Investigating the Short-Circuit Problem Using the Planarity Index of Complex q-Rung Orthopair Fuzzy Planar Graphs

Planar graphs play an effective role in many practical applications where the crossing of edges becomes problematic. This paper aims to investigate the complex q-rung orthopair fuzzy (CQROF) planar graphs (CQROFPGs). In a CQROFPG, the nodes and edges are based on complex QROF information that represents the uncertain knowledge in the range of unit circles in terms of complex numbers. The motivation in discussing such a topic is the wide flexibility of QROF information in the expression of uncertain knowledge compared to intuitionistic and Pythagorean fuzzy settings. We discussed the complex QROF graphs (CQROFGs), complex QROF multigraphs (CQROFMGs), and related terms followed by examples. Furthermore, the notion of strength and planarity index (PI) of the CQROFPGs is defined and exemplified followed by a study of strong and weak edges. We further defined the notion of complex QROF face (CQROFF) and complex QROF dual graph (CQROFDG) and exemplified these concepts. A study of isomorphism, coweak and weak isomorphism, is set up, and some results relating to the CQROFPG and isomorphisms are explored using examples. Furthermore, the problem of short circuits that results due to crossing is discussed because of the proposed study where an algorithm based on complex QROF (CQROF) information is presented for reducing the crossing in networks. Some advantages of the projected study over the previous study are observed, and some future study is predicted.

A study on regular picture fuzzy graph with applications in communication networks

Picture fuzzy graph (PFG) is an extended version of intuitionistic fuzzy graph (IFG) to model the uncertain real world problems, in which IFG may fail to model those problems properly. PFG is more precise, flexible and compatible than IFG to deal the real-life scenarios which consists of information these types: yes, abstain, no and refusal. The main focus of our study is to present the concept of isomorphic PFG, regular PFG (RPFG) and picture fuzzy multigraph. In this paper, we present the notation of RPFG. Many different types of RPFGs such as regular strong PFG, regular complete PFG, complete bipartite PFG and regular complement PFG are introduced. We also describe the concepts of dn and tdn-degree of a vertex in a RPFG. Based on those two types of degrees, we classify the regularity of PFG into 3 type’s namely, dn- RPFG, tdn-RPFG and n- highly irregular PFG. Several theorems of those RPFG are presented here. We define the busy vertex and free vertex in a RPFG. We present the notations of μ-complement, homomorphism, isomorphism, weak isomorphism and co weak isomorphism of RPFG. Some significant theorems on isomorphism and μ- complement of RPFG are derived here. We also introduce the notation of picture fuzzy multigraph. We present a mathematical model of communication network and transportation network by using picture fuzzy multigraph and real time data are collected so that the transportation network/communication network can work efficiently.

Planar Graphs under Pythagorean Fuzzy Environment

Graph theory plays a substantial role in structuring and designing many problems. A number of structural designs with crossings can be found in real world scenarios. To model the vagueness and uncertainty in graphical network problems, many extensions of graph theoretical ideas are introduced. To deal with such uncertain situations, the present paper proposes the concept of Pythagorean fuzzy multigraphs and Pythagorean fuzzy planar graphs with some of their eminent characteristics by investigating Pythagorean fuzzy planarity value with strong, weak and considerable edges. A close association is developed between Pythagorean fuzzy planar and dual graphs. This paper also includes a brief discussion on non-planar Pythagorean fuzzy graphs and explores the concepts of isomorphism, weak isomorphism and co-weak isomorphism for Pythagorean fuzzy planar graphs. Moreover, it presents a problem that shows applicability of the proposed concept.

Extending abelian groups to rings

For any abelian group G and any function f: G → G we define a commutative binary operation or ‘multiplication’ on G in terms of f. We give necessary and sufficient conditions on f for G to extend to a commutative ring with the new multiplication. In the case where G is an elementary abelian p–group of odd order, we classify those functions which extend G to a ring and show, under an equivalence relation we call weak isomorphism, that there are precisely six distinct classes of rings constructed using this method with additive group the elementary abelian p–group of odd order p2.