN graph G is a SVN detour self centered if and only if every
node of G is a SVN detour eccentric. detour eccentric node of a node a (say), means qaS.N.D .
Then there exist an extension of a − q SVN detour up to r or up to
Proof. Suppose G is a SVN detour self centered SVN graph and let
b be a node in G. Let ab . So p. But this contradicts the facts that qaS.N.D . Hence radS.N.D
S.N.D
(G) = diamS.N.D (G) and G is a SVN detour self centered SVN graph.
e bS.N.Da,b. Since G is a SVN detour self
S.N.D Theorem 4.2. If G is a SVN detour self centered SVN graph, then
centered SVN graph, eS.N.D (a) = eS.N.D (b) = S.N.D(a, b) and this radS.N.D (G) = diamS.N.D (G) = n − 1, here n is order of the graph G.
Journal of critical reviews 811
SINGLE VALUED NEUTROSOPHIC DETOUR DISTANCE

Proof. Let G be a SVN detour self centered SVN graph. If possible, Case 2: When b is internal node in G, then  a connection
let diamS.N.D (G) = l < n − 1. between b to x and b to y, because of connectedness of G. Then a
− b SVN detour can be extend to x or y. This is impossible,
Let U and U be two distinct SVN detour peripheral path. Let because b is a SVN detour eccentric node of a. Hence
1 2
e adiam G
pU , qU .So  a strong path between p and q, S.N.D S.N.D i.e, b is a SVN detour
1 2
peripheral node of G.
 U U
because of connectedness of G. Then nodes on 1and 2,
whose eccentricity > l, but this is impossible, because diamS.N.D (G) Conversely, we assume that b be a SVN detour peripheral node G.
U U U U So a SVN detour peripheral node, say a (distinct from b).
= l. Hence and are not distinct. Since and are
1 2 1 2 Therefore b is a SVN detour eccentric node of a.
arbitrary, so  node r in G such that r is a common in all SVN
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node, say c which is adjacent to b. So . Since G is
2 open Sets and Pre-closed Sets in Neutrosophic Topology,
P c International Journal of ChemTech Research, Vol.10 No.10,
connected, c is connected to a node of , say . So either
1 pp 449- 458, 2017
cP cPP 
or . Thus in any case the path from a
2 1 2
c P
to x or a to y through c and is longer than . But it is
2
impossible, since b is a SVN detour eccentric node of a. Hence
e adiam G
i.e, b is a SVN detour
S.N.D S.N.D
peripheral node of G.
Journal of critical reviews 812