Ratio Mathematica Volume 46, 2023 Strong interval – valued Pythagorean fuzzy soft graphs Mohammed Jabarulla Mohamed* Sivasamy Rajamanickam † Abstract A Strong interval – valued Pythagorean fuzzy soft sets (SIVPFSS) an extending the theory of Interval-valued Pythagorean fuzzy soft set (IVPFSS). Then we Propose Strong interval valued Pythagorean fuzzy soft graphs (SIVPFSGs). We also present several different types of operations on Strong interval- valued Pythagorean fuzzy soft graphs and explore of their analysis. Keywords: Strong Interval-valued Pythagorean fuzzy graph; Strong Interval-valued Pythagorean fuzzy soft graph; 2020 AMS subject classifications: 05C72, 06D72, 12D15. 1 *PG and Research Department of Mathematics, Jamal Mohamed College(Autonomous), (Affiliated to Bharathidasan University), Tiruchirappalli, Tamil Nadu, India. m.md.jabarulla@gmail.com. †PG and Research Department of Mathematics, Jamal Mohamed College(Autonomous), (Affiliated to Bharathidasan University), Tiruchirappalli, Tamil Nadu, India; sivasamyr1998@gmail.com. 1Received on September 15, 2022. Accepted on December 15, 2022. Published on March 20, 2023. DOI: 10.23755/rm.v46i0.1067. ISSN: 1592-7415. eISSN: 2282-8214. ©M. Mohammed Jabarulla et al. This paper is published under the CC-BY licence agreement. 127 M. Mohammed Jabarulla and R. Sivasamy 1 Introduction Fuzzy set is a analytical imitation to grips the exciting and insufficient details. consider a differentiating that uncertainty is also independently, FS was contin- ued to intuitionistic fuzzy set (IFS) by Atanassov and Gargov [1989]. If assigned a membership value α and a non membership value β to the conditions, satis- flying this results α + β ≤ 1 and uncertainty elements, γ = 1 − α − β. In decision-making problems, the membership value 0.7 and non membership value 0.4 for some information, then IF fails in this situation because 0.7 + 0.4 > 1, but (0.7)2 + (0.4)2 ≤ 1. To overcome this situation, the notion of Pythagorean fuzzy set (PFS) was satisfying the condition α2 + β2 ≤ 1. A PFS has more potential as compared to IFS is solving decision-making problems. The Pythagorean fuzzy number (PFG) was determinate by Zhang ( see S.Shahzadi and Akram [2020]). Zhang provided the Pythagorean fuzzy weighted averaging operator. The theory of IVFS was introduced by Zadeh [1965] as a perpetuation of fuzzy sets. Because they present more adequate description for uncertainty, interval- valued fuzzy sets more useful than conventional fuzzy sets. Soft set theory was started by Molodstov [1999] for the parameterized point of view for uncertainty modeling and soft computing. The iterpretation of IFSGs was given by Akram [2011]. The explanation of novel intuitionistic fuzzy soft multiple – decision- making methods of grips by Akram. Pythagorean fuzzy soft graphs with appli- cations was proposed by S.Shahzadi and Akram [2020].The SIVPFSG is defined and some results on SIVPFSG are studied. Also explore of their analysis. 2 Preliminaries Definition 2.1. An IVFSG over the set V is given by ordered 4 tuple ξ̃ = (ξ∗,X,Y,A) such that (i) A is of parameters. (ii) (X,A) is an IVFSS over V . (iii) (Y,A) is an IVFSS over E. (iv) (X(e),Y (e)) is an IVFSG for all e ∈ A. That is, α− Y (e) ((pq)) ≤ min(α− X(e) (p),α− X(e) (q)) and α+ Y (e) ((pq)) ≤ min(α+ X(e) (p),α+ X(e) (q))forall pq ∈ E. We denote ξ∗ = (V,E) a crisp graph H(e) = (X(e),Y (e)) an IVFSG and ξ̃ = (ξ∗,X,Y,A) an IVFSG. Definition 2.2. An IVFSG over the set V is defined to be a pair ξ = (X,Y ) where 1) The conditions α̃X : V → D[0, 1] and β̃X : V → D[0, 1] denote the degree of 128 Strong interval – valued Pythagorean fuzzy soft graphs membership and non membership of the element p ∈ V . such that 0 ≤ α̃X (p) + β̃X (p) ≤ 1∀(p,q) ∈ V. 2) The conditions α̃Y : E ⊆ V ×V → D[0, 1] and β̃Y : E ⊆ V ×V → D[0, 1] defined by α−Y L((p,q)) ≤ min(α − XL(p),α − XL(q)) and β − Y L((p,q)) ≥ max(β − XL(p),α − XL(q)), α+Y U ((p,q)) ≤ min(α + XU (p),α + XU (q)) and β + Y U ((p,q)) ≥ max(β + XU (p),α + XU (q)), such that 0 ≤ α2Y U (p,q) + β 2 Y U (p,q) ≤ 1∀(p,q) ∈ E. We the notation pq for (p,q) an element of E. Definition 2.3. An IVPFSG over the set V is given by ξ̃ = (ξ∗,X,Y,A) such that 1) The conditions α̃X : V → D[0, 1] and β̃X : V → D[0, 1] standered for the degree of membership and non membership of the element p ∈ V . such that 0 ≤ α̃X (p,q) + β̃X (p,q) ≤ 1∀(p,q) ∈ V. 2)(i) A is set of parameters (ii) (X,A) is an IVPFSS over V . (iii) (Y,A) is an IVPFSS over E. (iv) (X(e),Y (e)) is an IVPFSG for all e ∈ A. The conditions α̃Y : E ⊆ V ×V → D[0, 1] and β̃Y : E ⊆ V ×V → D[0, 1] defined by α+Y U ((p,q)) ≤ min(α + XU (p),β + XU (q)) and β + Y U ((p,q)) ≥ max(β + XU (p),β + XU (q)), α−Y L((p,q)) ≤ min(α − XL(p),β − XL(q)) and β − Y L((p,q)) ≥ max(β − XL(p),β − XL(q)), such that 0 ≤ α2Y U (p,q) + β 2 Y U (p,q) ≤ 1∀(p,q) ∈ E. 3 Strong intervel-valued Pythagorean fuzzy Graphs Definition 3.1. An SIVPFSG over the set V is given by ξ̃ = (ξ∗,X,Y,A) such that 1) The conditions α̃X : V → D[0, 1] and β̃X : V → D[0, 1] denote the degree of membership and non membership of the element x ∈ V . such that 0 ≤ α̃X (p,q) + β̃X (p,q) ≤ 1∀(p,q) ∈ V. 2)(i) A is set of parameters (ii) (X,A) is an SIVPFSS over V . (iii) (Y,A) is an SIVPFSS over E. (iv) (X(e),Y (e)) is an SIVPFSG for all e ∈ A. The conditions α̃Y : E ⊆ V ×V → D[0, 1] and β̃Y : E ⊆ V ×V → D[0, 1] defined by 129 M. Mohammed Jabarulla and R. Sivasamy α+Y U ((p,q)) = min(α + XU (p),β + XU (q)) and β + Y U ((p,q)) = max(β + XU (p),β + XU (q)), α−Y L((p,q)) = min(α − XL(p),α − XL(q)) and β − Y L((p,q)) = max(α − XL(p),β − XL(q)), such that 0 ≤ α2Y U (p,q) + β 2 Y U (p,q) ≤ 1∀(p,q) ∈ E. Example 3.1. If ξ∗ = (X,Y ) is a simple graph with X = {a,b,c,d} and Y = {ab,bc,cd,ad}. Let A = {e1,e2} be a parameter set and (X,A) be an SIVPFSS V determine X1(e) = { 〈a, [0.3, 0.4][0.2, 0.7]〉,〈b, [0.2, 0.5][0.3, 0.7]〉,〈c, [0.1, 0.6][0.2, 0.5]〉, and〈D[0.2, 0.7][0.3, 0.5]〉 } X2(e) = { 〈a, [0.2, 0.7][0.3, 0.5]〉,〈b[0.1, 0.6][0.2, 0.5]〉,〈c, [0.3, 0.4][0.2, 0.7]〉 } Take (Y,A) be an SIVPFSS E determine Y1(e) = { 〈ab[0.2, 0.5][0.3, 0.7]〉,〈bc[0.1, 0.6][0.3, 0.7]〉,〈ad[0.2, 0.7][0.3, 0.7]〉, and〈cd[0.1, 0.6][0.3, 0.5]〉 } Y2(e) = { 〈ab[0.1, 0.5][0.4, 0.6]〉,〈bc[0.1, 04][0.4, 0.8]〉,〈ac[0.1, 0.3][0.4, 0.8]〉 } It is clearly seen that H(e1) = (X(e1),Y (e1)) and H(e2) = (X(e2),Y (e2)) are SIVPFSGs comparable to the parameters e1 and e2 accordingly, by Figure 1. Hence ξ̃ = (ξ∗,X,Y,A) SIVPFSGs. Figure 1: SIVPFSGs Ǧ. Definition 3.2. If ξ̃1 = (ξ∗1,X1,Y1,A) and ξ̃2 = (ξ ∗ 2,X2,Y2,B) be double SIVPF- SGs of ξ∗1 = (X1,Y1) and ξ ∗ 2 = (X2,Y2) accordingly. The cross product of ξ̃1 and 130 Strong interval – valued Pythagorean fuzzy soft graphs ξ̃2 is denoted by ξ̃1 × ξ̃2 = (X1 ×X2,Y1 ×Y2) and is defined by 1) (αX1L ×αX2L)(p1,p2) = min(αX1L(p1),βX2L(p2)), (αX1U ×αX2U )(p1,p2) = min(αX1U (p1),αX2U (p2)), (βX1L ×βX2L)(p1,p2) = min(βX1L(p1),βX2L(p2)), (βX1U ×βX2U )(p1,p2) = max(βX1U (p1),βX2U (p2)),∀p1 ∈ V1,p2 ∈ V2. 2) (αY1L ×αY2L)(p,p2)(p,q2) = min(αY1L(p),αY2L(p2,q2)), (αY1U ×αY2U )(p,p2)(p,p2) = min(αY1U (p),αY2U (p2,q2)), (βY1L ×βY2L)(p,p2)(p,q2) = max(βY1L(p),βY2L(p2,q2)), (βY1U ×βY2U )(p,p2)(p,q2) = max(βY1U (p),βY2U (p2,q2)),∀p ∈ V1,p2q2 ∈ E2. 3) (αY1L ×αY2L)(p1,r)(q1,r) = min(αY1L(p1q1),αY2L(r)), (αY1U ×αY2U )(p1,r)(q1,r) = min(αY1U (p1q1),αY2U (r)), (βY1L ×βY2L)(p1,r)(q1,r) = max(βY1L(p1q1),βY2L(r)), (βY1U ×βY2U )(p1,r)(q1,r) = max(βY1U (p1q1),βY2U (r)),∀r ∈ V2,p1q1 ∈ E1. Example 3.2. Let Consider a graph ξ∗1 = (X1,Y1) and ξ ∗ 2 = (X2,Y2) be two graphs such that X1 = {a1,b1,c1,d1},Y1 = {a1b1,c1d1} and X2 = {a2,b2,c2,d2}, Y2 = {a2b2,c2d2}. Let A = e1 be a set of parameters and let (X1,A) and (Y1,A) be two SIVPFSSs over X1 and Y1 accordingly, defined by Figure 2: SIVPFSGs ξ̃1 and ξ̃2. X1(e) = { 〈a1[0.2, 0.5][0.4, 0.8]〉,〈b1[0.1, 0.4][0.4, 0.5]〉,〈c1[0.2, 0.6][0.3, 0.5]〉, and〈d1[0.1, 0.5][0.4, 0.6]〉 } Y1(e) = { 〈a1b1[0.1, 0.4][0.4, 0.8]〉,〈c1d1[0.1, 0.5][0.4, 0.6]〉 } 131 M. Mohammed Jabarulla and R. Sivasamy Figure 3: Cross product of ξ̃1 and ξ̃2. Take B = e2 be a set of parameters and let (X2,B) and (Y2,B) be two SIVPFSSs over X2 and Y2 accordingly, Find out X2(e) = { 〈a2[0.1, 0.4][0.2, 0.6]〉,〈b2[0.3, 0.3][0.7, 0.3]〉,〈c2[0.3, 0.7][0.4, 0.5]〉, and〈d2[0.3, 0.4][0.1, 0.6]〉 } Y2(e) = { 〈a2b2[0.1, 0.4][0.7, 0.6]〉,〈c2d2[0.3, 0.7][0.4, 0.6]〉 } Clearly H(e1) = (X(e1),Y (e1)) and H(e2) = (X(e2)),Y (e2)) are SIVPFSGs. Hence ξ̃1 = (ξ∗1,X1,Y1,A) and ξ̃2 = (ξ ∗ 2,X2,Y2,B) are SIVPFSGs ξ ∗ 1 and ξ ∗ 2 , accordingly, as shown in the Figure 2. Definition 3.3. If ξ̃1 = (ξ∗1,X1,Y1,A) and G̃2 = (ξ ∗ 2,X2,Y2,B) be two SIVPF- SGs of ξ∗1 = (X1,Y1) and ξ ∗ 2 = (X2,Y2) accordingly. The composition of ξ̃1 and ξ̃2 is standed by ξ̃1 ◦ ξ̃2 = (X1 ◦X2,Y1 ◦Y2) and is defined by 1) (αX1L ◦αX2L)(p1,p2) = min(αX1L(p1),βX2L(p2)), (αX1U ◦αX2U )(p1,p2) = min(αX1U (p1),αX2U (p2)), (βX1L ◦βX2L)(p1,p2) = min(βX1L(p1),βX2L(p2)), (βX1U ◦βX2U )(p1,p2) = max(βX1U (p1),βX2U (p2)),∀p1 ∈ V1,p2 ∈ V2. 2) (αY1L ◦αY2L)(p,p2) = min(αY1L(p),αY2L(p2,q2)), (αY1U ◦αY2U )(p,p2) = min(αY1U (p),αY2U (p2,q2)), (βY1L ◦βY2L)(p,q2) = max(βY1L(p),βY2L(p2,q2)), (βY1U ◦βY2U )(p,q2) = max(βY1U (p),βY2U (p2,q2)),∀p1 ∈ V1,p2q2 ∈ E2. 132 Strong interval – valued Pythagorean fuzzy soft graphs 3) (αY1L ◦αY2L)(p1,r)(q1,r) = min(αY1L(p1q1),αY2L(r)), (αY1U ◦αY2U )(p1,r)(q1,r) = min(αY1U (p1q1),αY2U (r)), (βY1L ◦βY2L)(p1,r)(q1,r) = max(βY1L(p1q1),βY2L(r)), (βY1U ◦βY2U )(p1,r)(q1,r) = max(βY1U (p1q1),βY2U (r)),∀r ∈ V2,p1q1 ∈ E1. 4) (αY1L ◦αY2L)(p1,p2)(q1,q2) = min(αX2L(p2),αX2L,αX1L(p1,q1)), (αY1U ◦αY2U )(p1,r)(q1,r) = min(αX2U (p2),αX2U (q2),αY1U (p1,q1)), (βY1L ◦βY2L)(p1,r)(q1,r) = max(βX2L(p2),βX2L(q2),βY1L(p1,q1)), (βY1U ◦βY2U )(p1,r)(q1,r) = max(βX2U (p2),βX2U (q2),βY1U )(p1,p2)(q1,q2)), ∀(p1,p2)(q1,q2) ∈ E◦ −E. where E◦ = E ∪{(p1,p2)(q1,q2)|p1q1 ∈ E1,p2 6= q2}. Definition 3.4. Let ξ̃1 = (ξ∗1,X1,Y1,A) and ξ̃2 = (ξ ∗ 2,X2,Y2,B) be two SIVPF- SGs of ξ∗1 = (X1,Y1) and ξ ∗ 2 = (X2,Y2) accordingly. If ξ̃1 and ξ̃2 is standed by ξ̃1 ∪ ξ̃2 = (G∗,X,Y,A∪B) where (X1 ∪X2,Y1 ∪Y2) and is replace 1) (i) (αX1L ∪αX2L)(p) = max(αX1L)(p),αX2L)(p))ifp ∈ V1 ∩V2 (αX1U ∪αX2U )(p) = max(αX1U )(p),αX2U )(p))ifp ∈ V1 ∩V2 (ii) (βX1L ∪βX2L)(p) = max(βX1U )(p),βX2L)(p))ifp ∈ V1 ∩V2 (βX1U ∪βX2U )(p) = max(βX1U )(p),βX2U )(p))ifp ∈ V1 ∩V2 2) (i) (αY1L ∪αY2L)(p,q) = max(αX1L(p,q),αX2L(p,q))if pq ∈ E1 ∩E2 (αY1U ∪αY2U )(p,q) = max(αX1U (p,q),αX2U (p,q))if pq ∈ E1 ∩E2 (ii) (βY1L ∪βY2L)(p,q) = max(βX1L(p),βX2L(q))if pq ∈ E1 ∩E2 (βY1U ∪βY2U )(p,q) = max(βX1U (p),βY2U (q))if pq ∈ E1 ∩E2 Definition 3.5. Let G̃1 = (ξ∗1,X1,Y1,A) and ξ̃2 = (ξ ∗ 2,X2,Y2,B) be two SIVPF- SGs of ξ∗1 = (X1,Y1) and ξ ∗ 2 = (X2,Y2) accordingly. If ξ̃1 and ξ̃2 is standed by ξ̃1 + ξ̃2 = (ξ ∗ 1,X1,Y1,A + B). Where ξ ∗ = (X1 + X2,Y1 + Y2) and is defined by 1) (αX1L + αX2L)(p) = (αX1L ∪αX2L))(p) (αX1U + αX2U )(p) = (αX1U ∪αX2U )(p)if p ∈ V1 ∪V2 (βX1L + βX2L)(p) = (βX1L ∪βX2L)(p) (βX1U + βX2U )(p) = (βX1U ∪βX2U )(p)if p ∈ V1 ∪V2 2)(αY1L + αY2L)(p,q) = (αY1L ∪αY2L)(p,q) (αY1U + αY2U )(p,q) = (αY1U ∪αY2U )(p,q)if p ∈ E1 ∩E2 (βY1L + βY2L)(p,q) = (βY1L ∪βY2L)(p,q) (βY1U + βY2U )(p,q) = (βY1U ∪βY2U )(p,q)if (p,q) ∈ E1 ∩E2. 3)(αY1L + αY2L)(p,q) = min(αX1L(p),αX2L(q)) (αY1U + αY2U )(p,q) = min(αX1U (p),αX2U (q)) (βY L + βY2L)(p,q) = max(βX1L(p),βX2L(q)) (βY1U + βY2U )(p,q) = max(βX1U (p),βX2U (q))if pq ∈ E Where E is the set of all edges joining the vertices of V1 and V2. Theorem 3.1. If ξ̃1 and ξ̃2 are SIVPFSGs, then so is ξ̃1 × ξ̃2. 133 M. Mohammed Jabarulla and R. Sivasamy proof Let ξ̃1 = (ξ∗1,X1,Y1,A) and ξ̃2 = (ξ ∗ 1,X1,Y1,B) be two SIVPFSGs of simple graphs ξ∗1 = (X1,Y1) and ξ ∗ 2 = (X2,Y2) accordingly. for all e1 ∈ A and e2 ∈ B, there are some results. Let ξ1 and ξ2 be SIVPFSGs Let E = {(p,p2)(p,q2)/p ∈ V1,p2q2 ∈ E2}∪{(p1,r)(q1,r)/r ∈ V2,p1q1 ∈ E1}. Consider (p,p2)(p,q2) ∈ E, we have (αY1L ×αY2L)(p,p2)(p,q2) = min(αX1L(p),αY2L(p2q2)) =min(αX1L(p),αX2L(p2).αX2L(q2)) =min(min(αX1L(p),αX2L(p2))min(αX1L(p),αX2L(q2))) (αY1L ×αY2L)(p,p2)(p,q2) = min((αX1L ×αX2L)(p,p2), (αX1L ×αX2L)(p,q2)) Similarly, (αY1U ×αY2U )(p,p2)(p,q2) = min((αX1U ×αX2U )(p,p2), (αY1U ×αY2U )(p,q2)) Now, (βY1L ×βY2L)(p,p2)(p,q2) = max((βX1L ×βX2L)(p,p2), (βX1L ×βX2L)(p,q2)) Similarly, (βY1U ×βY2U )(p,p2)(p,q2) = max((βX1U ×βX2U )(p,p2), (βX1U ×βX2U )(p,q2)) Consider, (p1,r)(q1,r) ∈ E, we have (αY1L ×αY2L)(p1,r)(q1,r) = min(αY1L(p1q1), (αX2L(r)) =min(αX1L(p1),αX2L(q1).αX2L(r)) =min(min(αX1L(p1),αX2L(r))min(αX1L(y1),αX2L(r))) (αY1L ×αY2L)(p1,r)(q1,r) = min((αX1L ×αX2L)(p1,r), (αX1L ×αX2L)(q1,r)) Similarly, (αY1U ×αY2U )(p1,r)(q1,r) = min((αX1U ×αX2U )(p1,r), (αX1U ×αX2U )(q1,r)) Now, (βY1L ×βY1U )(p1,r)(q1,r) = max((βX1L ×βX2L)(p1,r), (βX1L ×βX2L)(q1,r)) Similarly, (βY1U ×βY2U )(p1,r)(q1,r) = max((βX1U ×βX2U )(p1,r), (βX1U ×βX2U )(q1,r)) Hence ξ1 × ξ2 is an SIVPFSGs. 134 Strong interval – valued Pythagorean fuzzy soft graphs Theorem 3.2. If ξ̃1[ξ̃2] be SIVPFSGs ξ̃1 and ξ̃2 of ξ∗1 and ξ ∗ 2 is an SIVPFSGs. Proof Take (p,p2)(p,q2) ∈ E, we get (αY1L ◦αY2L)((p,p2)(p,q2)) = min((αX1L(p),αY2L)(p2q2) =min(αX1L(p),αX2L(p2),αX2L(q2)) =min(min(αX1L(p),αX2L(p2)),min(αX1L(p),αX2L(q2))) (αY1L ◦αY2L)((p,p2)(p,q2) = min(αX1L ◦αX2L)(p,p2), (αX1L ◦αX2L)(p,q2)). Similarly, (αY1U ◦αY2U )((p,p2)(p,q2) = min(αX1U ◦αX2U )(p,p2), (αX1U ◦αX2U )(p,q2)) Consider (p1,r)(q1,r) ∈ E, (αY1L ◦αY2L)((p1,r)(q1,r)) = min(αY1L(p1,q1),αX2L(r)) =min(αX1L(p1),αX1L(q1),αX2L(r)) =min(min(αX1L(p1),αX2L(r)),min(αX1L(q1),αX2L(r))) (αY1L ◦αY2L)((p1,r)(q1,r)) = min(αX1L ◦αX2L)(p1,r), (αX1L ◦αX2L)((q1,r)) Similarly, (αX1U ◦αX2U )((p1,r)(q1,r) = min(αX1U ◦αX2U )(p1,r), (αX1UαX2U )((q1,r)) Consider (p1,p2)(q1,q2) ∈ E, (αY1L ◦αY2L)((p1,p2)(q1,q2)) = min(αX2L(p2),αX2L(q2),αY1L(p1q1)) =min(αX2L(p2),αX2L(q2)),min(αX1L(p1),αX1L(q1)) =min(min(αX1L(p1),αX2L(p2)),min(αX1L(q1),αX2L(q2))) (αY1L ◦αY2L)((p1,p2)(q1,q2)) = min(αX1L ◦αX2L)(p1,p2), (αX1L ◦αX2L) ((q1,q2)) Hence ξ̃1[ξ̃2] be SIVPFSG . Theorem 3.3. If ξ̃1 ∪ ξ̃2 be SIVPFSGs ξ̃1 and ξ̃2 of ξ∗1 and ξ∗2 is an SIVPFSGs. Proof Take ξ̃1 and ξ̃2 be the SIVPFSGs of ξ̃1 and ξ̃2 accordingly. Since all conditions for X1∪X2 are obviously satisfied. It is enough to verify the conditions for Y1 ∪Y2, Consider (p,q) ∈ E1 ∪E2. Then (αY1L ∪αY2L)(p,q) = max(αY1L(p,q),αY2L(p,q)) =max(min(αX1L(p),αX1L(q)), (min(αX2L)(p),αX2L(q)) =min(max(αX1L(p),αX2L(p)), (max(αX1L(p),αX2L(q)))) =min((αY1L ∪αY2L)(p), (αY1L ∪αY2L)(q)) (αY1L ∪αY2L)(p,q) = min((αY1L ∪αY2L)(p), (αY1L ∪αY2L)(q)). 135 M. Mohammed Jabarulla and R. Sivasamy Similarly, (αY1U ∪αY2U )(p,q) = min((αY1U ∪αY2U )(p), (αY1U ∪αY2U )(q)) If (x,y) ∈ E1 and (x,y) /∈ E2, (αY1L ∪αY2L)(p,q) =min((αY1L ∪αY2L)(p), (αY1L ∪αY2L)(q)) (αY1U ∪αY2U )(p,q) =min((αY1U ∪αY2U )(p), (αY1U ∪αY2U )(q)). If (p,q) ∈ E2 and (p,q) ∈ E1, (αY1L ∪αY2L)(p,q) =min((αY1L ∪αY2L)(p), (αY1L ∪αY2L)(q)) (αY1U ∪αY2U )(p,q) =min((αY1U ∪αY2U )(p), (αY1U ∪αY2U )(q)). Theorem 3.4. If ξ̃1 + ξ̃2 be SIVPFSGs ξ̃1 and ξ̃2 of ξ∗1 and ξ ∗ 2 is an SIVPFSGs. Proof Take ξ̃1 + ξ̃2 be the SIVPFSGs of ξ∗1 and ξ ∗ 2 accordingly. , it is enough to find that ξ̃1 + ξ̃2 = (X1 + X2,Y1 + Y2) is an SIVPFSGs. Then Let (p,q) ∈ E (αY1L + αY2L)(p,q) =min(αX1L(p),αX2L(q)) =min((αX1L ∪αX2L)(p), ((αX1L ∪αX2L)(q))) (αY1L + αY2L)(p,q) =min((αX1L + αX2L)(p), ((αX1L + αX2L)(q))). Similarly, (αY1U + αY2U )(p,q) = min((αX1U + αX2U )(p), ((αX1U + αX2U )(q))). 4 Conclusions Graph theory is a very helpful mathematical tool for tackling challenging is- sues in a variety of disciplines. The IVPFSs model is appropriate for modeling issues involving uncertainty and inconsistent data when human understanding and evaluation are required. In contrast to IVFS models, IVIFS models, and, IVPFS models provide systems with sensitivity, flexibility, and conformance. SIVPFSGs are a novel idea that is introduced in this work. We also defined for the Cartesian product as well as some information about its composition on SIVPFSGs. We plan to use this data to create some algorithms and models shortly soon. References M. Akram. 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