Ratio Mathematica Volume 42, 2022 Mixed Picture Fuzzy Graph Myithili Kothandapani * Nandhini Chandrasekar † Abstract A new form of picture fuzzy graph has been identified and extended here as Mixed Picture Fuzzy Graph (MPFG). The picture fuzzy set is formed from the fuzzy set and the intuitionistic fuzzy set. It is help- ful when there are multiple options, such as yes, no, rejection and abstain. MPFG, which is dependent on the picture fuzzy relation, is defined in this paper. The properties of various types of degrees, order and size of MPFG are examined. Also some types of MPFG such as regular, strong, complete and complement of MPFG are introduced and their properties were analysed. As an application part, the con- cept of MPFG has been applied in instagram and the result has been discussed here. Keywords: picture fuzzy graph, MPFG, degree, size & order of MPFG, regular, strong, complete MPFG and complement of MPFG 2020 AMS subject classifications: 05C72 1 *Associate Professor and Head (Department of Mathematics(CA), Vellalar College for Women, Erode, India); myithili@vcw.ac.in. †Research Scholar (Department of Mathematics, Vellalar College for Women, Erode, India); c.nandhini@vcw.ac.in. 1Received on April 29th, 2022. Accepted on June 25th, 2022. Published on June 30th, 2022. doi: 10.23755/rm.v41i0.776. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 225 K. K. Myithili, C. Nandhini 1 Introduction Many decision-making problems in unpredictable environments have been modelled using fuzzy graphs. A variety of generalisations of fuzzy graphs have really been implemented to deal with the uncertainty of complex real-life circum- stances. Zadeh’s(19) fuzzy set theory played a significant role in decision making in unpredictable environments. Rosenfeld(14), developed the basic conception of fuzzy graph 10 years after Zadeh’s seminal article on fuzzy sets. As compared to the graph, the fuzzy graph seems to be a beneficial tool for modelling those problems because it is more efficient, flexible and compatible with any real-world problem. Mordeson & Nair(8) introduced the idea of a complement fuzzy graph, in which Sunitha & Kumar(17) expanded the concept. The principle of Atanassov’s(2) Intuitionistic Fuzzy Set (IFS) allocates a mem- bership and non-membership degree individually, with the sum of the two degrees not exceeding the value one. Shannon and Atanassov proposed a description for intuitionistic fuzzy relations and intuitionistic fuzzy graphs, as well as a list of properties in (16). Different operations on intuitionistic fuzzy graphs were defined by Parvathi et al.(10; 11). Nagoor Gani and Shajitha Begum(9) has characterised about degree, order and size of intuitionistic fuzzy graphs. The Picture Fuzzy Set (PFS) is a new idea that deals with uncertainties and is a direct continuation of the IFS. It can simulate uncertainty in circumstances in- cluding multiple types of answers: yes, abstention, rejection, and no. It is shown about one of the most fundamental concepts of degree of neutrality goes absent from the IFS principle. Cuong & Kreinovich(6) proposed PFS, a direct exten- sion of fuzzy set and IFS that integrates the principle of positive, negative, and neutral membership degree of an element. Cuong(4) investigated some PFS prop- erties and proposed distance measures between them. Phong and Co-authors(13) investigated some picture fuzzy relation compositions. Then, Cuong and Hai(5) extended some fuzzy logic operators for PFSs, including such conjunctions, com- plements, and disjunctions. Peng & Dai(12) proposed and implemented an algo- rithmic solution for PFS in a decision-making problem. New concepts of PFG with application was published by Cen Zuo et al.,(3). L. T. Koczy et al.(7) ana- lyzed the study of social networks and Wi-Fi networks using the concept of picture fuzzy graphs. Wei Xiao, Arindam Dey, and Le Hoang Son(18) spoke about their research on regular picture fuzzy graphs and how they can be used in communica- tion networks. Sankar Das and Ganesh Ghorai(15) investigated the creation of a road map based on a multigraph using picture fuzzy information. And Abdelkadir Muzey Mohammed(1) explained about mixed graph representation. 226 Mixed Picture Fuzzy Graph 2 Basic Definitions We stepped over some fundamental definitions in this section that are related to our main concept. Definition 2.1. (3) Let G∗pf = (V, E) be a graph. A pair Gpf = (A, B) is called a picture fuzzy graph on G∗ where A = (µA, ηA, νA) is a picture fuzzy set on V and B = (µB, ηB, νB) is a picture fuzzy set on E ⊆ V × V such that for each arc vu ∈ E. µB(v, u) ≤ min(µA(v), µA(u)) ηB(v, u) ≤ min(ηA(v), ηA(u)) νB(v, u) ≥ max(νA(v), νB(u))   (1) denotes the degree of positive membership, neutral membership & membership membership of the edge (v, u) ∈ E. Definition 2.2. (1) A mixed graph Gm = (V, E, A) is a graph consists from set of vertices V , set of undirected edges E & set of directed edges(or arcs) A. 2.1 Notations The following mathematical symbols were used throughout the paper: Gpf –picture fuzzy graph Gm–mixed graph Gmpf –mixed picture fuzzy graph µA(v), ηA(v), νA(v)–positive, neutral & negative membership of a vertex v in Gmpf µB(v, u), ηB(v, u), νB(v, u)–positive, neutral & negative membership of an edge vu in Gmpf µ→ B (v, u), η→ B (v, u), ν→ B (v, u)–positive, neutral & negative membership of an arc vu in Gmpf d(vi)–degree of a vertex vi in Gmpf δ(Gmpf)–minimum degree of a Gmpf ∆(Gmpf)–maximum degree of a Gmpf (Gmpf) c–complement of a Gmpf (Gcmpf) c–complement of complement Gmpf O(Gmpf)–order of a Gmpf S(Gmpf)–size of a Gmpf SP –strength of a path P H ′ –subgraph of Gmpf CDmpf(vi, vj)–circle-distance between vi and vj of Gmpf C(S)–centrality of a squad 227 K. K. Myithili, C. Nandhini 3 Mixed Picture Fuzzy Graph[MPFG] The popularity of social media sites and networks are growing every day. Pos- itive, neutral & negative membership of a vertex can be classified as good, neutral and bad activities in PFG. The situation now is why we should have to switch from PFG to MPFG? MPFG is the combination of both directed and undirected edges. Many real-life situations take the shape of MPFG. Further a real life prob- lem has been identified and resolved using this MPFG. Definition 3.1. Let G∗mpf = (V, E, → E) be a graph. An ordered triple Gmpf = (A, B, → B) is called mixed picture fuzzy graph on G∗mpf , where A = (µA, ηA, νA) is a picture fuzzy set on V, B = (µB, ηB, νB) is a picture fuzzy relation on the undirected edge E ⊆ V × V and → B= (µ→ B , η→ B , ν→ B ) is a picture fuzzy relation on the directed edge → E ⊆ V × V , which satisfies, µB(v, u) ≤ min(µA(v), µA(u)) ηB(v, u) ≤ min(ηA(v), ηA(u)) νB(v, u) ≥ max(νA(v), νA(u))   ∀(v, u) ∈ E & µ→ B (v, u) ≤ min(µA(v), µA(u)) η→ B (v, u) ≤ min(ηA(v), ηA(u)) ν→ B (v, u) ≥ max(νA(v), νA(u))   ∀(v, u) ∈ → E (2) Also → B must not have a symmetric relation. Figure 1: Mixed Picture Fuzzy Graph 228 Mixed Picture Fuzzy Graph Definition 3.2. Consider a graph H ′ = (V ′ , E′, → E ′ ) is Mixed Picture Fuzzy Sub- graph (MPFSG) of MPFG if V ′ ⊆ V, E′ ⊆ E and → E ′ ⊆ → E if, µ ′ A(v) ≤ µA(v), η ′ A(v) ≤ ηA(v), ν ′ A(v) ≥ νA(v), µ ′ B(v, u) ≤ µB(v, u), η ′ B(v, u) ≤ ηB(v, u), ν ′ B(v, u) ≥ νB(v, u), µ ′ → B (v, u) ≤ µ→ B (v, u), η ′ → B (v, u) ≤ η→ B (v, u), ν ′ → B (v, u) ≥ ν→ B (v, u). Theorem 3.1. A MPFG is a expandation of IFG. Proof. The statement becomes trivial by assuming the neutral membership/abstain is equal to zero. Hence MPFG can reduce to IFG. Similarly, the statement “A MPFG is a generalization of PFG” is also true. Theorem 3.2. If V = {v1, v2, ..., vn} is vertex set of MPFG, Gmpf = (V, E, → E ). Then total number of edges denoted by |Empf| in MPFG Gmpf is given by, |Empf| = 1/2 [ ∑ v∈V deg(v) + ∑ v∈V degin(v) ] or |Empf| = 1/2 [ ∑ v∈V deg(v) + ∑ v∈V degout(v) ] Proof. Let Gu = (V, E) be undirected subgraph of Gmpf and Gd = (V, → E) with directed edges which are disjoint MPFSGs of MPFG Gmpf = (V, E, → E) such that Empf = E ∪ → E Handshaking theorem and Elementary counting principle, which states that |E| = 1 2 ∑ deg(v) and | → E| = ∑ v∈V degin(v) = ∑ v∈V degout(v) (3) |Empf| = |E ∪ → E| = | → E| + |E| − |E ∩ → E| (4) since, Gu = (V, E) and Gd = (V, → E) are disjoint MPFSGs, |E ∩ → E|=0 then (4) is reduced to |Empf| = | → E| + |E| (5) substituting (3) in (5), we get |Empf| = 1/2 [ ∑ v∈V deg(v) + ∑ v∈V degin(v) ] or |Empf| = 1/2 [ ∑ v∈V deg(v) + ∑ v∈V degout(v) ] 229 K. K. Myithili, C. Nandhini [∵ ∑ v∈V degin(v)= ∑ v∈V degout(v)] Hence proved Definition 3.3. The degree of a vertex in a MPFG denoted as, d(vi) = (dµ(vi), dη(vi), dν(vi)) where, dµ(vi) = ∑ v ̸=u µB(v, u) + 1 2 [ ∑ v ̸=u µ→ Bin (v, u) + ∑ v ̸=u µ→ Bout (v, u) ] dη(vi) = ∑ v ̸=u ηB(v, u) + 1 2 [ ∑ v ̸=u η→ Bin (v, u) + ∑ v ̸=u η→ Bout (v, u) ] dν(vi) = ∑ v ̸=u νB(v, u) + 1 2 [ ∑ v ̸=u ν→ Bin (v, u) + ∑ v ̸=u ν→ Bout (v, u) ]   (6) From figure 1, we get, d(v1)=(0.45,0.3,0.3), d(v2)=(0.95,0.95,0.9), d(v3)=(0.4,0.45,0.4), d(v4)=(0.5,0.75,1.0), d(v5)=(0.8,0.9,0.85), d(v6)=(0.55,0.45,0.35), d(v7)=(0.45,0.3,0.3), δ(Gmpf)=(0.4,0.3,0.3) and ∆(Gmpf)=(0.95,0.95,1.0) Definition 3.4. Consider Gmpf = (V, E, → E) be a MPFG. The neighbourhood of a vertex is represented as, Nh(v) = (Nhµ(v), Nhη(v), Nhν(v)) where, Nhµ(v) = {u ∈ V/µB(v, u) = min(µA(v), µA(u)), µ→ B (v, u) = min(µA(v), µA(u))} Nhη(v) = {u ∈ V/ηB(v, u) = min(ηA(v), ηA(u)), η→ B (v, u) = min(ηA(v), ηA(u))} Nhν(v) = {u ∈ V/νB(v, u) = max(νA(v), νA(u)), ν→ B (v, u) = max(νA(v), νA(u))}   (7) and Nh[v] = Nh(v) ∪ {v} represents closed neighbourhood of a vertex. Definition 3.5. The neighbourhood degree of a vertex is represented as, dNh(v) = (dNhµ(v), dNhη(v), dNhν (v)) where, dNhµ(v) = ∑ u∈Nh(v) µA(u) dNhη(v) = ∑ u∈Nh(v) ηA(u) dNhν (v) = ∑ u∈Nh(v) νA(u)   (8) 230 Mixed Picture Fuzzy Graph Note: If a vertex is an isolated vertex then Nh(v) = ∅ Definition 3.6. The closed neighbourhood degree of a vertex is denoted as, dNh[v] = (dNhµ[v], dNhη[v], dNhν [v]) where, dNhµ[v] = ∑ u∈Nh(v) µA(u) + µA(v), dNhη[v] = ∑ u∈Nh(v) ηA(u) + ηA(v) dNhν [v] = ∑ u∈Nh(v) νA(u) + νA(v)   (9) Definition 3.7. A path in Gmpf = (A, B, → B) is a distinct vertices sequence v0, v1, v2, ..., vk one of the succeeding responses are satisfied with both directed & undi- rected edges, µB(vi−1, vi), ηB(vi−1, vi) > 0 and νB(vi−1, vi) = 0 µB(vi−1, vi), ηB(vi−1, vi) = 0 and νB(vi−1, vi) > 0 µB(vi−1, vi), ηB(vi−1, vi), νB(vi−1, vi) > 0 µ→ B (vi−1, vi), η→ B (vi−1, vi) > 0 and ν→ B (vi−1, vi) = 0 µ→ B (vi−1, vi), η→ B (vi−1, vi) = 0 and ν→ B (vi−1, vi) > 0 µ→ B (vi−1, vi), η→ B (vi−1, vi), ν→ B (vi−1, vi) > 0 i = 1, 2, ..., k. Where k denotes the length of the path. Definition 3.8. A MPFG Gmpf = (A, B, → B) seems to be connected, if each set of vertices possesses atleast 1 mixed picture fuzzy path connecting them, else it is said to be disconnected. Definition 3.9. If there is a path P = vn, v1, ..., vn for n ≥ 3 then it’s a cycle. Definition 3.10. The complement of a Gmpf = (A, B, → B) is a Gcmpf = (A c, Bc, → Bc) iff it follows, µA c = µA, ηAc = ηA, νAc = νA and µcB(v, u) = min(µA(v), µA(u)) − µB(v, u) ηcB(v, u) = min(ηA(v), ηA(u)) − ηB(v, u) νcB(v, u) = max(νA(v), νA(u)) − νB(v, u) µ→ B c(v, u) = min(µA(v), µA(u)) − µ→ B (v, u) η→ B c(v, u) = min(ηA(v), ηA(u)) − η→ B (v, u) ν→ B c(v, u) = max(νA(v), νA(u)) − ν→ B (v, u)   (10) 231 K. K. Myithili, C. Nandhini Figure 2: Complement of Mixed Picture Fuzzy Graph Theorem 3.3. If Gcmpf be a complement of MPFG, then (G c mpf) c = G Note: A MPFG is self-complementary if (Gcmpf) c = G Definition 3.11. The order of a Gmpf is represented by, O(Gmpf) = (Oµ(Gmpf), Oη(Gmpf), Oν(Gmpf)) where, Oµ(Gmpf) = ∑ u∈V µA(v) Oη(Gmpf) = ∑ u∈V ηA(v) Oν(Gmpf) = ∑ u∈V νA(v)   (11) Here Oµ(Gmpf), Oη(Gmpf) & Oη(Gmpf) are the order of positive, neutral & neg- ative membership degree respectively. Definition 3.12. Let Gmpf = (A, B, → B) is MPFG. The size of a Gmpf is repre- sented by, S(Gmpf) = (Sµ(Gmpf), Sη(Gmpf), Sν(Gmpf)) where, Sµ(Gmpf) = ∑ v,u∈V µB(v, u) + ∑ v,u∈V µ→ B (v, u) Sη(Gmpf) = ∑ v,u∈V ηB(v, u) + ∑ v,u∈V η→ B (v, u) Sν(Gmpf) = ∑ v,u∈V νB(v, u) + ∑ v,u∈V ν→ B (v, u), ∀j ̸= i.   (12) Here Sµ(Gmpf), Sη(Gmpf) and Sν(Gmpf) are the size of positive, neutral & negative membership respectively. 232 Mixed Picture Fuzzy Graph Definition 3.13. For a path P, Sµ = min v,u∈V {µB(v, u)} + min v,u∈V {µ→ B (v, u)} Sη = min v,u∈V {ηB(v, u)} + min v,u∈V {η→ B (v, u)} Sν = max v,u∈V {νB(v, u)} + max v,u∈V {ν→ B (v, u)}   (13) The strength of a path SP = (Sµ, Sη, Sν). Definition 3.14. A Gmpf = (A, B, → B) is said to be strong MPFG if, µB(v, u) = min(µA(v), µA(u)) ηB(v, u) = min(ηA(v), ηA(u)) νB(v, u) = max(νA(v), νA(u)), ∀(v, u) ∈ E & µ→ B (v, u) = min(µA(v), µA(u)) η→ B (v, u) = min(ηA(v), ηA(u)) ν→ B (v, u) = max(νA(v), νA(u)), ∀(u, v) ∈ → E.   (14) Figure 3: Strong Mixed Picture Fuzzy Graph Note:(Gcmpf) c = Gmpf iff G is strong MPFG Definition 3.15. A MPFG Gmpf = (A, B, → B) is said to be complete MPFG if, µB(v, u) = min(µA(v), µA(u)) ηB(v, u) = min(ηA(v), ηA(u)) νB(v, u) = max(νA(v), νA(u)) and µ→ B (v, u) = min(µA(v), µA(u)) η→ B (v, u) = min(ηA(v), ηA(u)) ν→ B (v, u) = max(νA(v), νA(u)), ∀v, u ∈ V   (15) 233 K. K. Myithili, C. Nandhini Figure 4: Complete Mixed Picture Fuzzy Graph Note: Every complete MPFG becomes a strong MPFG. But the contrary, does not have to be true. Theorem 3.4. The order of a complete MPFG is equal to the closed neighbour- hood degree of every vertex (i.e), Oµ(Gmpf) = {dNµ[v]|v ∈ V }, Oη(Gmpf) = {dNη[v]|v ∈ V }, Oν(Gmpf) = {dNν [v]|v ∈ V }. Proof. Consider Gmpf = (V, E, → E) be a complete MPFG. The µ, η and ν-order of Gmpf , is the sum of the positive, neutral and negative membership value of each vertex respectively. We know that, if Gmpf is a complete MPFG, then the closed neighbourhood µ, η and ν-degree of every vertex is the sum of the positive membership, neutral membership & negative membership values of the vertices respectively. There- fore, Oµ(Gmpf) = {dNµ[v]|v ∈ V }, Oη(Gmpf) = {dNη[v]|v ∈ V }, Oν(Gmpf) = {dNν [v]|v ∈ V }. Hence the result. Definition 3.16. A MPFG Gmpf = (A, B, → B) is defined as regular MPFG if, µB(v, u) = min(µA(v), µA(u)) and ∑ u̸=v µB(u, v) = constant, ηB(v, u) = min(ηA(v), ηA(u)) and ∑ u̸=v ηB(u, v) = constant, νB(v, u) = max(νA(v), νA(u)) and ∑ u̸=v νB(u, v) = constant, µ→ B (v, u) = min(µA(v), µA(u)) and ∑ u̸=v µ→ B (u, v) = constant, η→ B (v, u) = min(ηA(v), ηA(u)) and ∑ u̸=v η→ B (u, v) = constant, ν→ B (v, u) = max(νA(v), νA(u)) and ∑ u̸=v ν→ B (u, v) = constant.   (16) 234 Mixed Picture Fuzzy Graph Figure 5: Regular Mixed Picture Fuzzy Graph Theorem 3.5. Every complete MPFG is a regular MPFG. Proof. Consider Gmpf = (V, E, → E) be a MPFG. From the definition of complete MPFG we have, µB(v, u) = min(µA(v), µA(u)), ηB(v, u) = min(ηA(v), ηA(u)), νB(v, u) = max(νA(v), νA(u)) and µ→ B (v, u) = min(µA(v), µA(u)), η→ B (v, u) = min(ηA(v), ηA(u)), ν→ B (v, u) = max(νA(v), νA(u)) ∀v, u ∈ V. Then, the closed neighbourhood µ, η and ν-degree of every vertex is the sum of the positive membership, neutral membership & negative membership values of the vertices and itself respectively. As a result, the closed neighbourhood µ- degree, closed neighbourhood η-degree, & closed neighbourhood ν-degree were the same for all vertices. Therefore, min. closed neighbourhood degree is equal to max. closed neighbourhood degree. Hence Gmpf is a regular MPFG. Definition 3.17. Let Gmpf = (A, B, → B) be a MPFG. If two vertices v & u are linked by a length of a path k in Gmpf is P : v0, v1, v2, ..., vn−1, vn then µB(v, u), ηB(v, u), νB(v, u) and µ→ B (v, u), η→ B (v, u), ν→ B (v, u) are described as fol- lows µB k(v, u) = min{µB(v, v1), µB(v1, v2), ..., µB(vk−1, u)} ηB k(v, u) = min{ηB(v, v1), ηB(v1, v2), ..., ηB(vk−1, u)} νB k(v, u) = max{νB(v, v1), νB(v1, v2), ..., νB(vk−1, u)} µ→ B k(v, u) = min{µ→ B (v, v1), µ→ B (v1, v2), ..., µ→ B (vk−1, u)} η→ B k(v, u) = min{η→ B (v, v1), η→ B (v1, v2), ..., η→ B (vk−1, u)} ν→ B k(v, u) = max{ν→ B (v, v1), ν→ B (v1, v2), ..., ν→ B (vk−1, u)} Let µ∞(v, u), η∞(v, u), ν∞(v, u) is Strength of connectedness between the 235 K. K. Myithili, C. Nandhini two nodes v & u of MPFG. µ∞B (v, u) = sup{µB k(v, u)/k = 1, 2, ...} η∞B (v, u) = sup{ηB k(v, u)/k = 1, 2, ...} ν∞B (v, u) = inf{νB k(v, u)/k = 1, 2, ...} µ∞→ B (v, u) = sup{µ→ B k(v, u)/k = 1, 2, ...} η∞→ B (v, u) = sup{η→ B k(v, u)/k = 1, 2, ...} ν∞→ B (v, u) = inf{ν→ B k(v, u)/k = 1, 2, ...} here inf has been used to determine the minimum membership value and sup is used to determine the maximum membership value. Figure 6: Strength of connectedness Consider a conneted MPFG as shown in the figure 6 The possible paths between v1 to v4 are P1 : v1 − v4 along with the value of membership (0.4, 0.3, 0.2) P2 : v1 − v2 − v4 along with the value of membership (0.4, 0.3, 0.3) P3 : v1 − v3 − v4 along with the value of membership (0.3, 0.2, 0.3) P4 : v1 − v2 − v3 − v4 along with the value of membership (0.3, 0.2, 0.3) P5 : v1 − v2 − v3 − v4 along with the value of membership (0.3, 0.2, 0.3) P6 : v1 − v3 − v2 − v4 along with the value of membership (0.3, 0.2, 0.3) We’ve arrived to this conclusion through routine calculations, µ∞(v1, v4) = sup{0.4, 0.4, 0.3, 0.3, 0.3, 0.3} = 0.4 η∞(v1, v4) = sup{0.3, 0.3, 0.2, 0.2, 0.2, 0.2} = 0.3 ν∞(v1, v4) = inf{0.2, 0.3, 0.3, 0.3, 0.3, 0.3} = 0.2 The strength of connectedness between 2 vertices v1 & v4 of a MPFG is (0.4, 0.3, 0.2) 236 Mixed Picture Fuzzy Graph Definition 3.18. Consider Gmpf = (V, E, → E) be a MPFG & v, u be any two dis- tinct vertices. In Gmpf , eliminating an edge or arc (v, u) decreases the strength between some pair of vertices and is described to as a bridge. Definition 3.19. Let G ′ mpf = (A1, B1, → B1) and G ′′ mpf = (A2, B2, → B2) be two MPFGs. A homomorphism h : G ′ mpf → G ′′ mpf is a mapping function h from V1 to V2 if: • µA1(v1) ≤ µA2(h(v1)) ηA1(v1) ≤ ηA2(h(v1)) νA1(v1) ≥ νA2(h(v1)) • µB1(v1, u1) ≤ µB2(h(v1), h(v2)) ηB1(v1, u1) ≤ ηB2(h(v1), h(v2)) νB1(v1, u1) ≥ νB2(h(v1), h(v2)), ∀v1 ∈ V1 & v1, u1 ∈ E1 • µ→ B1 (v1, u1) ≤ µ→ B2 (h(v1), h(v2)) η→ B1 (v1, u1) ≤ η→ B2 (h(v1), h(v2)) ν→ B1 (v1, u1) ≥ ν→ B2 (h(v1), h(v2)), ∀v1 ∈ V1 & v1, u1 ∈ → E1 Definition 3.20. Let G ′ mpf = (A1, B1, → B1) and G ′′ mpf = (A2, B2, → B2) be two MPFGs. An isomorphism h : G ′ mpf → G ′′ mpf is a bijective mapping function h from V1 to V2 if: • µA1(v1) = µA2(h(v1)) ηA1(v1) = ηA2(h(v1)) νA1(v1) = νA2(h(v1)) • µB1(v1, u1) = µB2(h(v1), h(v2)) ηB1(v1, u1) = ηB2(h(v1), h(v2)) νB1(v1, u1) = νB2(h(v1), h(v2)), ∀v1 ∈ V1 & v1, u1 ∈ E1 • µ→ B1 (v1, u1) = µ→ B2 (h(v1), h(v2)) η→ B1 (v1, u1) = η→ B2 (h(v1), h(v2)) ν→ B1 (v1, u1) = ν→ B2 (h(v1), h(v2)), ∀v1 ∈ V1 & v1, u1 ∈ → E1 Theorem 3.6. Isomorphism of MPFG is an equivalence relation. 237 K. K. Myithili, C. Nandhini Proof. For show that MPFG isomorphism is an equivalence relation, we must first prove that it is reflexive, symmetric, and transitive. Reflexive: consider θ : Gmpf → Gmpf is a mapping, therefore θ is an identity function. Hence it is reflexive. Symmetric: In isomorphic MPFG Gmpf & Hmpf , there exist a 1-1 correspon- dence θ : Gmpf → Hmpf which sustains adjacency. From θ is 1-1 correspondence from Gmpf to Hmpf , here 1-1 correspondence θ−1 from Hmpf to Gmpf which sus- tains adjacency. Hence isomporphism of MPFG is symmetric. Transitive: If Gmpf is isomorpic to Hmpf and Hmpf is isomorphic to Kmpf , then there are 1-1 correspondences between θ & ϕ from Gmpf to Hmpf & Hmpf to Kmpf respectively, which sustains adjacency. It follows ϕ ◦ θ is a 1-1 correspon- dence between from Gmpf to Kmpf which sustains adjacency. Hence it is transi- tive. Therefore, isomorphism of MPFG is an equivalene relation. Definition 3.21. Let Gmpf = (A, B, → B) be a MPFG. A vertex v of Gmpf is said to be busy vertex if µA(v) ≤ dµ(v), ηA(v) ≤ dη(v), νA(v) ≥ dν(v). otherwise, it is called free vertex. Definition 3.22. Let Gmpf = (A, B, → B) be a MPFG. Then an edge (v, u) is de- fined as an effective edge iff µB(v, u) = min(µA(v), µA(u)), ηB(v, u) = min(ηA(v), ηA(u)), νB(v, u) = max(νA(v), νA(u)), µ→ B (v, u) = min(µA(v), µA(u)), η→ B (v, u) = min(ηA(v), ηA(u)), ν→ B (v, u) = max(νA(v), νA(u)). Note: When all edges in a graph are effective, the graph is complete. 4 Application of MPFG in instagram Social media has grown gaining popularity in latest years of its user-friendliness. Social media services such as Whatsapp, Facebook, Twitter, and Instagram allow people to communicate across long distances. To put it another way, social media has made the entire globe available at the touch of a button. Social media sites are also valuable resources for public awareness creation, as they rapidly distribute information about natural disasters and terrorist/criminal attacks to a mass audi- ence. Social network is a collection of vertices and edges. Persons, groups, coun- tries, associations, locations, business and other entities are represented by ver- tices, while edges define the relationship between vertices. We commonly use a classical graph to describe a social network, with vertices representing persons and edges representing relationships/flows between vertices. Several manuscripts 238 Mixed Picture Fuzzy Graph have been shared on social media platforms. However, a classical graph can- not accurately model a social network. Since all vertices in a classical graph are extremely significant. As a result, in today’s social networks, every social units (personal or organisational) are given equal weight. In fact, however, not all social units are equal in importance. In a classical graph, all edges (relationships) have the same weight. For example, a person may be well-versed in certain practises. On the other hand, they have no experience of certain activities, and he has a very little knowledge of others. We can easily represent these three kinds (positive, neutral and negative) of vertex and edge membership degrees with a picture fuzzy set, which has three membership values for each element. In Instagram, we can classify three activities namely good, neutral and bad activities which is represented in PFG as positive, neutral & negative membership values of a vertex. Similarly, edge membership value can be used to describe the strength of relationship between two vertices. Since social media has such a vast number of clients, it also contains mutual and single-sided relationships; it is not restricted to directed or undirected relationships. As a result, we have introduced a mixed picture fuzzy graph which includes both directed and undirected edges. It provides a more accurate result than previous methods. For example, in Instagram an undirected edge exists when two friends have a mutual relationship. Similarly, if a friend-1 follows friend-2 but friend-2 doesn’t then there occurs directed edge. The vertex effect on a social media platform is identified via centrality, which is one of the most significant concepts in social networking. The degree of cen- trality determines how closely a social squad is linked to other social squads. It essentially provides the social squad’s/person’s participation in the social net- work. A vertex’s centrality seems more central than that of other vertex’s. The centre people are muchis closer to the others and has access to more information. It should be noticed that a person’s information is shared by a friend of a friend. However, friends of friends communicate less information than direct friends. As a result, the importance of the relationship gradually decreases as it passes from one member to the next along a connected path. In MPFG, suppose a friend-1 directly connected to a friend-2, then we say v1 is circle distance-1(CD-1) friend of v2. The set of all CD-1 friends of v represented as cd1(v). i.e., cd1(v) = {vi ∈ V ; vi is a CD-1 friends of v}. Correspondingly, suppose there is a shortest path between v1 & v2 with m edges, then v1 is a CD-m friend of v2. That is, cdm(v) = {vi ∈ V ; vi is a CD-m friends of v}. Now, con- sider cd ′ m(v) = cdm(v) − cd ′ m−1(v), where m = 2, 3, ... and cd ′ m(v) = cdm(v). CD-1 friends are obviously more significant than CD-2 friends, and CD-2 friends are more significant than CD-3 friends, and so on. The linguistic term “more significant” could be denoted by weights(wm). Let 0 ≤ wm ≤ 1 have been the weights that gradually decreases, when the CD between the friends increases. Then w1 ≥ w2 ≥ ... ≥ wm ≥ .... 239 K. K. Myithili, C. Nandhini Let u1(= vi), u2, u3, ...um(= vj) are the vertices upon the path between vi and vj. We have to derive MPFCD CDmpf(vi, vj) between vi and vj with this path as CDmpf(vi, vj) = m−1∑ n=1 µ(un, un+1) + m−1∑ n=1 η(un, un+1) + m−1∑ n=1 ν(un, un+1) There could be several paths connecting two vertices in a networks. Let us assume these paths of equal length whose MPFCD (CDmpf ) seems to be the max- imum in Gmpf . Suppose these are n edgeds in this path of maximum MPFCD, we designate CDnmpf i.e, CD n mpf(vi, vj) represents the MPFCD between the vertices vi & vj in MPFG with the particular path having accurately n edges. We stated that social squad S with atmost CD-p friends. The centrality C(S) of a social squad is defined as follows: C(S) = ∑ u1∈cd11(v) w1CD 1 mpf(V, u1) + ∑ u2∈cd ′ 2(v) w2CD 2 mpf(V, u2) + ... + ∑ up∈cd ′ p(v) wpCD p mpf(V, up) (17) Close friends are valued more than the next closest friends, while the signifi- cance of the furthest friend gradually decreases. The significance is established by including the weight wi, which stands for CD-i friend, I = 1,2,3,... For example, MPFG of 7 people after 7 days is shown in figure 7. Also the link membership values are shown in same figure. In the definition of centrality of a social unit, p can be taken as fixed for a social network. Here we assumed that p = 3 and measure the centrality of social squad. Here, we take w1 = 1 and wi+1 = 1/2wi, i = 1,2,... 4.1 Centrality of v1 Here cd1(v1) = {v2, v4, v3} = cd ′ 1(v1), cd2(v1) = {v3, v5, v7, v2}, cd ′ 2(v1) = cd2(v1) − cd ′ 1(v1) = {v5, v7}, cd3(v1) = {v6, v7, v3}, cd ′ 3(v1) = cd3(v1) − cd ′ 2(v1) = {v6, v3}. 240 Mixed Picture Fuzzy Graph Figure 7: Mixed picture fuzzy network Now, ∑ u1∈(cd1) ′ (v1) (CDmpf) 1(v1, u1) = {membership values of(v1, v2)}+{membership values of(v1, v4)} + {membership values of (v1, v3)} = (µ(v1, v2), η(v1, v2), ν(v1, v2))+((µ(v1, v4), η(v1, v4), ν(v1, v4)) + (µ(v1, v3), η(v1, v3), ν(v1, v3)) = (0.37, 0.23, 0.26)+(0.3, 0.33, 0.27)+(0.4, 0.32, 0.2) = (1.07, 0.88, 0.73). ∑ u2∈(cd2) ′ (v1) (CDmpf) 2(v1, u2) = (CDmpf) 2(v1, v5) + (CDmpf) 2(v1, v7) = {membership values of{(v1, v5) + (v1, v5)}}+ {membership values of{(v1, v3) + (v3, v7)}} = {(0.3, 0.33, 0.27) + (0.2, 0.3, 0.2)}+ {(0.4, 0.32, 0.2) + (0.33, 0.3, 0.252)} = (1.23, 0.25, 0.922). 241 K. K. Myithili, C. Nandhini ∑ u3∈(cd3) ′ (v1) (CDmpf) 3(v1, u3) = (CDmpf) 3(v1, v6) + (CDmpf) 3(v1, v3) = {membership values of{(v1, v3) + (v3, v7) +(v7, v6)}}+{membership values of{(v1, v4) + (v4, v5) + (v5, v3)}} = {(0.4, 0.32, 0.2) + (0.33, 0.3, 0.252) + (0.4, 0.2, 0.23)} + {(0.3, 0.33, 0.27) + (0.2, 0.3, 0.2) + (0.52, 0.27, 0.25)} = (2.15, 1.72, 1.402). The centrality of v1 is C(v1) = ∑ u1∈cd ′ 1(v1) w1CD 1 mpf(v1, u1) + ∑ u2∈cd ′ 2(v1) 0.5 × CD2mpf(v1, u2) + ∑ u3∈cd ′ 3(v1) 0.25 × CD3mpf(v1, u3) = (2.2225, 1.935, 1.5415) Similarly, we can calculate centralities of other vertices. C(v2)=(1.8125,1.6125,1.363), C(v3)=(0.985,0.755,0.66975), C(v4)=(1.715,1.781,1.4775), C(v5)=(2.818,2.317,1.91225), C(v6)=(2.8005,2.1905,1.824), C(v7)=(3.442,2.33,2.042). 4.1.1 Disscussion Suppose there are more than one paths between two vertices, we have to choose the shortest distance path to calculate the centrality. From the results, we have centrality of v3 is comparatively less than other vertices. Because v3 has less number of mutual friends. So degree of centrality depends on mutual friends and friends of circle distance-i. Social networks are built on the backs of millions of users and massive amounts of data. We used a simple numerical example of a MPFG to describe a small social network problem in this study. The smaller examples are really useful in understanding the benefits of our suggested model. 242 Mixed Picture Fuzzy Graph 5 Conclusion The prime goal of this paper is to just introduce the terms and concepts of a MPFG and examined the various types of MPFG. Initially, we present a def- inition of an MPFG built from a picture fuzzy graph in this paper. Few types of degrees were discussed with its properties. We discuss about regular, strong, complete and complement of MPFG are some of the different forms of MPFG. The isomorphic property has also been analysed in MPFG. When comparing to picture fuzzy graph models, the MPFG can boost effectiveness, reliability, flexi- bility and comparability in modelling complex real-world scenarios. A model has been developed to represent a social network problem using MPFG. The concept of a MPFG can be used to a database system, a computer network, a traffic signal system, a social network, a transportation network and image processing among other things. 6 Acknowledgements Future work is to develop this concept in the field of transversals of MPFG. References [1] Abdelkadir Muzey Mohammed, Mixed graph representation and mixed graph isomorphism, Gazi University Journal of Science, 2017, 30(1), 303- 310. [2] K.T.Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 1986, 20(1), 87-96. [3] Cen Zuo, Anita Pal & Arindam Dey, New Concepts of picture fuzzy graphs with application, Mathematics, 2019, 7(5), 470, 1-18. [4] B.C.Cuong, Picture fuzzy sets, Journal of Computer Science and Cybernet- ics, 2014, 30(4), 409-420. [5] B.C.Cuong & P.V.Hai, Some fuzzy logic operators for picture fuzzy sets, Seventh International Conference on Knowledge and Systems Engineering, 2015 132-137. [6] B.C.Cuong & V.Kreinovich, Picture fuzzy sets-a new concept for compu- tational intelligence problems, Third World Congress on Information and Communication Technologies WIICT, 2013, 1-6. 243 K. K. Myithili, C. Nandhini [7] Laszlo T. Koczy, Naeem Jan, Tahir Mahmood & Kifayat Ullah, Analysis of social networks and Wi-Fi networks by using the concept of picture fuzzy graphs, Soft Computing - A Fusion of Foundations, Methodologies and Ap- plications, 2020, 24(21), 16551–16563. [8] J.N.Mordeson & P.R.Nair, Fuzzy graphs and fuzzy hypergraphs, 2012, Springer, Berlin, Germany. [9] A.Nagoor Gani & S.Shajitha Begum, Degree, order and size in intuitionistic fuzzy graphs, International Journal of Algorithms Computing and Mathe- matics, 2010, 3(3), 11-16. [10] R.Parvathi, M.G.Karunambigai & K.T.Atanassov Operations on intuition- istic fuzzy graphs, 2009 IEEE International Conference on Fuzzy Systems, 2009, 1396-1401. [11] R.Parvathi & M.G.Karunambigai, Intuitionistic fuzzy graphs, In Computa- tional Intelligence, Theory and Applications, Springer, Berlin, Germany, 2006, 139-150. [12] Peng & X.Dai, Algorithm for picture fuzzy multiple attribute decision- making based on new distance measure, International Journal for Uncer- tainty Quantification, 2017, 7(2), 177-187. [13] P.H.Phong, D.T.Hieu, R.H.Ngan & P.T.Them, Some compositions of picture fuzzy relations, Seventh National Conference on Fundamental and Applied Information Technology Research, FAIR’7, Thai Nguyen, 2014, 1-10. [14] A.Rosenfeld, Fuzzy graphs, fuzzy sets and their applications, 1975, Springer, Physica, Heidelberg, New York, USA. [15] Sankar Das & Ganesh Ghorai, Analysis of road map design based on multi- graph with picture fuzzy information, International Journal of Applied and Computational Mathematics, 2020, 6(3), 1-17. [16] A.Shannon & K.T.Atanassov, On a generalization of intuitionistic fuzzy graphs, Notes of Intuitionistic Fuzzy Sets, 2006, 12(1), 24-29 . [17] Sunitha & Vijayakumar, Complement of a fuzzy graph, Indian Journal of Pure and Applied Mathematics, 2002, 33(9), 1451-1464. [18] Wei Xiao, Arindam Dey & Le Hoang Son, A study on regular picture fuzzy graph with applications in communication networks, Journal of Intelligent & Fuzzy Systems, 2020, 39(3), 3633-3645. [19] L.A.Zadeh, Fuzzy sets, Information and Control, 1965, 8(3), 338-356. 244