Certain graphs under Pythagorean fuzzy environment
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Abstract
Graph theory plays crucial role in structuring many realworld problems including, medical sciences, control theory, expert systems and network security. Product in graphs, an operation that consider two graphs and produce a new graph by simple or complex changes, has wide range of applications in games theory, automata theory, structural mechanics and networking system. An intuitionistic fuzzy model is used to handle the vagueness and uncertainty in network problems. A Pythagorean fuzzy model is a powerful tool for describing vagueness and uncertainty more accurately as compared to intuitionistic fuzzy model. The objective of this paper is to apply the concept of Pythagorean fuzzy sets to graphs and then combine two Pythagorean fuzzy graphs (PFGs) using two new graph products namely, maximal product and the residue product. This research paper investigates the regularity for these products. Moreover, it discusses some eminent properties such as strongness, connectedness and completeness. Further, it proposes some necessary and sufficient conditions for \({\mathscr {G}}_{1}*{\mathscr {G}}_{2}\) and \({\mathscr {G}}_{1}\cdot {\mathscr {G}}_{2}\) to be regular. Finally, decisionmaking problems concerning evaluation of best company for investment and alliance partner selection of a software company are solved to better understand PFGs.
Keywords
Pythagorean fuzzy graphs Regular Pythagorean fuzzy graphs Maximal product Residue productIntroduction
Fuzzy set theory [1] is the most efficient tool having the capability to deal with imprecise and incomplete information in different disciplines, including engineering, mathematics, statistics, artificial intelligence, medical and social sciences. To cope with imprecise and incomplete information, consisting of doubts in human judgement, the fuzzy set shows some restrictions. So, for characterizing the hesitancy more explicitly, fuzzy sets were extended to intuitionistic fuzzy sets (IFSs) by Atanassov [2], which assign a membership grade (\(\mu \)) and a nonmembership grade (\(\nu \)) to the objects, satisfying the condition \(\mu + \nu \le 1\) and the hesitancy part \(\pi = 1  \mu  \nu \). The IFSs have gained extensive attentions and have been broadly applied in different areas of real life. The limitation \(\mu + \nu \le 1\) confines the choice of the membership and nonmembership grades in IFS. To evade this situation, Yager [3, 4, 5] initiated the idea of Pythagorean fuzzy set (PFS), depicted by a membership grade (\(\mu \)) and a nonmembership grade (\(\nu \)) with the condition \(\mu ^2 + \nu ^2 \le 1.\) Zhang and Xu [6] introduced the concept of Pythagorean fuzzy number (PFN) for interpreting the dual aspects of an element. In a decisionmaking environment, a specialist gives the preference information about an alternative with the membership grade 0.9 and the nonmembership grade 0.3; it is noted that the IFN fails to address this situation, as \(0.9 + 0.3 > 1\), but \((0.9)^2 + (0.3)^2 \le 1.\) Thus, the PFS has much stronger ability than IFS to model fuzziness in the practical MCDM problems. Under Pythagorean fuzzy environment, many researchers have initiated work in different directions and acquired various eminent results [7]. Some operations on PFSs [8] and Pythagorean fuzzy TODIM approach to multicriteria decisionmaking [9] have been discussed. Furthermore, the PFS has been investigated from different perspectives, including aggregation operators [10, 11]. Garg [12, 13, 14, 15, 16, 17, 18] explored applications of Pythagorean fuzzy sets in decisionmaking. Lately, the concept of Pythagorean fuzzy set has been extended to intervalvalued Pythagorean fuzzy set and hesitant Pythagorean fuzzy set. Garg [19, 20] elaborated exponential operational laws and their aggregation operators under intervalvalued Pythagorean fuzzy information. Yu et al. [21] discussed hesitant Pythagorean fuzzy Maclaurin symmetric mean operators and its applications to multiattribute decisionmaking.
Graphs are the pictorial representation that bond the objects and highlight their information. Graph theory is rapidly moving into the core of mathematics due to its applications in various fields, including physics, biochemistry, biology, electrical engineering, astronomy, operations research and computer science. To emphasize on a realworld problem, the bondedness between the objects occur due to some relations. But when there exists uncertainty and haziness in the bonding, then the corresponding graph model can be taken as fuzzy graph model. For example, a social network may be represented as a graph, where vertices represent members and edges represent relation between members. If the relations among the members are to be measured as good or bad, then fuzziness should be added to the representation. This and many other problems motivated to define fuzzy graph. The concept of fuzzy graphs was presented by Kaufmann [22], based on Zadeh’s fuzzy relation in 1971. Later, Rosenfeld [23] discussed several basic graph theoretical concepts, including paths, cycles, bridges and connectedness in fuzzy environment. Nagoor Gani and Radha [24] initiated the concept of regular fuzzy graph in 2008. Mordeson and Peng [25] introduced some operations on fuzzy graphs and studied their properties. Further, Nirmala and Vijaya [26] explored new operations on fuzzy graphs. Parvathi and Karunambigai [27] considered intuitionistic fuzzy graphs (IFGs). Later, Akram and Davvaz [28] discussed IFGs. An algorithm for computing sum distance matrix, eccentricity of vertices, diameter and radius in IFGs was presented by sarwar and Akram [29]. Akram and Dudek [30] described intuitionistic fuzzy hypergraphs with applications. Recently, Naz et al. [31] originally proposed the concept of PFGs, a generalization of the notion of Akram and Davvaz’s IFGs [28], along with its applications in decisionmaking. Akram and Naz [32] studied energy of PFGs with applications. Akram et al. [33] introduced the concept of Pythagorean fuzzy Planar graphs. Dhavudh and Srinivasan [35] dealt with IFGs2T. Ghorai and Pal [37, 38] studied some properties of mpolar fuzzy graphs. Recently, Akram et al. [34] introduced some new operations including rejection, symmetric difference, maximal product and residue product of PFGs. This paper describes two new operations namely, maximal product and residue product of PFGs that can allow the mathematical design of network. These products can be applied to construct and analyze several realworld networks such as road networks and communication networks. The work explores some significant properties such as regularity, strongness, completeness and connectedness. As regularity plays a substantial role in designing reliable communication networks, so the main focus is to familiarize the regularity of these product with illustrative examples. The work also proposes some necessary and sufficient conditions for these two products to be regular. Finally, it discusses some applications of PFGs in decisionmaking.
For better understanding, we present prerequisite terminologies and notations:
Definition 1.1
Definition 1.2
Definition 1.3
Definition 1.4
Definition 1.5
Definition 1.6
 1.
\({{\mathscr {A}}_{1}}\subseteq {{\mathscr {A}}_{2}}\) if and only if \(\mu _{{\mathscr {A}}_{1}}(u)\le \mu _{{\mathscr {A}}_{2}}(u)\) and \(\nu _{{\mathscr {A}}_{1}}(u)\ge \nu _{{\mathscr {A}}_{2}}(u),\)
 2.
\({{\mathscr {A}}_{1}} = {{\mathscr {A}}_{2}}\) if and only if \({{\mathscr {A}}_{1}}(u)\subseteq {{\mathscr {A}}_{2}}(u)\) and \({{\mathscr {A}}_{2}}(u)\subseteq {{\mathscr {A}}_{1}}(u),\)
 3.
\({{\mathscr {A}}_{1}}^{c} = \{\langle u, \nu _{{\mathscr {A}}_{1}}(u), \mu _{{\mathscr {A}}_{1}}(u)\rangle  u \in {\mathscr {X}}\},\)
 4.
\({{\mathscr {A}}_{1}} \cap {{\mathscr {A}}_{2}} =\{\langle u, \min \{\mu _{{\mathscr {A}}_{1}}(u),\mu _{{\mathscr {A}}_{2}}(u)\}, \max \{\nu _{{\mathscr {A}}_{1}}(u),\nu _{{\mathscr {A}}_{2}}(u)\} \rangle  ~u \in {\mathscr {X}} \},\)
 5.
\({{\mathscr {A}}_{1}} \cup {{\mathscr {A}}_{2}} =\{\langle u, \max \{\mu _{{\mathscr {A}}_{1}}(u),\mu _{{\mathscr {A}}_{2}}(u)\}, \min \{\nu _{{\mathscr {A}}_{1}}(u),\nu _{{\mathscr {A}}_{2}}(u)\} \rangle  ~u \in {\mathscr {X}}\}.\)
Definition 1.7
Definition 1.8
Example 1.1
Definition 1.9
Regular maximal product in Pythagorean fuzzy graphs
Definition 2.1
 (i)
 $$\begin{aligned}&\left\{ \begin{array}{ll}(\mu _{{\mathscr {A}}_1} *\mu _{{\mathscr {A}}_2})(u_1, u_2)=\mu _{{\mathscr {A}}_1}(u_1)\vee \mu _{{\mathscr {A}}_2}(u_2)\\ (\nu _{{\mathscr {A}}_1}*\nu _{{\mathscr {A}}_2})(u_1, u_2)=\nu _{{\mathscr {A}}_1}(u_1)\wedge \nu _{{\mathscr {A}}_2}(u_2)\end{array}\right. \\&\quad \hbox { for all } (u_1, u_2) \in V_1\times V_2, \end{aligned}$$
 (ii)
 $$\begin{aligned}&\left\{ \begin{array}{ll}(\mu _{{\mathscr {B}}_1}*\mu _{{\mathscr {B}}_2})((u,u_2)(u,v_2)){=}\mu _{{\mathscr {A}}_1}(u)\vee \mu _{{\mathscr {B}}_2}(u_2v_2)\\ (\nu _{{\mathscr {B}}_1} *\nu _{{\mathscr {B}}_2})((u,u_2)(u,v_2))= \nu _{{\mathscr {A}}_1}(u)\wedge \nu _{{\mathscr {B}}_2}(u_2v_2)\end{array}\right. \\&\quad \hbox {for all }u\in V_1\hbox { and }u_2v_2 \in E_2, \end{aligned}$$
 (iii)
 $$\begin{aligned}&\left\{ \begin{array}{ll}(\mu _{{\mathscr {B}}_1}*\mu _{{\mathscr {B}}_2})((u_1,z)(v_1,z))=\mu _{{\mathscr {B}}_1}(u_1v_1)\vee \mu _{{\mathscr {A}}_2}(z)\\ (\nu _{{\mathscr {B}}_1}*\nu _{{\mathscr {B}}_2})((u_1,z)(v_1,z))=\nu _{{\mathscr {B}}_1}(u_1v_1)\wedge \nu _{{\mathscr {A}}_2}(z)\end{array}\right. \\&\quad \hbox {for all }z\in V_2\hbox { and }u_1v_1 \in E_1. \end{aligned}$$
Definition 2.2
Example 2.1
By routine calculation, one can see from Fig. 2 that it is a strong Pythagorean fuzzy graph.
Theorem 2.1
If \({\mathscr {G}}_1=({\mathscr {A}}_1,{\mathscr {B}}_1)\) and \({\mathscr {G}}_2=({\mathscr {A}}_2,{\mathscr {B}}_2)\) are two strong PFGs, then their maximal product is also a strong PFG.
Proof
Remark 2.1
Converse of the Theorem 2.1 may not be true as it can be seen in the following example.
Here \(\mu _{{\mathscr {B}}_1}(u_1u_2) \ne \mu _{{\mathscr {A}}_1}(u_1) \wedge \mu _{{\mathscr {A}}_1}(u_2),~\nu _{{\mathscr {B}}_1}(u_1u_2) \ne \nu _{{\mathscr {A}}_1}(u_1) \vee \nu _{{\mathscr {A}}_1}(u_2)\) and \(\mu _{{\mathscr {B}}_2}(v_1v_2) \ne \mu _{{\mathscr {A}}_2}(v_1) \wedge \mu _{{\mathscr {A}}_2}(v_2),\nu _{{\mathscr {B}}_2}(v_1v_2) \ne \nu _{{\mathscr {A}}_2}(v_1) \vee \nu _{{\mathscr {A}}_2}(v_2)\). Hence, \({\mathscr {G}}_1\) and \({\mathscr {G}}_2\) are not strong PFGs.
But \((\mu _{{\mathscr {B}}_1} *\mu _{{\mathscr {B}}_2})((u,v)(x,y))= \mu _{{\mathscr {A}}_1}(u,v) \wedge \mu _{{\mathscr {A}}_2}(x,y)\), \((\nu _{{\mathscr {B}}_1} *\mu _{{\mathscr {B}}_2})((u,v)(x,y))= \nu _{{\mathscr {A}}_1}(u,v) \vee \nu _{{\mathscr {A}}_2}(x,y)\) for all edge (u, v)(x, y) in E. Thus, their maximal product \({\mathscr {G}}_1 *{\mathscr {G}}_2\) is a strong PFG.
Definition 2.3
Example 2.2
Theorem 2.2
The maximal product \({\mathscr {G}}_1 *{\mathscr {G}}_2\) of two connected PFGs is always a connected PFG.
Proof
Let \({\mathscr {G}}_1=({\mathscr {A}}_1,{\mathscr {B}}_1)\) and \({\mathscr {G}}_2=({\mathscr {A}}_2,{\mathscr {B}}_2)\) be two connected PFGs with underlying crisp graph \(G_1=(V_1,E_1)\) and \(G_2=(V_2,E_2)\), respectively. Let \(V_1=\{u_1,u_2,\ldots ,u_m\}\) and \(V_2=\{v_1,v_2,\ldots ,v_n\}\). Then \(\mu _1^\infty (u_iu_j)\,{>}\,0\), \(\nu _1^\infty (u_iu_j){>}\,0\) for all \(u_i,u_j\in V_1\) and \(\mu _2^\infty (v_iv_j)\,{>}\,0\), \(\nu _2^\infty (v_iv_j)\,{>}\,0\) for all \(v_i,v_j \in V_2\).
The maximal product of \({\mathscr {G}}_1=({\mathscr {A}}_1,{\mathscr {B}}_1)\) and \({\mathscr {G}}_2=({\mathscr {A}}_2,{\mathscr {B}}_2)\) can be taken as \({\mathscr {G}}=({\mathscr {A}},{\mathscr {B}})\). Consider the k subgraphs of \({\mathscr {G}}=({\mathscr {A}},{\mathscr {B}})\) with the vertex sets \(V_2=\{u_iv_1,u_iv_2,\ldots ,u_iv_n\}\) for \(i=1, 2,\ldots , m.\) Each of these subgraphs of \({\mathscr {G}}=({\mathscr {A}},{\mathscr {B}})\) is connected as \(u_i's\) are same.
Since \({\mathscr {G}}_1=({\mathscr {A}}_1,{\mathscr {B}}_1)\) and \({\mathscr {G}}_2=({\mathscr {A}}_2,{\mathscr {B}}_2)\) are connected, each \(u_i\) and \(v_i\) are adjacent to at least one of the vertices in \(V_1\) and \(V_2\). Therefore, there exists at least one edge between any pair of the above k subgraphs.
Definition 2.4
Example 2.3
By routine calculation, one can see from Fig. 6 that it is a complete Pythagorean fuzzy graph.
Remark 2.2
By routine calculation, one can see from Fig. 7 that \({\mathscr {G}}_1\) and \({\mathscr {G}}_2\) are complete PFGs. While notice that \({{\mathscr {G}}_1}~*~{{\mathscr {G}}_2}\) is not a complete PFG, as the case \(u_1u_2 \in E_1\) and \(v_1v_2 \in E_2\) is not included in the definition of the maximal product. Further, one can notice that the maximal product of two complete PFGs is a strong PFG (Fig. 9).
Definition 2.5
Example 2.4
Since \(d_{\mu }({u_i})=0.6\) and \(d_{\nu }({u_i})=1.4\) \({\mathrm{for ~all ~u_i} \in V}\) and \(i=1,2,\ldots ,6,\) \({\mathscr {G}}\) is a regular Pythagorean fuzzy graph of degree (0.3, 0.7) or (0.3, 0.7)regular PFG.
Remark 2.3
Definition 2.6
Let \({\mathscr {G}}=({\mathscr {A}},{\mathscr {B}})\) be a Pythagorean fuzzy graph on underlying crisp graph \(G=(V,E).\) Then, \({\mathscr {G}}\) is said to be a partially regular Pythagorean fuzzy graph if \(G=(V,E)\) is a regular graph.
Example 2.5
Consider a Pythagorean fuzzy graph \({\mathscr {G}} = ({\mathscr {A}},{\mathscr {B}})\) as displayed in the Fig. 12.
Since \(d_{\mu }({a})=1.65 \ne 2 =d_{\mu }({a})\) and \(d_{\nu }({a})=2.35 \ne 1.8 = d_{\nu }({b}).\) Hence, \({\mathscr {G}}\) is not a regular Pythagorean fuzzy graph but G is a regular graph as the degree of each vertex is equal. Thus, \({\mathscr {G}}\) is a partially regular Pythagorean fuzzy graph.
The following theorems explain the conditions for the maximal product of two regular PFGs to be regular.
Theorem 2.3
If \({\mathscr {G}}_1=({\mathscr {A}}_1,{\mathscr {B}}_1)\) is a partially regular PFG and \({\mathscr {G}}_2=({\mathscr {A}}_2,{\mathscr {B}}_2)\) is a PFG such that \(\mu _{{\mathscr {A}}_1}\le \mu _{{\mathscr {B}}_2}\), \(\nu _{{\mathscr {A}}_1}\ge \nu _{{\mathscr {B}}_2}\) and \(\mu _{{\mathscr {A}}_2}\), \(\nu _{{\mathscr {A}}_2}\) are constant functions of values \(c_1\) and \(c_2\), respectively, then their maximal product is regular if and only if \({\mathscr {G}}_2=({\mathscr {A}}_2,{\mathscr {B}}_2)\) is regular PFG.
Proof
Let \({\mathscr {G}}_1=({\mathscr {A}}_1,{\mathscr {B}}_1)\) be a partially regular PFG such that \(G_1=(V_1,E_1)\) is rregular and \({\mathscr {G}}_2=({\mathscr {A}}_2,{\mathscr {B}}_2)\) be any PFG with \(\mu _{{\mathscr {A}}_1}\le \mu _{{\mathscr {B}}_2}\), \(\nu _{{\mathscr {A}}_1}\ge \nu _{{\mathscr {B}}_2}\) then \(\mu _{{\mathscr {A}}_2}\ge \mu _{{\mathscr {B}}_1}\), \(\nu _{{\mathscr {A}}_2}\le \nu _{{\mathscr {B}}_1}\) and \(\mu _{{\mathscr {A}}_2}\), \(\nu _{{\mathscr {A}}_2}\) are constant functions of values \(c_1\) and \(c_2\), respectively.
\(\square \)
Theorem 2.4
If \({\mathscr {G}}_1=({\mathscr {A}}_1,{\mathscr {B}}_1)\) and \({\mathscr {G}}_2=({\mathscr {A}}_2,{\mathscr {B}}_2)\) are two partially regular PFG such that \(\mu _{{\mathscr {A}}_1}\ge \mu _{{\mathscr {B}}_2}\), \(\nu _{{\mathscr {A}}_1}\le \nu _{{\mathscr {B}}_2}\) , \(\mu _{{\mathscr {A}}_2}\ge \mu _{{\mathscr {B}}_1}\), \(\nu _{{\mathscr {A}}_2}\le \nu _{{\mathscr {B}}_1}\) and \(\mu _{{\mathscr {A}}_2}\), \(\nu _{{\mathscr {A}}_2}\) are constant functions of values \(c_1\) and \(c_2\), respectively, then their maximal product is regular if and only if \(\mu _{{\mathscr {A}}_1}\) and \(\nu _{{\mathscr {A}}_1}\) are constant functions.
Proof
Definition 2.7
Let \({\mathscr {G}}=({\mathscr {A}},{\mathscr {B}})\) be a Pythagorean fuzzy graph on underlying crisp graph \(G=(V,E).\) Then \({\mathscr {G}}\) is said to be a full regular Pythagorean fuzzy graph if \({\mathscr {G}}\) is both regular and partially regular graph.
Example 2.6
Since \(d_{\mu }({u_i})=1.5\) and \(d_{\nu }({u_i})=2\) for all \(u_i \in V\), where \(i=1,\ldots ,6.\) Hence, \({\mathscr {G}}\) is a regular Pythagorean fuzzy graph of degree (1.5, 2). Also, G is a regular graph as the degree of each vertex is equal. Thus, \({\mathscr {G}}\) is a full regular Pythagorean fuzzy graph.
Remark 2.4
By routine calculation, one can see that \({\mathscr {G}}_1\) and \({\mathscr {G}}_2\) are full regular PFGs. But \((d)_{{\mathscr {G}}_1*{\mathscr {G}}_2}(a,d)\ne (d)_{{\mathscr {G}}_1*{\mathscr {G}}_2}(b,d)\). Hence, \({{\mathscr {G}}_1}~*~{{\mathscr {G}}_2}\) is not full regular PFG.
Remark 2.5
By routine calculation, one can see that \({\mathscr {G}}_1\) and \({\mathscr {G}}_2\) are regular PFGs. But \((d)_{{\mathscr {G}}_1*{\mathscr {G}}_2}(u_1,u_2)\ne (d)_{{\mathscr {G}}_1*{\mathscr {G}}_2}(v_1,v_2)\). Hence, \({{\mathscr {G}}_1}~*~{{\mathscr {G}}_2}\) is a partially regular PFG as crisp graph is regular.
Regular residue product in Pythagorean fuzzy graphs
Definition 3.1
 (i)

\(\left\{ \begin{array}{ll} (\mu _{{\mathscr {A}}_1} \cdot \mu _{{\mathscr {A}}_2})(u_1,u_2)~=~\mu _{{\mathscr {A}}_1}(u_1)\vee \mu _{{\mathscr {A}}_2}(u_2)\\ (\nu _{{\mathscr {A}}_1} \cdot \nu _{{\mathscr {A}}_2})(u_1,u_2)~=~\nu _{{\mathscr {A}}_1}(u_1)\wedge \nu _{{\mathscr {A}}_2}(u_2) \\ \quad {\mathrm{~ for~ all~}} (u_1,u_2) \in V_1 \times V_2, \end{array}\right. \)
 (ii)

\(\left\{ \begin{array}{ll} (\mu _{{\mathscr {B}}_1} \cdot \mu _{{\mathscr {B}}_2})(u_1,u_2)(v_1,v_2)~=~\mu _{{\mathscr {B}}_1}({u_1}{v_1})\\ (\nu _{{\mathscr {B}}_1} \cdot \nu _{{\mathscr {B}}_2})(u_1,u_2)(v_1,v_2)~=~\nu _{{\mathscr {B}}_1}({u_1}{v_1})\\ \quad {\mathrm{~ for~ all~}} {u_1}{v_1} \in E_1, u_2 \ne v_2. \end{array}\right. \)
Remark 3.1
Here \(\mu _{{\mathscr {B}}_1}(u_1u_2) = \mu _{{\mathscr {A}}_1}(u_1) \wedge \mu _{{\mathscr {A}}_1}(u_2),~\nu _{{\mathscr {B}}_1}(u_1u_2) = \nu _{{\mathscr {A}}_1}(u_1) \vee \nu _{{\mathscr {A}}_1}(u_2)\) for all \(u_1u_2 \in E_1\) and \(\mu _{{\mathscr {B}}_2}(v_1v_2) = \mu _{{\mathscr {A}}_2}(v_1) \wedge \mu _{{\mathscr {A}}_2}(v_2),~\nu _{{\mathscr {B}}_2}(v_1v_2) = \nu _{{\mathscr {A}}_2}(v_1) \vee \nu _{{\mathscr {A}}_2}(v_2)\) for all \(v_1v_2 \in E_2\). Hence, \({\mathscr {G}}_1\) and \({\mathscr {G}}_2\) are strong PFGs.
But \((\mu _{{\mathscr {B}}_1} \cdot \mu _{{\mathscr {B}}_2})((u_1,v_2)(u_2,v_1))= ~0.6 \ne \mu _{{\mathscr {A}}_1}(u_1,v_2) \wedge \mu _{{\mathscr {A}}_2}(u_2,v_1) = 0.7\) and \((\nu _{{\mathscr {B}}_1} \cdot \nu _{{\mathscr {B}}_2})((u_1,v_2)(u_2,v_1))\) \(= ~0.5 \ne \nu _{{\mathscr {A}}_1}(u_1,v_2) \wedge \nu _{{\mathscr {A}}_2}(u_2,v_1) = 0.4\). Thus, their residue product \({\mathscr {G}}_1 \cdot {\mathscr {G}}_2\) is not a strong PFG.
The following theorem gives the condition under which the residue product is strong.
Theorem 3.1
The residue product of a strong PFG \({\mathscr {G}}_1=({\mathscr {A}}_1,{\mathscr {B}}_1)\) with any PFG \({\mathscr {G}}_2=({\mathscr {A}}_2,{\mathscr {B}}_2)\) is a strong PFG if \(\mu _{{\mathscr {A}}_1}\ge \mu _{{\mathscr {A}}_2}\), \(\nu _{{\mathscr {A}}_1}\le \nu _{{\mathscr {A}}_2}\).
Proof
Theorem 3.2
The residue product \({\mathscr {G}}_1 *{\mathscr {G}}_2\) of two connected PFGs is always a connected PFG.
Proof
The residue product of \({\mathscr {G}}_1=({\mathscr {A}}_1,{\mathscr {B}}_1)\) and \({\mathscr {G}}_2=({\mathscr {A}}_2,{\mathscr {B}}_2)\) can be taken as \({\mathscr {G}}=({\mathscr {A}},{\mathscr {B}})\). Consider the k subgraphs of \({\mathscr {G}}=({\mathscr {A}},{\mathscr {B}})\) with the vertex sets \(V_2=\{u_iv_1,u_iv_2,\ldots ,u_iv_n\}\) for \(i=1, 2, \ldots , m.\) Each of these subgraphs of \({\mathscr {G}}=({\mathscr {A}},{\mathscr {B}})\) is connected as \(u_i's\) are same.
Since \({\mathscr {G}}_1=({\mathscr {A}}_1,{\mathscr {B}}_1)\) and \({\mathscr {G}}_2=({\mathscr {A}}_2,{\mathscr {B}}_2)\) are connected, each \(u_i\) and \(v_i\) are adjacent to at least one of the vertices in \(V_1\) and \(V_2\). Therefore, there exists at least one edge between any pair of the above k subgraphs.
Remark 3.2
By routine calculation, one can see from Fig. 20 that \({\mathscr {G}}_1\) and \({\mathscr {G}}_2\) are complete PFGs. While notice that \({\mathscr {G}}_1 \cdot {\mathscr {G}}_2\) is not a complete PFG as the only case \(u_1u_2 \in E_1,\) is included in the definition of the residue product.
Remark 3.3
If \({\mathscr {G}}_1=({\mathscr {A}}_1,{\mathscr {B}}_1)\) and \({\mathscr {G}}_2=({\mathscr {A}}_2,{\mathscr {B}}_2)\) are two regular PFGs, then their residue product \({\mathscr {G}}_1 \cdot {\mathscr {G}}_2\) may not to be regular PFG as it is explained in this example. Consider two PFGs \({{\mathscr {G}}_1}=({{\mathscr {A}}_1},{{\mathscr {B}}_1})\) and \({{\mathscr {G}}_2}=({{\mathscr {A}}_2},{{\mathscr {B}}_2})\) on \(V_1=\{a,b\}\) and \(V_2=\{c,d\}\), respectively, as shown in Fig. 22. Their residue product \({{\mathscr {G}}_1}~\cdot ~{{\mathscr {G}}_2}\) is shown in Fig. 23.
By routine calculation, one can see from Fig. 22 that \({\mathscr {G}}_1\) and \({\mathscr {G}}_2\) are regular PFGs. While notice that \((d)_{{\mathscr {G}}_1\cdot {\mathscr {G}}_2}(a,d)\ne (d)_{{\mathscr {G}}_1*{\mathscr {G}}_2}(a,c)\). Therefore, \({{\mathscr {G}}_1}~\cdot ~{{\mathscr {G}}_2}\) is not regular PFG.
The following theorems explain the conditions for the residue product of two regular PFGs to be regular.
Theorem 3.3
The residue product \({\mathscr {G}}_1 \cdot {\mathscr {G}}_1\) of any PFG \({\mathscr {G}}_1=({\mathscr {A}}_1,{\mathscr {B}}_1)\) with a PFG \({\mathscr {G}}_2=({\mathscr {A}}_2,{\mathscr {B}}_2)\) such that \(V_2=1\), is always a PFG with no edge.
Proof
The proof follows from the Definition 3.1. \(\square \)
Theorem 3.4
If \({\mathscr {G}}_1=({\mathscr {A}}_1,{\mathscr {B}}_1)\) and \({\mathscr {G}}_2=({\mathscr {A}}_2,{\mathscr {B}}_2)\) are two PFGs such that \(V_2>1\), then their residue product is regular if and only if \({\mathscr {G}}_1\) is regular.
Proof
Let \({\mathscr {G}}_1=({\mathscr {A}}_1,{\mathscr {B}}_1)\) be a (k, l)regular PFG and \({\mathscr {G}}_2=({\mathscr {A}}_2,{\mathscr {B}}_2)\) be any PFG with \(V_2>1\).
Example 3.1
Since \((d)_{{\mathscr {G}}_1}(u_i)~=~(0.3,0.7)\) for i=1,2 and \((d)_{{\mathscr {G}}_2}(v_1) \ne (d)_{{\mathscr {G}}_2}(v_2)\), therefore \({\mathscr {G}}_1~=~({\mathscr {A}}_1,{\mathscr {B}}_1)\) is a regular PFG and \({\mathscr {G}}_2~=~({\mathscr {A}}_2,{\mathscr {B}}_2)\) is not regular PFG with \(V_2>1\). Now, \((d)_{{\mathscr {G}}_1 \cdot {\mathscr {G}}_2}(u_i,v_j)~=~(0.6,1.4)\) for \(i=1,2\) and \(j=1,2,3.\) Thus, the residue product \({{\mathscr {G}}_1 \cdot {\mathscr {G}}_2}\) is a regular PFG.
Theorem 3.5
Proof
Definition 3.2
Example 3.2
Theorem 3.6
If \({\mathscr {G}}_1~=~({\mathscr {A}}_1,{\mathscr {B}}_1)\) is a totally regular PFG and \({\mathscr {G}}_2~=~({\mathscr {A}}_2,{\mathscr {B}}_2)\) is a PFG such that \(\mu _{{\mathscr {A}}_1}\ge \mu _{{\mathscr {A}}_2}\), \(\nu _{{\mathscr {A}}_1}\le \nu _{{\mathscr {A}}_2}\) and \( V_2  > 1\), then the residue product is totally regular PFG.
Proof
Applications to decisionmaking
In this section, we apply the concept of PFGs to decisionmaking problems. Two decisionmaking problems concerning the ‘evaluation of best company for investment’ and ‘alliance partner selection of a software company’ are solved to illustrate the applicability of the proposed concept of PFGs in realistic scenario based on Pythagorean fuzzy preference relations (PFPRs) [31]. The algorithm for the alliance partner selection of a software company within the framework of PFPR is outlined in Algorithm 1.
Evaluation of best company for investment
PFPR of the decisionmaker
R  \(x_{1}\)  \(x_{2}\)  \(x_{3}\)  \(x_{4}\)  \(x_{5}\)  \(x_{6}\) 

\(x_{1}\)  (0.5, 0.5)  (0.4, 0.8)  (0.3, 0.6)  (0.7, 0.2)  (0.8, 0.6)  (0.1, 0.8) 
\(x_{2}\)  (0.8, 0.4)  (0.5, 0.5)  (0.9, 0.4)  (0.5, 0.8)  (0.6, 0.8)  (0.5, 0.7) 
\(x_{3}\)  (0.6, 0.3)  (0.4, 0.9)  (0.5, 0.5)  (0.7, 0.6)  (0.5, 0.8)  (0.9, 0.3) 
\(x_{4}\)  (0.2, 0.7)  (0.8, 0.5)  (0.6, 0.7)  (0.5, 0.5)  (0.9, 0.3)  (0.5, 0.6) 
\(x_{5}\)  (0.6, 0.8)  (0.8, 0.6)  (0.8, 0.5)  (0.3, 0.9)  (0.5, 0.5)  (0.8, 0.5) 
\(x_{6}\)  (0.8, 0.1)  (0.7, 0.5)  (0.3, 0.9)  (0.6, 0.5)  (0.5, 0.8)  (0.5, 0.5) 
Alliance partner selection of a software company
PFPR of the expert from the business process outsourcing department
\(R_{1}\)  \(x_{1}\)  \(x_{2}\)  \(x_{3}\)  \(x_{4}\)  \(x_{5}\) 

\(x_{1}\)  (0.5, 0.5)  (0.5, 0.7)  (0.8, 0.4)  (0.7, 0.6)  (0.3, 0.6) 
\(x_{2}\)  (0.7, 0.5)  (0.5, 0.5)  (0.2, 0.9)  (0.7, 0.5)  (0.8, 0.3) 
\(x_{3}\)  (0.4, 0.8)  (0.9, 0.2)  (0.5, 0.5)  (0.2, 0.6)  (0.9, 0.2) 
\(x_{4}\)  (0.6, 0.7)  (0.5, 0.7)  (0.6, 0.2)  (0.5, 0.5)  (0.3, 0.7) 
\(x_{5}\)  (0.6, 0.3)  (0.3, 0.8)  (0.2, 0.9)  (0.7, 0.3)  (0.5, 0.5) 
PFPR of the expert from the operation management department
\(R_{2}\)  \(x_{1}\)  \(x_{2}\)  \(x_{3}\)  \(x_{4}\)  \(x_{5}\) 

\(x_{1}\)  (0.5, 0.5)  (0.9, 0.3)  (0.7, 0.2)  (0.3, 0.8)  (0.5, 0.8) 
\(x_{2}\)  (0.3, 0.9)  (0.5, 0.5)  (0.6, 0.7)  (0.1, 0.5)  (0.8, 0.6) 
\(x_{3}\)  (0.2, 0.7)  (0.7, 0.6)  (0.5, 0.5)  (0.7, 0.5)  (0.3, 0.7) 
\(x_{4}\)  (0.8, 0.3)  (0.5, 0.1)  (0.5, 0.7)  (0.5, 0.5)  (0.8, 0.3) 
\(x_{5}\)  (0.8, 0.5)  (0.6, 0.8)  (0.7, 0.3)  (0.3, 0.8)  (0.5, 0.5) 
PFPR of the expert from the engineering management department
\(R_{3}\)  \(x_{1}\)  \(x_{2}\)  \(x_{3}\)  \(x_{4}\)  \(x_{5}\) 

\(x_{1}\)  (0.5, 0.5)  (0.7, 0.6)  (0.5, 0.8)  (0.3, 0.9)  (0.7, 0.6) 
\(x_{2}\)  (0.6, 0.7)  (0.5, 0.5)  (0.8, 0.6)  (0.1, 0.7)  (0.3, 0.8) 
\(x_{3}\)  (0.8, 0.5)  (0.6, 0.8)  (0.5, 0.5)  (0.4, 0.8)  (0.5, 0.7) 
\(x_{4}\)  (0.9, 0.3)  (0.7, 0.1)  (0.8, 0.4)  (0.5, 0.5)  (0.8, 0.4) 
\(x_{5}\)  (0.6, 0.7)  (0.8, 0.3)  (0.7, 0.5)  (0.4, 0.8)  (0.5, 0.5) 
The decision results of the alternatives using the different methods
Methods  Score of alternatives  Ranking of alternatives 

Zhao et al. [40]  − 0.6800 −2.1800 − 0.9400 3.8200 − 0.0200  \({A_{4} \succ A_{5} \succ A_{1} \succ A_{3} \succ A_{2}}\) 
Our proposed method  0.3367 0.2276 0.3276 0.6067 0.3595  \({A_{4} \succ A_{5} \succ A_{1} \succ A_{3} \succ A_{2}}\) 
 \(m_{1}:\)

\(p^{(1)}_{1}=(0.6139,0.5502)\), \(p^{(1)}_{2}=(0.6457,0.5078)\), \(p^{(1)}_{3}=(0.7312, 0.3949)\), \(p^{(1)}_{4}=(0.5180,0.5094)\), \(p^{(1)}_{5}=(0.5152,0.5036)\);
 \(m_{2}:\)

\(p^{(2)}_{1}=(0.6720,0.4536)\), \(p^{(2)}_{2}=(0.5574,0.6239)\), \(p^{(2)}_{3}=(0.5459,0.5933)\), \(p^{(2)}_{4}=(0.6639,0.3160)\), \(p^{(2)}_{5}=(0.6295,0.5448)\);
 \(m_{3}:\)

\(p^{(3)}_{1}=(0.5761,0.6645)\), \(p^{(3)}_{2}=(0.5574,0.6518)\), \(p^{(3)}_{3}=(0.5985,0.6454)\), \(p^{(3)}_{4}=(0.7789,0.2993)\), \(p^{(3)}_{5}=(0.6371,0.5305)\).
Comparative Analysis: We individually compute and compare the decision results using the method of Ref. [40] and our proposed method. They are shown in Table 5. From Table 5, the decision results of the method of Ref. [40] are consistent with our proposed method. It implies that our proposed method considers all the Pythagorean fuzzy evaluation information.
Conclusions and further work
Graph theory has vast range of applications in solving various networking problems encountered in different fields such as signal processing, transportation and errorcorrecting codes. For modeling the obscurity and uncertainties in practical decisionmaking and graphical networking problems, Pythagorean fuzzy graphs (PFGs) have better ability due to the increment of spaces in membership and nonmembership grades. This paper has introduced two new graph products specifically, maximal product and residue product, for uniting two Pythagorean fuzzy graphs. Using these products, different types of structural models can be combined to produce a better one. They may be useful for the configuration processing of space structures. Further, the paper has explored some crucial properties like strongness, connectedness, completeness and regularity. A special focus on the regularity has been given as it can be applied widely in designing reliable communication and computer networks. Applications of PFGs in decisionmaking concerning evaluation of best company for investment and alliance partner selection of a software company have been presented.
Further work We are working to extend our study to : (1) Pythagorean fuzzy soft graphs; (2) Rough Pythagorean fuzzy graphs; (3) Simplified intervalvalued Pythagorean fuzzy graphs and; (4) Hesitant Pythagorean fuzzy graphs
Notes
Acknowledgements
The authors are highly thankful to Associate Editor anonymous referees for their valuable comments and suggestions for improving the paper.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest regarding the publication of the research article.
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