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Symmetry properties are of vital importance for graphs. The famous Cayley graph is a good mathematical model as its high symmetry. The normality of the graph can well reflect the symmetry of the graph. In this paper, we characterize the normality of the direct product of Cayley graphs and give a sufficient and necessary condition for the direct product of two graphs to be a Cayley graph. Moreover, a sufficient and necessary condition for judging the normality of the direct product of two graphs is given. [ABSTRACT FROM AUTHOR]

In this research, we introduce the notion of an L -soft and investigate several universal properties of it. Then, the family of L -softs is established and denoted by Soft (). So, we tried to build an algebraic structure on Soft (). First, we showed that Soft () with sub- L operation is a poset. Also, we proved that Soft () with some operations made a complete distributive bounded lattice, bounded hoop, bounded L-algebra, CKL-algebra, BCK-algebra, commutative integral quantum B-algebra and Boolean algebra. Finally, we presented a congruence relation on Soft () and we investigated the quotient of it. [ABSTRACT FROM AUTHOR]

Humanity's efforts are manifested in the creation of novel solutions to complex problems in diverse fields. Traditional mathematical methods fail to solve real-world problems due to their complexity. Researchers have come up with new mathematical theories like fuzzy set theory and rough set theory to help them figure out how to model the uncertainty in these fields. Soft set theory is a novel approach to real-world problem solving that does not require the membership function to be specified. This aids in the resolution of a wide range of issues, and significant progress has recently been made. After Jun et al. came up with a hybrid system that combined fuzzy and soft set concepts, many people came up with hybrid ideas in different algebraic structures. In this paper, we introduce the concepts of subsystem and strong subsystem of a hybrid finite state machine (HFSM) and investigate a portion of their significant properties. We also provide an example that shows that every subsystem does not need to be a strong subsystem. Additionally, we study the cyclic subsystem of HFSMs and also obtain their equivalent results and examples. Finally, we define the notions of homomorphism of subsystems and strong subsystems of HFSMs. [ABSTRACT FROM AUTHOR]

Differential geometry of a kind of helical hypersurface family that depends on six parameters within the seven‐dimensional Euclidean space is explored. The curvatures of these hypersurfaces are determined, and their minimality is examined, along with illustrative examples being provided. Lastly, the Laplace–Beltrami operator for that kind of hypersurfaces is calculated. [ABSTRACT FROM AUTHOR]

The single‐valued neutrosophic set (SVNS) is a more common platform for representing the degree of truth, indeterminacy, and falsity memberships. Indeterminacy is an important aspect of SVNS, and the score function (SF) is important in ranking alternatives in decision‐making scenarios. When the score and accuracy values for different SVNNs were equal, existing SFs were unable to rank the alternatives. To address the limitations, this paper introduces the perfect score function (PSF) as a solution for ranking single‐valued neutrosophic numbers (SVNNs) without any errors and presents the concepts of strong and weak SVNSs. This article has emphasized the drawbacks of the existing SF in SVNNs. Additionally, we propose an optimized weighted geometric operator (OWGO) for SVNNs, and its properties such as idempotency, boundedness, monotonicity, and commutativity are explored. Furthermore, the decision‐making approach has been depicted based on the proposed score function and OWGO to select the finest faculty for a university. [ABSTRACT FROM AUTHOR]

An essential and quite different class of functions is the triangular norm and its related operators (uninorms, nullnorms, and associative copulas). They are used widely in many disciplines, including fuzzy set theory, probability and statistics, decision sciences, and others. This paper proposes the notion of hybrid Ξ -norm and defines the concepts of Ξ -hybrid ideals, Ξ -hybrid h -ideals in a hemiring . Some equivalent conditions are obtained for a hybrid structure to be a Ξ -hybrid left h -ideal, and it is proved that every imaginable Ξ -hybrid left h -ideal of hemiring is a hybrid left h -ideal, but not conversely by an example. [ABSTRACT FROM AUTHOR]

Spectral graph theory explores the relationship between the eigenvalues and the structural properties of graphs. This study focuses on the fundamental matrices associated with simple graphs, specifically the adjacency matrix. This work contributes to the spectral graph theory literature by introducing a new family of simple connected graphs and investigating their eigenvalue properties. We examine conditions under which these graphs satisfy specific spectral properties, such as the strong antireciprocal eigenvalue property. Additionally, we construct another family of graphs from this new family, proving that each graph within this derived family satisfies the strong antireciprocal eigenvalue property. Through these contributions, we aim to deepen the understanding of the spectral characteristics of graph families and their applications in theoretical and applied contexts. [ABSTRACT FROM AUTHOR]

The notion of θ-generalized int-soft subfields, θ-generalized int-soft algebras over θ-generalized int-soft subfields, and θ-generalized int-soft hypervector spaces are introduced, and their properties and characterizations are considered. In connection with linear transformations, θ-generalized int-soft hypervector spaces are discussed.

In a right regular L A -semigroup, the hybrid ideals, hybrid interior ideals, hybrid (1,2)-ideals, hybrid generalized bi-ideals, and hybrid quasi-ideals are all introduced in this work. A right regular L A -semigroup has also been described in terms of its hybrid right and hybrid left ideals. Moreover, we obtain some equivalent conditions in terms of hybrid two-sided ideal, hybrid bi- (respectively, generalized bi-)ideal, hybrid quasi-ideal, hybrid interior ideal and hybrid (1 , 2) -ideal over right regular L A -semigroups. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of (∈, ∈ ∨(κ⋇, q...))-anti-fuzzy b-ideal (AFBI) in BCI-algebra and explores its properties. It can be extended the notion of(∈ ∨(... ⋇, q...), ∈ ∨(... ⋇, q...))-AFBI is then presented and related properties are explored. We discuss the relations between (∈, ∈ ∨(... ⋇, q...))-AFBI and (∈ ∨(... ⋇, q...), ∈ ∨(... ⋇, q...))-AFBI. Furthermore, the concept of (∈, ∈ ∨(... ⋇, q...))-level subset applied to anti-fuzzy b-ideal. [ABSTRACT FROM AUTHOR]

One of the most critical concepts in the study of groups is the notion of nilpotency. Nilpotent groups arise in Galois theory, as well as in the classification of groups. In this paper, we apply the notion of nilpotency on B C K -algebras by using fuzzy ideals. First we define the concept of nilpotent fuzzy ideal in B C K -algebras and investigate their properties. Furthermore, we find some relation between fuzzy nilpotent ideals and quotient B C K -algebras. Specially, we construct a nilpotent B C K -algebra by using a fuzzy nilpotent ideal. Finally, we prove that any minimal ideals of a B C K -algebra X , is contained in the center of any fuzzy nilpotent ideals of X. [ABSTRACT FROM AUTHOR]

In this paper, the notion of closed cubic intuitionistic ideals, cubic intuitionistic p-ideals and cubic intuitionistic a-ideals in BCI-algebras are introduced, and several related properties are investigated. Relations between cubic intuitionistic subalgebras, closed cubic intuitionistic ideals, cubic intuitionistic q-ideals, cubic intuitionistic p-ideals and cubic intuitionistic a-ideals are discussed. Conditions for a cubic intuitionistic ideal to be a cubic intuitionistic p-ideal are provided. Characterizations of a cubic intuitionistic a-ideal are considered. The cubic intuitionistic extension property for a cubic intuitionistic a-ideal is established.

In this paper, we apply the m -polar cubic structures on BCI-algebras. We first present the ideas of m -polar cubic p -ideals, m -polar cubic q -ideals and m -polar cubic a -ideals in BCI-algebras. Then we prove that m -polar cubic p (q and a)-ideals are m -polar cubic ideals, but the converse assertion is not true, and an example is provided in this support. We define the conditions under which m -polar cubic ideals coincide with m -polar cubic p (q and a)-ideals. The associated properties of m -polar cubic p -ideals, m -polar cubic q -ideals, and m -polar cubic a -ideals are also examined. Furthermore, we consider m -polar cubic p (q and a)-ideals in terms of p (q and a)-ideals of BCI-algebras. [ABSTRACT FROM AUTHOR]

In this paper, we use IVI-octahedron ideals and IVI-octahedron bi-ideals to discuss some characterizations of special semigroups, e.g., an intra-regular semigroup, a semigroup which is semilattice of left [respectively, right] simple semigroups or simple left groups and a semigroup which is a semilattice of groups and a [respectively, left and right] simple semigroups. Finally, we define an IVIO-simple [IVIOL-simple and IVI) R-simple] semigroup and study some of its properties and we give a characterization of a semisimple semigroup by IVIOIs. [ABSTRACT FROM AUTHOR]

Our aim is to explore the idea of neutrosophic N−ideals in near-subtraction semigroups in this article and obtain some outcomes that are equivalent to them. We also illustrate the notion of a neutrosophic κ− intersection. Additionally, in a near-subtraction semigroup, we examine the term homomorphism of a neutrosophic κ− structure and establish some conclusions based on a homomorphic neutrosophic κ− structure preimage of a neutrosophic κ− left (respectively, right) ideal. [ABSTRACT FROM AUTHOR]

An AG-groupoid is the midway between commutative semigroup and groupoid. The core structure of Flock theory is an AG-groupoid, which focuses on motion replication and distance optimization and has numerous applications in physics and biology. Unfortunately, in many cases, modelling real-world problems in domains like computer science, operations research, artificial intelligence, control engineering, and robotics can be risky. Different theories, such as fuzzy sets, intuitionistic fuzzy sets, probability, soft sets, neutrosophic sets, and others, have been created to deal with similar situations. In this paper, We define the notions of neutrosophic κ-ideal structures in an AG-groupoid and investigate their properties. We also obtain equivalent assertion of neutrosophic κ-ideals and product of neutrosophic κ-structures in AG-groupoid. [ABSTRACT FROM AUTHOR]

Fuzzy sets, soft sets, and their generalisations have always been important tools for mathematicians and researchers working with uncertainty. Jun proposed a hybrid structure that combined the concepts of a fuzzy set and a soft set. It should be noted that hybrid structures are a combination of soft set and fuzzy set speculation. Our aim is to explore the concept of hybrid ordered ideals and hybrid interior ideals in ordered semirings, as well as look at some of their related properties, which is one of the important aspects of this paper. In order to investigate the structure theory of hybrid ideals in ordered semirings, we define hybrid composition and hybrid addition. We also establish and characterise the regularity of ordered semirings in terms of hybrid structures. [ABSTRACT FROM AUTHOR]

In this paper, by considering the notion of fuzzy hyperoperation, we define the concept of fuzzy hyper B C K (B C I) -algebras as a generalization of B C K (B C I) -algebras and study related properties. Moreover, we introduce the notions of fuzzy hyper B C I -ideal and weak (strong) fuzzy hyper B C I -ideal, and then examine the relations among them. Finally, we investigate the relation between fuzzy hyper B C I -algebras with fuzzy hypersemigroup, fuzzy hypergroup and hyper B C I -algebras. [ABSTRACT FROM AUTHOR]

In this paper, the notions of ideals in NBCK-algebras are presented and some properties of such ideals are analyzed. Essentially, our approach is to define NBCK-algebras of all cosets by considering an operation on the set of all cosets defined via ideals. Afterwards, our aim is to study ideal theorems on NBCK-algebras. [ABSTRACT FROM AUTHOR]

The purpose of multicriteria clustering is to locate groups of alternatives that have comparable qualities and have been examined across multiple criteria. An ordered profile clustering is a well-known problem, and the fuzzy c-means clustering (FCM) technique is one of the most broadly used in every field of life. At present, FCM is for the partitioning of information into numerous clusters which are still lacking priority relations. To address the problem of finding ranking in clusters based on multicriteria in the fuzzy environment, we propose a multicriteria ordered clustering algorithm based on the partial net outranking flow of the preference organization for enrichment evaluations method (PROMETHEE) and fuzzy c-means. Lastly, we apply the proposed algorithm to solve a real-world targeted clustering problem regarding the human development indexes. To test the efficacy of the proposed algorithm, a comparative analysis of ordered K-means clustering (OKM) and FCM is carried out with it. [ABSTRACT FROM AUTHOR]

In this article. we propose the idea of hybrid congruences and hybrid strong ŋ -ideals of hemirings and we build an example that every liybrid strong ŋ-ideal is an hybrid h-ideal. but not conversely. The hybrid hemiring homomorphisins and quotient rings through the hybrid strong 11-ideals are also discussed. In addition. we look into the relationship between hybrid congruence and hybrid strong [1-ideals of hemirings. [ABSTRACT FROM AUTHOR]

In this article, we construct the generating functions for new families of special polynomials including two parametric kinds of Bell-based Bernoulli and Euler polynomials. Some fundamental properties of these functions are given. By using these generating functions and some identities, relations among trigonometric functions and two parametric kinds of Bell-based Bernoulli and Euler polynomials, Stirling numbers are presented. Computational formulae for these polynomials are obtained. Applying a partial derivative operator to these generating functions, some derivative formulae and finite combinatorial sums involving the aforementioned polynomials and numbers are also obtained. In addition, some remarks and observations on these polynomials are given. [ABSTRACT FROM AUTHOR]

Many uncertainties arise in real-world problems, making them impossible to solve using conventional approaches. Researchers all over the world have developed new mathematical theories like fuzzy set theory and rough set theory to better understand the uncertainties that occur in various fields. Soft set theory, which was recently introduced, offers a novel approach to real-world problem solving by removing the need to set the membership function. This is helpful in resolving a variety of issues, and much progress is being made these days. Recently, Jun introduced the concept of a hybrid structure, which blends the concepts of a fuzzy set as well as a soft set. In this paper, we define the hybrid k-sum and hybrid k-product of k-ideals of semiring and investigate their properties. We illustrate with an example that the hybrid sum and hybrid product of two k-ideals are not always hybrid ideals. We also describe semiring regularity constraints in terms of hybrid k-ideal structures. [ABSTRACT FROM AUTHOR]

In general, models of universe problems in almost all fields, such as engineering, mathematics, medical sciences, computer science, physics, management sciences, artificial intelligence, and operations research, are practically full of complexities and include various types of uncertainties when dealing with them on numerous occasions. Different theories, such as probability, rough sets, fuzzy sets, soft ideals, and so on, have been created to deal with these uncertainties. An algebraic structure, AG-groupoid, is an intermediate structure between two types of structures: commutative semigroup and groupoid. This structure has a very close relationship to a commutative semigroup since a commutative AG-groupoid is always a semigroup. These structures have so many applications in flocks theory, geometry, topology, and many more. We explore several structural properties of an AG-groupoid by using hybrid structures in this paper. The main motivation behind this paper is to present the concepts of hybrid ideals, hybrid bi-ideals and hybrid interior ideals of an AG-groupoid and characterize AG-groupoid in terms of hybrid structures. Also, we show that the hybrid intersection and hybrid product structures will coincide under certain conditions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new kinds of graphs based on equality algebras. First of all, by using the meet operation we define the notion of zero divisors on equality algebra and study related properties. Then we introduce a meet graph on equality algebra by using zero divisors. In addition, we investigate some conditions for a meet graph to be a complete, connected and a star graph. Also, by using the notion of filters in equality algebras, we define an equivalence relation on equality algebras and then by using the equivalence classes we introduce two kinds of graphs on them. Finally, conditions for a graph based on these classes to be connected, or bipartite, or complete, or planar or an outer-planar graph are provided. [ABSTRACT FROM AUTHOR]

In this research article, we present the notion of a cubic planar graph and investigate its related properties. The cubic graphs are more effective than both interval-valued and fuzzy graphs as it represents the level of participation (membership degree) of vertices and edges both in interval form and as a fuzzy number. Moreover, it handles the uncertainty and vagueness more efficiently than both interval-valued fuzzy graph and fuzzy graph. The interval indicates a continuous process, whereas the point indicates a specific process. We introduce the terms cubic multigraph, cubic strong and weak edges, and degree of planarity for cubic planar graphs. Some fundamental theorems based on these concepts are also elaborated. We also propose the idea of a cubic strong and weak fuzzy faces and cubic dual graph. Some results related to these concepts are also established. Comparison with the existing method shows the worth of our proposed work. [ABSTRACT FROM AUTHOR]

The coronavirus (COVID-19) pandemic, which began in China and is fast spreading around the world, has increased the number of cases and deaths. Governments have suffered substantial damage and losses not only in the health sector but also in a variety of other areas. In this situation, it is critical to determine the most crucial vaccine that doctors and specialists should implement. In order to evaluate the many vaccines to control the COVID-19 epidemic, a decision problem based on the decisions of many experts, with some contradicting and multiple criteria, should be taken into account. This decision process is characterized as a multiattribute group decision-making (MAGDM) problem that includes uncertainty in this study. T -spherical fuzzy sets are utilized for this, allowing decision-experts to make evaluations over a larger area and better deal with complicated data. The T -spherical fuzzy set is a useful tool for dealing with uncertainty and ambiguity, especially where additional answers of the type "yes," "no," "abstain," and "refusal" are required, and the 2-tuple linguistic terms are useful for the qualitative evaluation of uncertain data. From the perspective of the uncertainty surrounding the problems of MAGDM, we propose the notion of 2-tuple linguistic T -spherical fuzzy numbers (2TL T -SFNs) generated with the integration of T -spherical fuzzy numbers and 2-tuple linguistic terms. Then, the assessment based on distance from average solution (EDAS) for the ranking of alternatives based on the 2TL T -SFNs is investigated as a new decision-making strategy. This study provides the following significant contributions: (1) the procedure for constructing a 2TL T -SFNs is described, together with their aggregation operators, ranking criteria, relevant attributes, and some operational laws. (2) The traditional Maclaurin symmetric mean (MSM) operator is useful for modeling attribute interrelationships and aggregating 2TL T -SF information to tackle the MAGDM problems. A few recent MSM and dual MSM operators are being built to evaluate the 2TL T -SF information. Thus, 2-tuple linguistic T -spherical fuzzy Maclaurin symmetric mean (2TL T -SFMSM) operator, 2-tuple linguistic T -spherical fuzzy weighted Maclaurin symmetric mean (2TL T -SFWMSM) operator, 2-tuple linguistic T -spherical fuzzy dual Maclaurin symmetric mean (2TL T -SFDMSM) operator, and 2-tuple linguistic T -spherical fuzzy weighted dual Maclaurin symmetric mean (2TL T -SFWDMSM) operator are proposed. (3) We incorporate the 2TL T -SFNs into the EDAS approach and develop a new 2TL T -SF-EDAS method for solving the MAGDM problems based on the proposed aggregation operators in a 2TL T -SF environment. A case study for the selection of an optimal vaccine to control the outbreak of the COVID-19 epidemic is also presented to show the validity of the proposed methodology. Furthermore, the comparative analysis with existing approaches shows the advantages and superiority of the proposed framework. [ABSTRACT FROM AUTHOR]

The cubic set, introduced as a combination of a fuzzy set and an interval-valued fuzzy set, provided researchers with more flexibility than the previous two sets in dealing with complex and uncertain problems. Fuzzy graphs, based on this type of set, are among the emerging fuzzy graphs that have a great potential to model the surrounding phenomena. Consistent with the special role that cubic graphs play in decision-making and selecting superior options, dominating these graphs is of great importance and value. In this paper, we introduce the domination of the cubic graphs in terms of strong edges and examine their properties. In addition, we examine domination in terms of independent sets and since many of the phenomena surrounding us are hybrid, we also discuss the domination concept on its fuzzy operations. Finally, we present an application of this graph on the subject of domination. [ABSTRACT FROM AUTHOR]

The fuzzy set is highly beneficial for expressing people's hesitations in their everyday lives, and it is a great tool for dealing with uncertainty, which can be described precisely and perfectly from the decision-maker's point of view. Soft set theory has been developed in recent years to address realworld issues. Jun et al. merged fuzzy and soft sets to produce hybrid structures. Hybrid structures are soft set and fuzzy set speculations. The concept of hybrid ideals in near-subtraction semigroups is introduced in this paper, and their equivalent results are obtained. Additionally, we demonstrate the concept of hybrid intersection. Moreover, we define the concept of homomorphism of a hybrid structure in a near-subtraction semigroup. [ABSTRACT FROM AUTHOR]

The notions of hybrid ideals and k-hybrid ideals in a ternary semiring are introduced in this paper, and a substantial amount of effort has been made to study some of their features. In terms of characteristic function, we show some properties of k-hybrid ideals and give some characterizations of hybrid intersection with respect to these k-hybrid ideals. Finally, results based on a k-hybrid ideal's homomorphic hybrid preimage are provided. With respect to k-hybrid ideals, we give certain characterizations of hybrid intersection. [ABSTRACT FROM AUTHOR]

Natural genetic material may shed light on gene expression mechanisms and aid in the detection of genetic disorders. Single Nucleotide Polymorphism (SNP), small insertions and deletions (indels), and major chromosomal anomalies are all chromosomal abnormality-related disorders. As a result, several methods have been applied to analyze DNA sequences, which constitutes one of the most critical aspects of biological research. Thus, numerous mathematical and algorithmic contributions have been made to DNA analysis and computing. Cost minimization, deployment, and sensitivity analysis to many factors are all components of sequencing platforms built on a quantitative framework and their operating mechanisms. This study aims to investigate the role of DNA sequencing and its representation in the form of graphs in the analysis of different diseases by means of DNA sequencing. [ABSTRACT FROM AUTHOR]

In general, ordered algebraic structures, particularly ordered semigroups, play an important role in fuzzification in many applied areas, such as computer science, formal languages, coding theory, error correction, etc. Nowadays, the concept of ambiguity is important in dealing with a variety of issues related to engineering modeling problems, network theory, decision-making problems in real-life situations, and so on. Several theories have been developed by various researchers to overcome the difficulties that arise from uncertainty, including fuzzy sets, intuitionistic fuzzy sets, probability, soft sets, neutrosophic sets, and many more. In this paper, we focus solely on neutrosophic set theory. In ordered semigroups, we define and investigate the properties of neutrosophic ℵ-ideals and neutrosophic ℵ-interior ideals. We also use neutrosophic K-ideals and neutrosophic ℵ-interior ideals to characterize ordered semigroups. [ABSTRACT FROM AUTHOR]

Models of universe problems are brimming with complexities and uncertainties in almost every field of study, including engineering, mathematics, medical sciences, computer science, physics, management sciences, artificial intelligence, and operations research. To address these uncertainties, various theories have been developed, including probability, rough sets, fuzzy sets, soft ideals, and neutrosophic sets. Neutrosophic set theory is the focus of this paper. In this paper, we introduce the notions of neutrosophic N-filters and neutrosophic N-bi-filters in a semigroup and investigate several properties. Moreover, the relations of prime biideal subset and prime neutrosophic N-bi-ideal structure; neutrosophic N-bi-filter and neutrosophic N-bi-ideal structure; left (resp., right) filter and neutrosophic N-left (resp., right) filter; neutrosophic N-left(resp., right) filter and prime neutrosophic N-left (resp., right) ideals in semigroups are discussed. Finally we prove that: let X be a semigroup and XN be any neutrosophic structure. Then XN is a neutrosophic N-bi-filter of X if and only if XNc is a prime neutrosophic N-bi-ideal of X. [ABSTRACT FROM AUTHOR]

By combining the notions of interval-valued m-polar fuzzy graphs and m-polar fuzzy graphs, the notion of m-polar cubic graphs is first introduced. Then, the degree of a vertex in m-polar cubic graphs and complete m-polar cubic graphs is defined. After that, the concepts of direct product and strong product of m-polar cubic graphs are given. Moreover, weak isomorphism and co-weak isomorphism are defined, and examples are given to prove that weak isomorphism and co-weak isomorphism are not an isomorphism. Finally, the notion of complement m-polar cubic graphs and weak complement m-polar cubic graphs is presented. [ABSTRACT FROM AUTHOR]