- Hi!I'm Dr. Shahbaz Ali working in the department of Mathematics, the Islamia University of Bahawalpur, Rahim Yar Khan... moreHi!I'm Dr. Shahbaz Ali working in the department of Mathematics, the Islamia University of Bahawalpur, Rahim Yar Khan Campus, Pakistan. I have completed my Ph.D in the field of number theory and graph theory. I have published more than 45 research articles in well reputated journals. Best regards,Dr. Shahbaz Aliedit
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The term metric or distance of a graph plays a vital role in the study to check the structural properties of the networks such as complexity, modularity, centrality, accessibility, connectivity, robustness, clustering, and vulnerability.... more
The term metric or distance of a graph plays a vital role in the study to check the structural properties of the networks such as complexity, modularity, centrality, accessibility, connectivity, robustness, clustering, and vulnerability. In particular, various metrics or distance-based dimensions of different kinds of networks are used to resolve the problems in different strata such as in security to find a suitable place for fixing sensors for security purposes. In the field of computer science, metric dimensions are most useful in aspects such as image processing, navigation, pattern recognition, and integer programming problem. Also, metric dimensions play a vital role in the field of chemical engineering, for example, the problem of drug discovery and the formation of different chemical compounds are resolved by means of some suitable metric dimension algorithm. In this paper, we take rotationally symmetric and hexagonal planar networks with all possible faces. We find the sequ...
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Polygroups are an extended form of groups and a subclass of hypergroups that follow group-type axioms. In this paper, we define a triplet single-valued neutrosophic set, which is a generalization of fuzzy sets, intuitionistic fuzzy sets,... more
Polygroups are an extended form of groups and a subclass of hypergroups that follow group-type axioms. In this paper, we define a triplet single-valued neutrosophic set, which is a generalization of fuzzy sets, intuitionistic fuzzy sets, and neutrosophic sets, and we combine this novel concept with hypergroups and polygroups. Firstly, the main goal of this paper is to introduce hypergroups, polygroups, and anti-polygroups under a triplet single-valued neutrosophic structure and then present various profound results. We also examine the interaction and properties of level sets of triplet single-valued neutrosophic polygroups and (normal) subpolygroups. Secondly, we rank the alternatives and select the best ones in a single-valued neutrosophic environment using the weighted cosine similarity measure between each alternative and the ideal alternative. Finally, we provide an example that clearly shows how the proposed decision-making method is applied.
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In order to make quantitative structure-movement/property/danger relations, topological indices (TIs) are the numbers that are related to subatomic graphs. Some fundamental physicochemical properties of chemical compounds, such as... more
In order to make quantitative structure-movement/property/danger relations, topological indices (TIs) are the numbers that are related to subatomic graphs. Some fundamental physicochemical properties of chemical compounds, such as breaking point, protection, and strain vitality, correspond to these TIs. In the compound graph hypothesis, the concept of TIs was developed in view of the degree of vertices. In investigating minimizing exercises of Star of David, these indices are useful. In this study, we explore the different types of Zagreb indices, Randić indices, atom-bond connectivity indices, redefined Zagreb indices, and geometric-arithmetic index for the Star of David. The edge partitions of this network are tabled based on the sum of degrees-of-end vertices and the sum of degree-based edges. To produce closed formulas for some degree-based network TIs, these edge partitions are employed.
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In order to make quantitative structure-movement/property/danger relations, topological indices (TIs) are the numbers that are related to subatomic graphs. Some fundamental physicochemical properties of chemical compounds, such as... more
In order to make quantitative structure-movement/property/danger relations, topological indices (TIs) are the numbers that are related to subatomic graphs. Some fundamental physicochemical properties of chemical compounds, such as breaking point, protection, and strain vitality, correspond to these TIs. In the compound graph hypothesis, the concept of TIs was developed in view of the degree of vertices. In investigating minimizing exercises of Star of David, these indices are useful. In this study, we explore the different types of Zagreb indices, Randić indices, atom-bond connectivity indices, redefined Zagreb indices, and geometric-arithmetic index for the Star of David. The edge partitions of this network are tabled based on the sum of degrees-of-end vertices and the sum of degree-based edges. To produce closed formulas for some degree-based network TIs, these edge partitions are employed.
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Research Interests:
In this paper, we investigate two new Dombi aggregation operators on bipolar neutrosophic set namely bipolar neutrosophic Dombi prioritized weighted geometric aggregation (BNDPWGA) and bipolar neutrosophic Dombi prioritized ordered... more
In this paper, we investigate two new Dombi aggregation operators on bipolar neutrosophic set namely bipolar neutrosophic Dombi prioritized weighted geometric aggregation (BNDPWGA) and bipolar neutrosophic Dombi prioritized ordered weighted geometric aggregation (BNDPOWGA) by means of Dombi t-norm (TN) and Dombi t-conorm (TCN). We discuss their properties along with proofs and multi-attribute decision making (MADM) methods in detail. New algorithms based on proposed models are presented to solve multi-attribute decision-making (MADM) problems. In contrast, with existing techniques a comparison analysis of proposed methods are also demonstrated to test their validity, accuracy and significance.
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Introduction: In this paper, we present a novel hybrid model m-polar Diophantine fuzzy N-soft set and define operations on it. Methods: We generalize the concepts of fuzzy sets, soft sets, N-soft sets, fuzzy soft sets, intuitionistic... more
Introduction: In this paper, we present a novel hybrid model m-polar Diophantine fuzzy N-soft set and define operations on it. Methods: We generalize the concepts of fuzzy sets, soft sets, N-soft sets, fuzzy soft sets, intuitionistic fuzzy sets, intuitionistic fuzzy soft sets, Pythagorean fuzzy sets, Pythagorean fuzzy soft sets and Pythagorean fuzzy N-soft sets by incorporating our proposed model. Additionally, we define three different sorts of complements for Pythagorean fuzzy Nsoft sets and examine few outcomes which do not hold in Pythagorean fuzzy N-soft sets complements unlike to crisp set. We further discuss about (α, β, γ) -cut of m-polar Diophantine fuzzy N-soft sets and their properties. Lastly, we prove our claim that the defined model is a generalization of soft set, N-soft set, fuzzy N-soft set, intuitionistic fuzzy N soft set and Pythagorean fuzzy N-soft set. Results: m-polar Diophantine fuzzy N-soft set is more efficient and an adaptable model to manage uncertainties ...
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Research Interests:
Adding new classes of integers to literature is both challenging and charming. Until a new class is completely characterized, mathematics is never going to be worth it. While it's absurd to play with integers without intended... more
Adding new classes of integers to literature is both challenging and charming. Until a new class is completely characterized, mathematics is never going to be worth it. While it's absurd to play with integers without intended consequences. In this work, we introduce and investigate four new classes of integers namely, anti-totient numbers, half anti-totient numbers, near Zumkeller numbers and half near Zumkeller numbers by using the notion of non-coprime residues of $n$ including $n$. We formulate and propose relations of these new classes of numbers with previous well-known numbers such as perfect, totient, triangular, pentagonal, and hexagonal numbers. These new classes of integers have been completely characterized. Finally, as an application of these new classes of numbers, a new graph labeling is also proposed on anti-totient numbers.
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A positive integern is called super totient if the residues of n which are prime ton can be partitioned into two disjoint subsets of equal sums. LetG be a given graph withV, the set of vertices and E is the set of its edges. An injective... more
A positive integern is called super totient if the residues of n which are prime ton can be partitioned into two disjoint subsets of equal sums. LetG be a given graph withV, the set of vertices and E is the set of its edges. An injective function g defined onV into subset of integers will be termed as super totient labeling of the graph G, if the functiong∗ : E → N defined byg∗(xy) = g(x)g(y) assigns a super totient number for all edgesxy ∈ E, wherex, y ∈ V. A graph admits this labeling is called a super totient graph. In the current manuscript, the authors investigate a novel labeling algorithm, called super totient labeling, for several classes of graphs such as friendship graphs, wheel graphs, complete graphs and complete bipartite graphs. AMS (MOS) Mathematics subject classification (2000): 05C25, 11E04, 20G15
For any positive integer $m, ~\varphi(m)$ find out how many residues of $m$ thats are co-prime to $m$, where $\varphi$ is theEuler's totient function. In this work, we introduce the notion oftotient, super totient and hyper totient... more
For any positive integer $m, ~\varphi(m)$ find out how many residues of $m$ thats are co-prime to $m$, where $\varphi$ is theEuler's totient function. In this work, we introduce the notion oftotient, super totient and hyper totient numbers and discuss their relations. Many postulates and characterizations of these numbers have been proposed with straight forward proofs. Finally, applications of these numbers in graph labeling have been demonstrated over a family of well known graph.}}
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In this study, we investigate two graphs, one of which has units of a ring Z n as vertices (or nodes) and an edge will be built between two vertices u and v if and only if u 3 ≡ v 3 mod n . This graph will be termed as cubic residue... more
In this study, we investigate two graphs, one of which has units of a ring Z n as vertices (or nodes) and an edge will be built between two vertices u and v if and only if u 3 ≡ v 3 mod n . This graph will be termed as cubic residue graph. While the other is called Gaussian quadratic residue graph whose vertices are the elements of a Gaussian ring Z n i of the form α = a + i b , β = c + i d , where a , b , c , d are the units of Z n . Two vertices α and β are adjacent to each other if and only if α 2 ≡ β 2 mod n . In this piece of work, we characterize cubic and Gaussian quadratic residue graphs for each positive integer n in terms of complete graphs.
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In this paper, a new family of rotationally symmetric planar graphs is described based on an edge coalescence of planar chorded cycles. Their local fractional metric dimension is established for those ones arisen from chorded cycles of... more
In this paper, a new family of rotationally symmetric planar graphs is described based on an edge coalescence of planar chorded cycles. Their local fractional metric dimension is established for those ones arisen from chorded cycles of order up to six. Their asymptotic behaviour enables us to ensure the existence of new families of rotationally symmetric planar graphs with either constant or bounded local fractional dimension.
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The understanding of a computer based software or self-designed tool is extremely difficult to analyze without some appropriate mathematics. In any algorithm, it is always possible to visualize its core steps with the help of a flow chart... more
The understanding of a computer based software or self-designed tool is extremely difficult to analyze without some appropriate mathematics. In any algorithm, it is always possible to visualize its core steps with the help of a flow chart or a graph. Thus, drawing graphs and digraphs of any structure based algorithm or computer oriented outputs is much more helpful in achieving a better understanding. In discrete mathematics, digraphs based on modular arithmetic are becoming a core interest of many computer scientists and number theorist. As these digraphs are easy to label using the residues of any given integer. Thus, algorithms based on integral mathematics are easy to evaluate with the help of a power digraph. A power digraph can be constructed from the congruence equation $a^{m}\equiv b(\text{mod}\ n)$, where $a, b, \in Z_{n}$. In this note, we highlight some fields of interest where power digraphs can be utilized in a more friendly way to entertain a layman without knowing muc...
In this research article, we motivate and introduce the concept of possibility belief interval-valued N-soft sets. It has a great significance for enhancing the performance of decision-making procedures in many theories of uncertainty.... more
In this research article, we motivate and introduce the concept of possibility belief interval-valued N-soft sets. It has a great significance for enhancing the performance of decision-making procedures in many theories of uncertainty. The N-soft set theory is arising as an effective mathematical tool for dealing with precision and uncertainties more than the soft set theory. In this regard, we extend the concept of belief interval-valued soft set to possibility belief interval-valued N-soft set (by accumulating possibility and belief interval with N-soft set), and we also explain its practical calculations. To this objective, we defined related theoretical notions, for example, belief interval-valued N-soft set, possibility belief interval-valued N-soft set, their algebraic operations, and examined some of their fundamental properties. Furthermore, we developed two algorithms by using max-AND and min-OR operations of possibility belief interval-valued N-soft set for decision-making...
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In this paper, we investigate two new Dombi aggregation operators on bipolar neutrosophic set namely bipolar neutrosophic Dombi prioritized weighted geometric aggregation (BNDPWGA) and bipolar neutrosophic Dombi prioritized ordered... more
In this paper, we investigate two new Dombi aggregation operators on bipolar neutrosophic set namely bipolar neutrosophic Dombi prioritized weighted geometric aggregation (BNDPWGA) and bipolar neutrosophic Dombi prioritized ordered weighted geometric aggregation (BNDPOWGA) by means of Dombi t-norm (TN) and Dombi t-conorm (TCN). We discuss their properties along with proofs and multi-attribute decision making (MADM) methods in detail. New algorithms based on proposed models are presented to solve multi-attribute decision-making (MADM) problems. In contrast, with existing techniques a comparison analysis of proposed methods are also demonstrated to test their validity, accuracy and significance.
Research Interests:
In this paper, we establish some theorems of fixed point on multivalued mappings satisfying contraction mapping by using gauge function. Furthermore, we use Q - and R -order of convergence. Our main results extend many previous existing... more
In this paper, we establish some theorems of fixed point on multivalued mappings satisfying contraction mapping by using gauge function. Furthermore, we use Q - and R -order of convergence. Our main results extend many previous existing results in the literature. Consequently, to substantiate the validity of proposed method, we give its application in integral inclusion.