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REGULARITY OF ALTERNATE QUADRA SUBMERGING POLAR FUZZY GRAPH AND ITS
APPLICATION
1 * 0
Anthoni Amali A1 , J . Jesintha Rosline 2
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Auxilium College(Autonomous), Vellore - 632006, Affiliated to Thiruvalluvar University, Serkadu, Tamil Nadu, India, [email protected], [email protected]
Abstract
Fuzzy soft sets and graphs are invented to solve uncertain problems in the field of Applied mathematics. It is a general mathematical tool introduced with many parameters to model the vagueness of the changing world. The insight learning of the AQSP fuzzy soft graphs paved the way to discover the extension of the AQSP fuzzy soft graph. In this research article we introduce the Regularity of AQSP fuzzy soft graph with definitions, theorems, properties, and real-life applications. The aim of this invention is mainly to obtain the parametric values in submerging level of confidence [-0.5, 0.5] C [-1,1]. The scope of this new AQSP fuzzy soft graph is to solve the imprecise problems in the field of Mathematical Engineering, Bio Mathematics, Economics, Medical Science, Artificial Intelligence and Machine learning. The regularity of AQSP fuzzy soft graph is combined with the concepts of regular, totally regular, and perfectly regular. The application of this new graph is developed for governing of the women safety vehicle network in different spots with membership submerging values. The future extension can be applied in Approximate reasoning, Mathematical psychology, Decision making for medical diagnosis.
Keywords: Regular AQSP fuzzy soft graph, Totally regular, Perfectly regular AQSP fuzzy soft graph, Alternate Quadra Submerging level of confidence.
1. Introduction
The concept of graph theory was introduced by Euler in 1736. He concreted the way to find the solution of Konigsberg bridge problem. In 1965 Zadeh[20] invented Fuzzy set theory as a mathematical fuzzy tool for handling uncertainties like vagueness, ambiguity, and imprecision in linguistic variables. Fuzzy set has resulted as a potential area of interdisciplinary exploration and the fuzzy graph theory is of modern inducement. The first definition of fuzzy graph was determined by Kaufmann[10] in 1973, based on Zadeh's fuzzy relation in 1971. In 1975, Rosenfeld[16] introduced the concept of fuzzy graph. The structure of fuzzy graphs, using fuzzy relations, obtaining contrasts of several graph hypothetical concepts are the masterpiece of Rosenfeld. Operations on fuzzy graphs were exposed by J.N.Moderson[14] and C.S.Peng. A.Nagoorgani[8] and K.Radha[9] invented the concept of regular fuzzy graphs in 2008.
In 1999, D.Molodtsov[12] intended the notion of soft set theory to solve complicated uncertain problems in Applied Mathematics, Engineering and Environmental studies. In 2001, P.K.Maji[11], initiated the concept of fuzzy soft sets. Zou and Xio discussed the application of the fuzzy soft sets in an imprecise scenario. Later, Akram[4] and Nawaz[15] presented new ideas known as fuzzy soft graphs. A.Pouhassani[24] and H.Doostie studied degree, total degree, regularity and total regularity of fuzzy soft graph and its properties. Regular fuzzy soft graphs and its related properties are investigated by B.Akhilandeswari. The concepts of fuzzy bipolar
soft sets and bipolar fuzzy soft sets have been introduced by Naz and Shabir. Aslam et al studied some basic operations on bipolar fuzzy soft sets.
In this article, we portray a new mathematical fuzzy graph model AQSP Fuzzy Soft graph for dealing imprecise information by integrating the concepts of fuzzy graph and fuzzy soft graphs. We estimate the regularities of AQSP fuzzy soft graphs and some of their characteristics and properties. Here Regular AQSP fuzzy soft graphs, and totally regular AQSP fuzzy soft graphs and perfectly regular AQSP fuzzy soft graphs are examined. Total degree of an AQSP fuzzy soft graph is designed. Theorems for regular AQSP fuzzy soft graphs and totally regular AQSP fuzzy soft graphs are presented. A necessary condition under which they are equivalent is provided. Some properties of regular AQSP fuzzy soft graphs, perfectly regular AQSP fuzzy soft graphs are reviewed with real life applications.The perception of AQSP fuzzy soft graph membership values with submerging level of confidence is applicable in Machine learning and medical psychology.We explored the AQSP fuzzy soft graph module in Governing of women safety police vehicle network with membership score functions.
2. Preliminaries
2.1. Fuzzy Graph [16]
Let U is a non-empty set. A fuzzy graph is a set of two of functions G : (a, p) where a is a fuzzy subset of U, p is a symmetric fuzzy relation on a, where a : U ^ [0,1] and the edge set p : U x U ^ [0,1] such that, p (x,y) < min (p (x), p (y)) Vx,y e U. The underlying crisp graph of fuzzy graph G : (a, p) is with the notion G* : (a*, p*) where a* is denoted as the non-empty set U of vertices and p* = E e V x V.
2.2. Fuzzy Soft graph [13]
A fuzzy soft graph G = (G*, F, K, A) is a four tuple such that
1. G* = (V, E) is a simple graph.
2. A is a non empty set of parameters.
3. (F, A) is a fuzzy soft vertex set V.
4. (K, A) is a fuzzy soft edge set E.
5. F(a), K(a) is a fuzzy soft graph of G*V a e A.
Then it satisfies the condition, K(a)(x, y) < F(a)(x) A F(a)(Y) Va e A and (x, y) e V.
2.3. Fuzzy soft graph degree of a vertex [4]
Let G = (G*, F, K, A) be a fuzzy soft graph on G*. The fuzzy soft graph degree of a vertex a is defined as degG (a) = EeeA Ex=y K(e)(x, y)Va e A and (x, y) e V.
2.4. Regular Fuzzy soft graph [4]
Let G = (G*,F, K, A) be a regular fuzzy soft graph if (F(e),K(e)) is regular fuzzy graph of degree k for all e, e A then G is a k- regular fuzzy soft graph.
2.5. Order of fuzzy soft graph [4]
Let Ga,v = ((A, a,), (A, p,)) be a fuzzy soft graph. Then the order of fuzzy soft graph GA,V = EeeA TjxeA ae(x).
2.6. AQSP Fuzzy Soft Graph [18]
Let V =(K(x),of(x),pP(x),pf(x)), (of (x), of (x),pP2 (x),pf(x))...(of(x), of(x),p2(x),pf(x))) be a nonempty AQSP fuzzy set. E (Parameters set) and AAOs2 C E. Also let,
(i) a2 : AAOS2 —>• FAOS2(V)(Collection of all AQSP fuzzy subsets in V), e -—> of, and
of : V —> [0,1], Vi -—> then (AAOS2, a2) : AQSP fuzzy soft vertex set.
(ii) oN : AAOS2 —> FAOS2(V)(Collection of all AQSP fuzzy subsets in V), e -—> of, and
of : V —> [-1,0], vi -—> of then (AAOS2, oN) : AQSP fuzzy soft vertex set.
(iii) p2 : AAOS2 —> FAOS2(V)(Collection of all AQSP fuzzy submerge subsets in V), e i—> pp,
and pp : V —> [0,0.5], vi -—> pp then ( AAOS2, p2 ) : AQSP fuzzy soft vertex set.
(iv) pN : AAOS2 —> FAOS2(V) (Collection of all fuzzy submerge subsets in V), e i—> pN, and
pN : V~—> [-0.5,0], vi -—> pN then (AAOS2,pN) : AQSP fuzzy soft vertex set.
(v) p2 : AAOS2 —> FAOS2(V x V) (Collection of all AQSPfuzzy subsets in V x V), e i—> p2e,
p2e : V x V —> [0,1], (Vi, Vj ) -—> p2e (Vi, Vj ) then ( Aaos2, p2 ) : AQSP fuzzy soft membership edge set.
(vi) pN : AAOS2 —> FAOS2(V x V)(Collection of all AQSPfuzzy subsets in V x V), e -—> pN,
and pN : V x V [-1,0], (Vi, Vj ) -—^ pN (Vi, Vj ) then ( AAOS2, pN ) : AQSP fuzzy soft non - membership edge set.
(vii) y2 : AAOS2 —> FAOS2(V x V)(Collection of all AQSPfuzzy subsets in V x V), e -—> y2,
and y2~~: V x V -4 [0,0.5], (Vi, Vj) -—^ y2(Vi, Vj) then (AAOS2, y2) : AQSP fuzzy soft submerge membership edge set.
(viii) yN : AAOS2 —> FAOS2(V x V)(Collection of all AQSPfuzzy subsets in V x V), e -—> yN, and yN ^ V x V —> [-0.5,0], (Vi, Vj ) -—^ yN (Vi, Vj ) then ( AAOS2, yN ) : AQSP fuzzy soft submerge membership edge set. Then the AQSP fuzzy soft graph is,
((AAOS2), (a2,aN,p2,pN)), ((AAOS2), (p2, pN, y2, YN)) if the conditions are satisfied (a) p2 (x, y) < o2 (x) A o2 (y), (b)~ pN (x, y) > of (x) V oN (y), (c) y2 (x, y) < p2 (x) A p22 (y), (d) yN (x, y) > pN (x) V pN (y), for all e G Aaqs2 and for all values of x,y = 1,2,3,...,n and this AQSP fuzzy soft graph is denoted as GAOS2(A, V).
3. Method
The essential definition of AQSP fuzzy soft graph method is deliberated with an examples.
3.1. Alternate Quadra Sub - merging Polar(AQSP) Fuzzy Graph
An Alternate Quadra - Submerging Polar (AQSP) Fuzzy Graph G = (oAQS2, pAQS2) is a fuzzy graph with crisp graph G* = (oAOS2 , pAOS2 ) is given as V = (o2 (x), oN (x), p2 (x), pN (x)) which is the membership value of vertices along with the uncertain membership value of edges is given as, E = V x V = (p2 (x,y), pN (x,y), y2 (x,y), yN (x,y)).
Here the vertex set V is defined with the given condition in a unique method which is an alternate contrast submerging polarized uncertain transformation.Here o2 = V 4 [0,1], oN =
V 4 [-1,0], p2 = d I 0.5, o2 (x) I and pN = -d |-0.5, oN (x)|. Here (-0.5, 0.5) is the fixation of uncertain alternate contrast polarized
submerging transformation into certain consistent preferable position. And the edge set E satisfies the following sufficient conditions.
(i) p2 (x,y) < min (o2 (x), o2 (y) ), (ii) pN (x, y) > max (oN (x), oN (y) )
(iii) y2 (x, y) < min (p2 (x), p2 (y) ) (iV) yn (x, y) > max (pN (x), pN (y) ),
V(x, y) G E. By definition, p2 = V x V 4 [0,1] x [1,0], pN = V x V 4 [-1,0] x [0, -1] and the submerging mappings, y2 = V x V 4 [0,0.5] x [0.5,0],
YN = V x V ^ [-0.5,0] x [0, -0.5], which denotes the impact of the alternate quadrant polarized fuzzy mapping.
The maximum of submerging presumption to be at the level of confidence [0,0.5] C [0,1] and the minimum of submerging presumption level of confidence is [-0.5,0] C [-1,0] extension of the graph with its membership and non - membership values portrait the unique level of submerging destination in an AQSP fuzzy graph.
Also it must satisfy the condition, -1 < aP (x) + aN (x) < 1 and |pP (x) + pN (x) | < 1 with constrains 0 < aP (x) + aN (x) + |pP (x) + pN (x)| < 2 such that the uncertain status of submerging presumption, transform into its precise consistent level with fixation mid - value 0.5, which implies that level of confidence 0.5 in an AQSP as the valuable membership of its position which is real and valid in the fuzzification. The example of AQSP fuzzy graph is given in Figure.1.
Figure 1: AQSP Fuzzy Graph G = (uaqsp, paqsp)
3.2. Example of AQSP Fuzzy Soft Graph
Consider an AQSP fuzzy soft graph GAQSP(A, V), where V = (vi, v2, v3, v4) and E = (e1,e2,e3). Here Gaqsp(A, V) is described in Table.1. and
pe(v,, Vj) = 0, V(v,, Vj) e V x V {(V1, V2), (v2, V3), (V3, V4), (V1, V4), (V1, V3)} for all e e E.
Table 1: Tabw/ar represeniai/on of AQSP Fuzzy Soft Graph parameter vertex set.
(a, p) v1 V2 v3 V4
e1 ( 0.6, - 0.7, (0.7, - 0.8, ( 0.8, - 0.9, ( 0.6, - 0.7,
0.1,-0.2) 0.2, -0.3) 0.3, - 0.4 ) 0.1,- 0.2)
e2 ( 0.7, - 0.6, (0.8, - 0.7, ( 0.9, - 0.8, ( 0.8, - 0.8,
0.2,- 0.1) 0.3, -0.2) 0.4, - 0.3 ) 0.3,- 0.3)
e3 ( 0.8, - 0.6, (0.9, - 0.7, ( 0.8, - 0.8, ( 0.9, - 0.9,
0.3,- 0.1) 0.4, -0.2) 0.3, - 0.3 ) 0.4,- 0.4)
Table 2: Tabw/ar representation of AQSP Fuzzy Soft Graph parameter edge set.
Vi, V2 V,V3 V3, V4 V4, Vi V1, V3
ei ( 0.6, - 0.7, ( 0.7, - 0.8, ( 0.6, -0.7, ( 0.6, - 0.7, ( 0.6, - 0.7,
0.1,- 0.2) 0.2, -0.3) 0.1,-0.2 ) 0.1,- 0.2) 0.1,- 0.2)
e2 ( 0.7, - 0.6, ( 0.7, - 0.7, ( 0.8, - 0.8, ( 0.7, - 0.6, ( 0.6, - 0.6,
0.2,- 0.1) 0.2, -0.2) 0.3, - 0.3 ) 0.2,- 0.1) 0.1,-0.1)
£3 ( 0.8, - 0.6, ( 0.8, - 0.7, ( 0.8, - 0.7, ( 0.7, - 0.6, ( 0.8, - 0.6,
0.3,- 0.1) 0.3, -0.2) 0.3, - 0.2 ) 0.2,- 0.1) 0.3,- 0.1)
Table. 2. represents the AQSP fuzzy graph with parametric membership and non - membership with submerge values.
4. Descriptions of the regularity of AQSP fuzzy soft graph
4.1. Regular AQSP Fuzzy Ssoft Graph
Let G* = (a*, p*) be a crisp graph and GAQSP(A, V) be an regular AQSP fuzzy soft graph of G*. Then Gaqsp(A, V) is said to be an regular AQSP soft graph, if RAQSP(e,) is an regular AQSP fuzzy soft graph of degree k for all e, € Aaqsp, then Gaqsp (A, V) is a k - regular AQSP fuzzy soft graph.
V, (0.7, -0.8,0.2, -0.3)
v3 (0.7, -0.8,0.2, -0.3)
Figure 2: Gaqsp (A, V) - Corresponding to the parameter ei
v1 (0.8,-0.7,0.3,-0.2)
v3 ( 0.3, - 0.9, 0.4, -0.4)
Figure 3: Gaqsp (A, V) - Corresponding to the parameter e2
4.2. Example of an AQSP Fuzzy Soft Graph
Consider, an AQSP fuzzy soft graph, Gaosp(A, V), the vertex set V = (vi, v2, v3, v4) and let the corresponding parameters E = (e1, e2).
Here Gaosp(A, V) = ((Aaqsp), (aP,aN,pP,pN)), ((Aaqsp), (/,pN,YP, YN)) is described by Table.3 and Table. 4 (v1, v2, v3, v4)).
4.3. Remark on Regular AQSP Fuzzy Graph
Fron Figure.2 and Figure.3 we get the result that the regular AQSP fuzzy graph which can not be a totally regular AQSP fuzzy graph. Table 3. represents the AQSP fuzzy soft graph vertex set.
Table 3: Tabw/ar representation of AQSP Fuzzy Soft Graph parameter vertex set.
(Ц, Y)
Vi
V2
V3
v4
ei e2
( 0.7, - 0.8, 0.2,- 0.3) ( 0.8, - 0.7, 0.3,- 0.2)
(0.8, - 0.9, 0.3, - 0.4) (0.7, - 0.9, 0.2, - 0.4)
( 0.7, - 0.7, 0.2, - 0.2 ) ( 0.9, -0.9, 0.4, - 0.4 )
(0.8, - 0.8, 0.3,- 0.3) (0.8, - 0.8, 0.3 ,- 0.3)
Table 4: Tabw/ar representation of AQSP Fuzzy Soft Graph parameter edge set.
V1V2 V2 V3 V3 V4 v4v1
e1 ( 0.6, - 0.7, ( 0.7, - 0.8, ( 0.6, - 0.7, ( 0.7, - 0.8,
0.1,- 0.2) 0.2, - 0.3) 0.1, - 0.2 ) 0.2,- 0.3)
e2 ( 0.7, - 0.7, ( 0.6, - 0.6, ( 0.7, - 0.7, ( 0.6, - 0.6,
0.2,- 0.2) 0.1,-0.1) 0.2, - 0.2 ) 0.1,- 0.1)
Table. 4 represents the corresponding edges, (v1, v2), (v2, v3), (v3, v4), (v4, v1) , for all values of e G Aaqsp.
4.4. Totally Regular AQSP Fuzzy Soft Graph
Let G* = (a, p) be a simple graph and Gaosp(A, V) be an AQSP fuzzy soft graph of G*. Then Gaosp (A, V) is said to be a totally regular AQSP fuzzy soft graph if Raosp (A, V) is totally regular fuzzy soft graph for all values of e, G Aaosp, then Gaosp (A, V) is called k totally regular AQSP fuzzy soft graph.
Theorem 1. If Gaosp (A, V) satifies the condition of regular and totally regular AQSP fuzzy soft graph, then we prove that ((Aaosp), (aP,aN,pP,pN)) is a constant AQSP fuzzy soft function in HAQSP (A, V) of G* for all values of e G AAQSP.
Proof. Let Gaosp (A, V) satifies the condition of regular and totally regular AQSP fuzzy soft graph. Then we have the degree of vertices as,
(i) degap(a) = k1, dega^(a) = k2, degpp(a) = k3, degpN(a) = k4 and
(ii) tdegap(a) = /1, tdega^(a) = /2, tdegpp(a) = /3, tdegpN(a) = /4.
In AQSP fuzzy subgraphs HAQSP (A, V) for all values of e G Aaosp, a G V. This implies that, degap(a) + Aaqsp aeP(a) = /1, dega^(a) + Aaqsp aeN(a) = /2, degpp (a) + Aaqsp Pp (a) = /3,
degpN(a) + Aaqsp PN(a) = /4 G Haqsp(A, V), V e G Aaqsp, a G V.
Aaqsp op (a) = ¡1 - ki
Aaqsp of (a) = ¡2 - k2
A AQSP Pp (a) = ¡3 - k3
Aaqsp Pn(a) = ¡4 - k4 G Haqsp (A, V),
V e G Aaqsp, a G V.
Hence, ((Aaqsp), (oP,oN,pP,pN)) is a constant AQSP fuzzy soft function in Haqsp (A, V) of G* for all values of e G Aaqsp.
V1 (0.8,-0.8,0.3,-0.3)
V3 (0.7, -0.7, 0.2, -0.2)
Figure 4: Gaqsp(A, V) - Corresponding to the parameter ei
V, (0.8,-0.8,0.3,-0.3)
V3 (0.9, -0.8, 0.4, -0.4)
Figure 5: Gaqsp(A, V) - Corresponding to the parameter e2
4.5. Example Totally Regular AQSP Fuzzy Soft Graph
Consider, an AQSP fuzzy soft graph, GAQSP(A, V), the vertex set V = (v1, v2, v3, v4) and let the corresponding parameters E = (ei, e2) is shown in the Figure.4 and Figure.5.
Here Gaqsp(A, V) = ((Aaqsp), (oP,oN,pP,pN)), ((Aaqsp), (/, FN, YP, YN)) is described Figure.6 (v1, v2, v3, v4)). Figure. 7 represents the corresponding edges, (v1, v2), (v2, v3), (v3, v4), (v4, v1) , for all values of e G Aaqsp.
Table 5: Tabular representation of AQSP Fuzzy Soft Graph parameter vertex set.
(V, Y) V1_V2_V3_V4_
e1 ( 0.8, - 0.8, ( 0.7, - 0.7, ( 0.7, - 0.7, ( 0.8, - 0.8, 0.3,-0.3) 0.2,-0.2) 0.2,-0.2) 0.3,-0.3) e2 ( 0.8, - 0.8, ( 0.9, - 0.9, ( 0.9, -0.9, ( 0.8, - 0.8, 0.3,- 0.3) 0.4, - 0.4) 0.4, - 0.4 ) 0.3 ,- 0.3)
Table 6: Tabular representation of AQSP Fuzzy Soft Graph parameter edge set.
(V, Y) V1V2_V2V_V3V4_V4 V1
e1 ( 0.6, - 0.6, ( 0.7, - 0.7, ( 0.6, - 0.6, ( 0.6, - 0.6, 0.1,- 0.1) 0.2, - 0.2) 0.1, - 0.1 ) 0.1,- 0.1) e2 ( 0.7, - 0.7, ( 0.6, - 0.6, ( 0.7, - 0.7, ( 0.7, - 0.7, 0.2,- 0.2) 0.1, - 0.1) 0.2, - 0.2 ) 0.2,- 0.2)
4.6. Example of AQSP Fuzzy Soft Graph
Consider, an AQSP fuzzy soft graph, GAQSP(A, V), the vertex set V = (v1, v2, v3, v4) and let the corresponding parameters E = (e1, e2).
Here Gaqsp(A, V) = ((Aaqsp), (oP, oN,pP,pN)), ((Aaqsp), (vp, VN, YP, YN)) is described by Table.5 and Table. 5 such as, (v1, v2), (v2, v3), (v3, v4), (v1, v3), (v1, v4), (v4, v1), (v1, v1) , for all values of e G Aaqsp.
4.7. Remark on Regular AQSP Fuzzy Soft Graph
From Theorem.5.7. we get the result if GAQSP (A, V) is a regular AQSP fuzzy soft graph and
(( Aaqsp ), (oP, oN, pP, pN)) is a constant AQSP fuzzy soft function, then Gaqsp (A, V) is a regular AQSP fuzzy soft graph.
4.8. Remark on Totally Regular AQSP Fuzzy Soft Graph
From Theorem.5.7. similarly we get the result if Gaqsp (A, V) is a totally regular AQSP fuzzy soft graph and ((Aaqsp), (oP,oN,pP,pN)) is a constant AQSP fuzzy soft function, then Gaqsp(A, V) is a totally regular AQSP fuzzy soft graph.
Theorem 2. Let Gaqsp (A, V) = ((Aaqsp ), (oP, oN, pP, pN)), ((Aaqsp ), (vp, VN, YP, YN )),forall values of e G Aaqsp. be an AQSP fuzzy soft graph with the vertex and edge membership and non - membership submerging values. Then we prove that,
(i) LagAtdegGAQSp(A,V) (oP(a) = 2S(Gaqsp(A, V)) + 0(Gaqsp(A, V))
(ii) LagAtdegGAQSp(A,V) (oN(a) = 2S(Gaqsp(A, V)) + 0(Gaqsp(A, V))
(iii) EaGA tdegGAQSp(a,v) (pP(a) = 2S(Gaqsp(A, V)) + 0(Gaqsp(A, V))
(iv) EagAtdegGAQSp(AV) (pN(a) = 2S(Gaqsp(A, V)) + 0(Gaqsp(A, V))
Proof. (i) tdegoAQSp(A,V) (op(a)) = EeG Aaqsp (Ea GV(Vp (a, b) + 0p(a),
EaGV tdegGAQSP(A,V) (op(a)) = EaGV(EeGAAQSP (EaGV(Vp (a, b) + ^^
= EaeAtdegGAQSp(a?(a)) = 2S(Gaosp(A, V)) + 0(Gaosp(A, V)). For non - membership AQSP fuzzy soft graph values are,
(ii) tdegGAQSP(A,V) (aN(a)) = LeeAAQSP (EieV (a, b) + aN (a),
EaeV tdegGAQSP(A,V) (aN(a)) = E aeV (Eee Aaqsp (Ea ev (a, b) + aeN (a),
= EaeAtdegGAQSp(A,V) (aeN(a)) = 2S(Gaosp(A, V)) + 0(Gaosp(A, V). Now, the Submerging membership values are,
(iii) tdegGAQSp(A,V) (pp(a)) = EeeAaqsp (EaeV(7p(a, b) +P (a)),
EaeV tdegGAQSP(A,V) (Pp(a)) = EieV EeeAAQSP (Ea eV (Yp (a, b) + Pp (a)),
= Eae a tdegGAQSP ( a,v) (pp (a)) = 2S(Gaosp ( A, V )) + 0(Gaosp ( A, V )). For the Submerging non - membership values are,
(iv) idegGAQSp( a,v ) (pN (a)) = Eee Aaqsp (EaeV Y (a, b) +eN (a)),
EaeV tdegGAQSP(A,V) (PN(a)) = EaeV EeeAAQSP (EaeV(?N(a, b) + PN(a)), = Eae a tdegGAQSP ( a,v) (pN (a)) = 2S(Gaqsp ( A, V)) + 0(Gaqsp ( A, V )). ■
5. Properties of Regular and Totally Regular AQSP Fuzzy Soft Graph
Theorem 3. The size of the (ki,k2,k3,k4) regular AQSP fuzzy soft graph (GAoSP(A, V) on G* = (V, E) is (i) Pk1, (ii) pk2, (iii) pr and (iv) pr where p = |V| and deg a?(a) = ki,
deg aN (a) = k2, deg p? (a) = k3 and deg pN (a) = k3
Proof.
(i) S(GaosP(A, V) = EeeAAQSP(Ea=b b))
since GAOSP (A, V) is a k1 regular AQSP fuzzy soft graph we get,
dega? (a) = k1, Va e V ,
Now, S (Gaosp ( A, V)) = Eee aaqsp (Ea=b (a, b)) EaeV
deg aeP (a)
V- deg aeP (a) = y- k1
EaGV 2 = EaGV 2 '
(ii) S(Gaqsp(A, V)) = EeeAAQSP(Ea=b PeN(a,b))
since (Gaosp(A, V)) is a (k1,k2,k3,k4) regular AQSP fuzzy soft graph we get, degaN(a) = k2, Va G V ,
Now, S (Gaosp (A, V)) = Eee Aaqsp (Ea=b pN (a, b))
V- deg aeN (a)
EaGV -2-
V- deg aeN (a) = v kg.
EaGV 2 = EaGV 2
(iii) S(Gaqsp(A, V)) = EeGAAQSP(Ea=b Yp(a,b))
since (Gaosp (A, V)) is a k3 regular AQSP fuzzy soft graph we get, degpp (a) = k3, Va G V ,
Now, S(Gaosp (A, V)) = EeGAaqsp (Ea=b Yp(a, b))
E deg pp(a)
EaeV -2-,
L deg pp(a) _ L кз LagV 2 _ LagV 2 '
(iv) S(Gaosp(A, V)) _ EeeAAQSP(Ea_b yn(a, b))
since (Gaosp(A, V)) is a k4 regular AQSP fuzzy soft graph we get,
degpp (a) _ k4, Va G V ,
Now, S(Gaosp (A, V)) _ EeeAaqsp (La_b yN(a, b))
L deg PN (a) LaGV -2-
L deg pN (a) _ L k4 LaGV 2 _ LaGV 2
Hence, The size of the (ki, k2, k3, k4) regular AQSP fuzzy soft graph (Gaosp(A, V) on G* _ (V, E) is p2i, Pr, PJT and ^ where p = |V| ~ ■
Theorem 4. If Gaosp (A, V) _ ((Aaqsp ), (aP, aN, pP, pN)), ((Aaosp ), (цР, UN, YP, YN)) be an regular AQSP fuzzy on G* _ (a*, ц*) is a k-totally regular AQSP fuzzy soft graph. Then, 2S(Gaosp(A, V) + 0(Gaosp(A, V)) _ (aPpk,aNpk, jPpk, jNpk) where, (aPp,aNp,YPp,YNp) _ |V|.
Proof. Since, Gaosp (A, V) is a k-totally regular AQSP fuzzy soft graph,
tdegQAQSP(A,V)aPa _ kb tdegGAQSp(A,V)aNa _ ^ tdegGAQSP(A,V)pPa _ k3 and tdegGAQSP(A,V)pNa _ k2, Va G V.
degGAQSP(A,V)+ EeGA ap (a), degGAQSP(A,V)^a + EeGA (a), degGAQSP(A,V)pPa + EeGA Pp (a) and degGAQSP(A,V)pNa + EeGA PN(а), Va G V.
EaGV degGAQSP(A,V)aPa + EaGV Leg_AAQSP aP(a _ EaGV, EaGV degGAQSP(A,V)°~Na + EaGV LegAAQSP a _ EaGV .
For submerging AQSP fuzzy soft graph values are,
EaGV degGAQSP(A,V)pP(a + EaGV EeGAAqSp pPa _ EaGV, EaGV degGAQSP(A,V)pNa + EaGV LegAAQSP pNa _ EaGV . tdegGAQSP(A,V)aPa _ tdegGAQSP(A,V)aNa _ k2, tdegGAQsp(A,V)pPa _ k3 and tdegGAQSP(A,V)pNa _ k2 Va G V.
Hence, 2S(Ga,v(AQSP) + 0(GA,v(AQSP)) _ (aPpk,aNpk,YPpk,YNpk).
■
Theorem 5. If (Gaosp(A, V)) _ ((Aaqsp), (aP,aN,pP,pN)), ((Aaqsp), (цР, ЦN, YP, YN)) be an AQSP fuzzy soft graph on G* _ (а*, ц*) is a k-regular AQSP fuzzy soft graph.Then (i)O(GAQSP(a, V)) = n(li - ki), (u)O(Gaosp(A, V)) = n(l2 - k2), (iii) 0(Gaosp(A, V))= п(1з - k3) and (iv) 0(Gaosp(A, V)) = n(l4 - k4) where n _ |V|.
Proof. Since (Gaosp(A, V)) is an k-regular AQSP fuzzy soft graph, then we have
degGAQsp(A,V)aPa _ kb degGAQsp(A,V)aNa _ k2, degGAQsp(A,V)pPa _ k3 and
degGAQSP(a,v)pNa _ k4, Va G V. Here, (Gaosp(A, V)) is totally regular AQSP fuzzy soft graph, then we consider,
tdegGAQSP(A,V)aPa _ Ь tdegGAQSP(A,V)aNa _ l2, tdegGAQSP(A,V)pPa _ l3 and tdegGAQSp(Ay)pNa _ l4, Va G V. Now we have,
LaeA tdegGAQsp(A,V)aPa _ aP Pk1,
EaGA tdegGAQSP(A,V)aNa _ aN Pk2
LaGA tdegGAQSP(A,V)pPa _ pP Pk3,
EaGA tdegGAQSP(A,V)aNa _ aN Pk4.
(i) The AQSP fuzzy soft graph membership value is,
LagV l1 _ LagVdegGAQSP(A,V)aPa + O(GAOSP (A, V)) EaGVli _ LagVki + 0(GAQSP(A, V))
nil = nki + 0(Gaosp(A, V)) 0(Gaosp(A, V)) = nki - nil 0(Gaosp(A, V)) = n(ki - li) 0(Gaosp(A, V)) = n(1i - ki).
(ii) The AQSP fuzzy soft graph non-membership value is,
= £aeVdegGAQSP(A,V)^N« + 0(Gaosp(A, V)) EaeV^ = Eae^ + 0(Gaosp(A, V)) nl2 = nk2 + 0(Gaosp(A, V)) 0(Gaosp(A, V)) = nk2 - nl2 0(Gaosp(A, V)) = n(k2 - l2) 0(Gaosp(A, V)) = n(l2 - k2).
(iii) The AQSP fuzzy soft graph submerrging membership value is,
EaeV1? = E«eVdegGAQSP(A,V)pp« + 0(Gaosp(A V)) E«eVl3 = £«eyk3 + 0(Gaosp(A, V)) n/3 = nk3 + 0(Gaosp(A, V)) 0(Gaosp(A, V)) = nk3 - n/3 0(Gaosp(A, V)) = n(k3 - /3) 0(Gaqsp(A, V)) = n(l3 - k3).
(iv) The AQSP fuzzy soft graph submerrging non- -membership value is,
EaeV^ = EaeVdegcAQSP(a,v)Pn« + 0(Gaqsp(A, V)) E«eyl3 = E«eyk3 + 0(Gaqsp(A, V)) n/3 = nk3 + 0(Gaosp(A, V)) 0(Gaosp(A, V)) = nk3 - n/3 0(Gaqsp(A, V)) = n(k3 - /3) 0(Gaosp(A, V)) = n(/3 - k3). Hence the result.
6. Perfectly regular AQSP fuzzy soft graph
Let Gaosp (A, V) be an AQSP fuzzy soft graph on V. Then Gaosp (A, V) is called as perfectly regular AQSP fuzzy soft graph if GAQSP (A, V) = ((AAQSP), (aP, , pP, pN)), ((AAQSP), (pP, pN, yp, yn)) is a regular and totally regular AQSP fuzzy soft graph Ve, G AAQSP.
Table 7: Tabw/ar representation of AQSP Fuzzy Soft Graph parameter vertex set.
(p, Y) Vi_V2_V3_V4_
ei ( 0.8, - 0.8, ( 0.8, - 0.8, ( 0.8, - 0.8, ( 0.8, - 0.8, 0.3,- 0.3) 0.3,- 0.3) 0.3,- 0.3) 0.3,- 0.3) e2 ( 0.9, - 0.9, ( 0.9, - 0.9, ( 0.9, - 0.9, ( 0.9, - 0.9, _0.4,- 0.4) 0.4,- 0.4) 0.4,- 0.4) 0.4,- 0.4)
Table 7. represent the AQSP Fuzzy Soft Graph corresponding parameteric vertex set
Table 8: Tabw/ar representation of AQSP Fuzzy Soft Graph parameter edge set.
(p, y) Vi V2_V2V3__V4 Vi
ei ( 0.7, - 0.7, ( 0.7, - 0.7, ( 0.7, - 0.7, ( 0.7, - 0.7, 0.2, - 0.2) 0.2, - 0.2) 0.2, - 0.2) 0.2, - 0.2) e2 ( 0.8, - 0.7, ( 0.8, - 0.7, ( 0.8, - 0.7, ( 0.8, - 0.7, _0.3,- 0.2) 0.3, - 0.2) 0.3, - 0.2 ) 0.3,- 0.2)
Table 8. explains the AQSP Fuzzy Soft Graph corresponding parameteric edge set
6.1. Example of AQSP Fuzzy Soft Graph
From the Figure.8 we get the result of AQSP fuzzy soft graph with the condition, tdegGAQSP ( A, V) = 2S(Gaosp(A, V) + 0(Gaosp(A, V)) : 5.6 + 3.2 = 8.8
where, 2S(Gaosp(A, V) = 5.6 and 0(Gaosp(A, V)) = 3.2, then, tdegGAQSp(A, V) = 8.8. Using Figure.6 and Figure.7 we can get the same result of AQSP fuzzy soft graph.
U1 ( 0.8, -0.8, 0.3, -0.3)
v3 (0.8, -0.8,0.3, -0.3)
Figure 6: Perfectly regular AQSP fuzzy soft graph Corresponding to the parameter ei
v.j (0.9, - 0.9,0.4, -0.4)
(0.8,-0.7,0.3, -( 0.9, - 0.9, 0.4, -0.4) v.
(0.8, -0.7,0.3, -
V3 ( 0.9, - 0.9, 0.4, -0.4) Figure 7: Perfectly regw/ar AQSP fuzzy soft graph Corresponding to the parameter ei
Theorem 6. For a perfectly regular AQSP fuzzy soft graph Gaosp (A, V) we have ((AAQSP), (aP, aN, pP, pN)) is a constant function.
Proof. From Theorem .4 and Theorem. 5 we prove that
Gaosp(A, V) = ((Aaqsp), (oP,aN,pP,pN)), ((Aaqsp), (pP, pN, YP, YN)) is perfectly regular AQSP fuzzy soft graph. ■
Theorem 7. Let GAQSP (A, V) be an AQSP fuzzy soft graph. Then we prove that GAQSP (A, V) is perfectly regular AQSP fuzzy soft graph if and only if the given conditions are satisfied for edges and vertices with membership values.
(i) Lx=y Pp (x y) = Ez=y Pp (z y)
(ii) E*=y PN (x y) = Ez=y PN (z y)
(iii) E*=y yp (x y) = Ez=y yp (z, y)
(iv) Ex=y YeN(x, y) = Ez=y YN (z, y) Vx, y G V, G Aaosp.
(v) of (x) = of (z), (vi) o-N(x) = aN(z)
(vii) pP(x) = pP(z), (viii) pN(x) = pN(z), Vx,y G V, G Aaqsp.
Proof. Consider, Gaosp (A, V) is perfectly regular AQSP fuzzy soft graph. By definition Gaosp (A, V) is regular AQSP fuzzy soft graph, hence it trivially satifies (i), (ii), (iii) and (iv).
Therfore we have the following,
degcAQSP(A,v) °p (x) = deScAQSP (a,v) °p (z), àegGAQSP(A,v) °N (x) = degGAQSP(A,v) °N (z), degGAQSp(A,v) PP (x) = degGAQSP (a,v) PP (z),
degGAQSP(A,v)PN(x) = degGAQSP(a,v)PN(z), Vx,Z G v, ei G aaosp-Thus implies the results by proposition 8.2.in the following,
Ex=y ¥p(x y) = Ez=y Fp (z y)
Lx=y FN y) = Lz=y FN (z, y) Lx=y Ye (x y) = Lz=y Ye (z, y)
Ex=y Yn (x, y) = Ez=y YN (z, y) Vx, y G v, ei G Aaosp. by Theorem.6, (v), (vi), (vii) and (viii) also holds.
Conversely, suppose that GAQSP (A, v) is an AQSP fuzzy soft graph such that it satisfies the conditions from (i), (ii), (iii) and (iv).
deggaqsp(A,v) 0 (x) = degGAQsP(A,v) °P (z) = r1, deg Gaqsp(A,v) VN (x) = deg Gaqsp(A,v ) VN (z), r2 deggaqsp(a,v)pP (x) = degGAQsP(A,v)pP (z), r3
degGAQSP(A,v)PN (x) = degGAQSP(A,v)PN (z) = ^ Vx, z G V, ei G AAQSP.
This implies that GAQSP( A, v) is a regular AQSP fuzzy soft graph. From, (v), (vi), (vii) and (viii) we get the result,
(v) of (x) = aP(z) = ki,
(vi) oN(x) = of (z) = k2
(vii) pp (x) = pp (z) = k3,
(viii) PN(x) = PN(z) = k4, Vx, z G v, ei G Aaqsp .
Thus, ((AAQSP), (oP,oN,pP,pN)) is a constant AQSP fuzzy soft function.
tdegGAQsP(A,v) °P (z) = degGAQsP(A,v) °P (z) + °P (z) = r1 + kV tdegGAQsP(A,v)°P (w) = degGAQsp(A,v)°P (w) + °P (w) = r1 + k1, tdegGAQSP (A,v) °N (z) = degGAQSP (A,v) °N (z) + °N (z) = r2 + k2, tdegGAQSP (A,v) °N (w) = degGAQSp(A,v) °N (w) + °N (w) = r2 + k2, tdegGAQSP (A,v) PP (z) = degGAQSP (A,v) PP (z) + PP (z) = r3 + k3, tdegGAQsP(A,v)PP (w) = degGAQsp(A,v)PP (w) + PP (w) = r3 + ^ tdegGAQsP (A,v) PN (z) = degGAQsp (A,v) PN (z) + PN (z) = r4 + ^
tdegGAQSP(A,v)PN(w) = degGAQSP(A,v)PN(w) + PN(w) = r4 + k4, Vx,z G v ei G AAQSP. The toally regular AQSP fuzzy soft graph is,
tdegGAQsP (A,v)oP (z) = tdegGAQsP (A,v)oP (w) = k1,
tdegGAQsp (A,v) °N (z) = tdegGAQsp ( A,v) °N (w) = k2,
tdegGAQsP(A,v)pP (z) = tdegGAQsP(A,v)pP (w) = k3,
tdegGAQSP(A,v)PN(z) = tdegGAQSP(A,v)PN(w). = k4 Vx, z G v ei G AAQSP. Hence Gaqsp (A, v) is toally regular AQSP fuzzy soft graph. This implies that Gaqsp (A, v) = ((Aaqsp ), (oP, oN, pP, pN )), ((Aaqsp ), (fp, FN, YP, YN )) is perfectly regular AQSP fuzzy soft graph and ((AAQSP ), (oP, oN, pP, pN)) is a constant function. therefore,
tdegGAQSP(A,v)PP (z) = tdegGAQSP(A,v)PP (w) = k1, k2, k3, and k4, Vx z G v ei G AAQSP.
7. Application of AQSP Fuzzy Soft Graph
AQSP fuzzy soft graph can be used in the governing of women safety police network (WSPN) of a city or a district or any Non safety area region. The WSPN can be utilized using AQSP fuzzy soft graph, where the police vehicle depots are the vertices (v1, v2, v3, ...vn) and the route connecting two police vehicles are considered as corresponding edges.
For women safety police inspectors are positioned and the objective of the Governing problem is to find the minimum number of women inspectors required who will inspect the police vehicle for a particular time and particular bus stop or any region. The following description of AQSP fuzzy soft graph will help to find the solution of Patrolling of Police vehicle Network.
7.1. Method of AQSP Fuzzy Soft Graph Women Safety Police Vehicle Network
1. Let V = (v1, v2, v3...vn) be the vertices of AQSP fuzzy soft graph police vehicle depots in a particular women safety vehicle network corresponding to the women Institutions, Companies, Colleges and Working places especially in bus stops.
2. We consider the edges as women working regions E = (v1v2, v1 v3, v2v4, ...vmvn). The vertices membership and non-membership values of the police vehicle V, is determine as
V, G Aaqsp for/ = 1,2, ...n.
3. Now, define a term safety of women work is satisfied, which is the minimum number of
women saved from particular people who distubs them while they stay or travel or work in different places.It is denoted as S vertices.The vehicle route is denoted by edges R = V/Vj in Alternate quadra submerging polar fuzzy soft graph.
4. Find the membership values of the women safety vehicle route v,vj between the range [-1,1] using AQSP fuzzy graph soft graph with the given conditions if
(i) S > R,for AQSP membership values (ii) S < R,for AQSP non-membership values.
5. (a) pP(x, y) < oP(x) A of (y), (b) pN(x, y) > oN(x) V of (y),
(c) 7P(x,y) < pp(x) A pP(y), (d) yN(x,y) > pN(x) V pN(y), for all e G Aaqsp and for all values of x, y = 1,2,3,..., n.
6. Let the capacity of five women police vehicle depots as vertices v1 = 4, v2 = 3, v1 = 5, v4 = 4,, number of women exist in the spot facing dangerous situation denoted as edges, v1, v2 = 55 , v2, v3 = 95, v3, v4 = 100 , v4, v1 =92 are tabulated below.
7. The score values are measured by the AQSP score formula which gives the result of low and high self-esteem influential person, n (iP E — i1/ E )
Table 9: Tafrw/ar represeniai/on of AQSP Fuzzy Soft Graph parameter vertex set.
(o, p) v1 V2 v3 v4
e1 ( 0.6, - 0.8, ( 0.7, - 0.7, ( 0.8, - 0.9, ( 0.6, - 0.9,
0.1,- 0.3) 0.2, -0.2) 0.3, - 0.4 ) 0.1,- 0.4)
e2 ( 0.7, - 0.9, ( 0.8, - 0.6, ( 0.9, - 0.8, ( 0.8, - 0.8,
0.2,- 0.4) 0.3, -0.1) 0.4, - 0.3 ) 0.3,- 0.3)
Score 0.500 0.900 0.925 0.900
The score values of the women needed safety in different spots are given with membership and non membership values of the edges are v1, v2 = 0.550 , v2, v3 = 0.950, v3, v4 = 1.000 , v4, v1 = 0.925. The police vehicle v3 = 3 is the important vehicle to be in the spot v3, v4 = 1.000 where women in that area need safety. The bar diagram given below shows the result.
Table 10: Tabular representation of AQSP Fuzzy Soft Graph parameter edge set.
V1V2 V2V3 V3V4 V4V1
e1 ( 0.7, - 0.7, ( 0.7, - 0.7, (1.0, - 0.7, ( 0.7, - 0.7,
0.2, - 0.2) 0.2, - 0.2) 0.5, - 0.2) 0.2, - 0.2)
e2 ( 0.8, - 0.7, ( 0.8, - 0.7, ( 0.9, -1.0, ( 0.8, - 0.7,
0.3,- 0.2) 0.3, - 0.2) 0.4, - 0.5 ) 0.3,- 0.2)
Score 0.550 0.950 1.000 0.925
8. Conclusion
The Alternate Quadra Submerging Polar (AQSP) fuzzy graph is introduced with the basic perception of Fuzzy soft graphs. In this article, we introduce the new module AQSP fuzzy soft graphs with suitable definitions, theorems, examples, and properties. The membership and non-membership values of AQSP fuzzy soft graph is introduced with submerging level of confidence [-0.5,0.5]. The introduction of this new module AQSP fuzzy soft graph is an indispensable concept that can be rather developed into interdisciplinary subjects. The main purpose of this new graph is to find the reliable corresponding parametric membership values. The regular, totally regular, and perfectly regular AQSP fuzzy soft graph combinatoric concepts and properties can be applied in Combinatoric subjects, Applied Mathematics, Statistics, Probability, Artificial intelligence, Approximate reasoning, Teaching learning projects and Mathematical psychology. Different types of AQSP fuzzy soft graphs and the Network method of Governing the women safety vehicle in different spots are presented specifically. Finding the important police vehicle, connected routes and spots are the extent of the AQSP fuzzy soft graph. In future the extension of the AQSP fuzzy soft graph can be developed in Decision making analysis, medical diagnosis, and machine learning. The regularity of AQSP fuzzy soft sets and graphs are applicable in real life situations which are uncertain. The combinatoric membership and non-membership submerging values can be found using corresponding parameters in different fuzzy fields.
Declarations Acknowledgements
The authors do thankful to the editor for giving an opportunity to submit our research article in this esteemed journal. And grateful to the Institution for providing SEED Money and MATLAB software for the research purpose. Conflict of interest
The authors declared that they have no conflict of interest regarding the publication of the research article.
Contributions
The authors worked equally regarding the publication of the research article.
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