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Giving the proper citations, it is shown that, except of Lemma 2.4 and Theorem 2.6, almost all the results of the paper in the title have been previously published for ordered hypersemigroups in Eur. J. Pure Appl. Math. and they are not new. There are also two results obtained from ordered semigroups just putting a "o" instead of "o" (that isn't a correct way to work), without reference to ordered semigroups on which the results on ordered hypersemigroups are based. One of them can be obtained as corollary to a theorem in Eur. J. Pure Appl. Math. as well, and it is not new. [ABSTRACT FROM AUTHOR]

Thomas L. Saaty, a world-renowned scholar and our beloved Advisory Editor, passed away on August 14, 2017 at the age of 91. He has made many fundamental contributions to operations research, analytics, business and mathematics. Despite his heavy illness for 14 months, as passionate about science, he has also enthusiastically published his research. We review his final works published in 2017 and 2018. We are also honored to reveal and publish his final statements on neural firing and synthesis in making comparisons & life satisfaction, respectively, in his final two papers; the previous one submitted by him just before his passing and the other one submitted by his co-authors after his death to European Journal of Pure and Applied Mathematics (EJPAM). We just would like to remember and honor Saaty's memory by publishing his final papers at EJPAM and this humble remarks dedicated to the memory of Thomas L. Saaty on the occasion of the first anniversary of his passing, August 14, 2018. [ABSTRACT FROM AUTHOR]

Although Selberg-type single positive definite symmetric matrices gamma and beta integrals have been evaluated by several authors, see e.g., Askey and Richards [1], Gupta and Kabe [2, 4], Mathai [8], and elsewhere in the vast multivariate statistical analysis literature. However, several other types of Selberg-type integrals appear to have been neglected in the literature. Thus e.g., Selberg-type integrals associated with inverse Wishart densities, inverse multivariate beta densities, their noncentral counterparts, etc, have not been explored as yet. The present paper records Selberg-type generalized quadratic forms gamma and beta integrals. Our methodology is based on hypercomplex (HC) multivariate normal distribution theory, Kabe [6]. [ABSTRACT FROM AUTHOR]

The nervous system uses its own kind of mathematical function patterns for both external and internal realities. The conscious part of the nervous system is there to respond to what happens outside by regulating externally received information signals from the senses and the skin and muscles of the body itself. To do that, it needs to communicate with its subconscious using the familiar language of neural firing. In this paper, we show that because reciprocal pairwise comparisons are performed at the neural level, the division algebra of the octonions, in which commutativity and associativity are not satisfied, provides the structure needed to represent mental processes and that these processes could be represented in G2-manifolds. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of the direct product of sets that involve Fermatean neutrosophic (FN) elements in structures called INK-algebras. It defines terms like the direct product of Fermatean neutrosophic INK-ideals (FNINK-Is) in INK-algebras and Fermatean neutrosophic sets (FNSs), FNINK-Is, and Fermatean neutrosophic closed INK-ideals (FNCINK-Is). The proof of theorems illustrating the relationships between these ideas is included in the paper. It also defines the INK-subalgebra embedded in an INK-algebra and gives a theorem elucidating the connection between the direct product of FNINK-Is and the images of these subalgebras. In essence, the paper investigates and establishes connections between different mathematical ideas concerning INK-algebras and FNSs. [ABSTRACT FROM AUTHOR]

In this paper, a numerical method for solving the fractional order Fredholm integrodifferential equations via the Caputo-Liouville derivative is presented. The method uses the wellknown shifted Chebyshev expansion and a truncated series to represent the unknown function. It also incorporates numerical integration techniques like the Trapezoidal, Simpson's 1/3, and Simpson's 8/3 methods. The paper also provides an approximation for the derivative of an integer. The procedure converts the provided problem into a system of algebraic equations using shifted Chebyshev coefficients and collocation points. The coefficients are found by solving this system using well-known techniques like Newton's method. Numerical results are presented graphycally to illustrate the applicability, efficacy, and accuracy of the approach presented in this work. All calculations in this study were performed using the MATHEMATICA software program. [ABSTRACT FROM AUTHOR]

Let G be a simple undirected graph of order n. In this paper, we introduce a new distance matrix called the universal distance matrix of G, denoted as UD (G) and it is defined as UD (G) = αTr (G) + βD(G) + γJ + δI, where Tr (G) is the diagonal matrix whose elements are the vertex transmissions, and D(G) is the distance matrix of G. Here J is the all-ones matrix, and I is the identity matrix and α, β, γ, δ ∈ R and β ̸= 0. This unified definition enables us to derive the spectra of different matrices associated with the distance matrix of graphs. The set of eigenvalues of the universal distance matrix namely, {ρ1, ρ2,.. ., ρn} is known as the universal distance spectrum of G. As a consequence, by taking appropriate values for α, β, γ, δ ∈ R and β ̸= 0, we obtain the eigenvalues of distance matrix, distance Laplacian matrix, distance signless Laplacian matrix, generalized distance matrix, distance Seidal matrix and distance matrices of graph complements. In this paper, we obtain the universal distance spectra of regular graph, join of two regular graphs, joined union of three regular graphs, generalized joined union of n disjoint graphs with one arbitrary graph H using the Schur complement of a block matrix. [ABSTRACT FROM AUTHOR]

This paper introduces the notion of a B-filter in a B-algebra (X, ∗, 0) and presents characteristics of its properties: for any a ∈ X, the set ⟨a⟩B = {ak: k ∈ Z} forms a B-ideal and B-filter of X. Moreover, this paper showns some properties of exponents on B-algebra. [ABSTRACT FROM AUTHOR]

In this paper, we consider various Stirling numbers of both kinds, including the unsigned degenerate Stirling numbers of the first kind, the degenerate Stirling numbers of the second kind, the unsigned degenerate r-Stirling numbers of the first kind and the degenerate r-Stirling numbers of the second kind. The aim of this paper is by using generating functions to further study explicit expressions, some identities and equivalent relations for those Stirling numbers. [ABSTRACT FROM AUTHOR]

This paper explores the solutions of certain fractional partial differential equations using two methods; the first method involves separation of variables, which is a common technique for solving partial differential equations. However, since many equations cannot be separated in this way, the tensor product of Banach spaces method is applied to find the atomic solutions. To solve the resulting ordinary differential equations, the reproducing Kernel Hilbert space method is used to find numerical solutions, which are then used to find the numerical solution of the partial differential equation. The residual errors indicate that this method is effective and powerful. In summary, this paper presents a study on the solutions of certain fractional partial differential equations using two methods and demonstrates the effectiveness of these methods in finding numerical solutions. [ABSTRACT FROM AUTHOR]

This paper introduces a novel trigonometric B-spline collocation method for solving a specific class of second-order boundary value problems. The study showcases the method’s practicality and effectiveness through various numerical examples. Furthermore, it evaluates the technique’s performance by calculating maximum errors for different step sizes in the spatial domain. The paper also conducts a comparative analysis with alternative methods, demonstrating the superior accuracy of the trigonometric B-spline approach. [ABSTRACT FROM AUTHOR]

This paper defines a new class of graphs using the spanning subgraphs of a cycle graph as vertices. This class of graphs is called j-edge intersection graph of cycle graph, denoted by EC(n,j) . The vertex set of EC(n,j) is the set of spanning subgraphs of cycle graph with j edges where n ≥ 3 and j is a nonnegative integer such that 1 ≤ j ≤ n. Two distinct vertices are adjacent if they have exactly one edge in common. EC(n,j) is considered as a simple graph. Furthermore, EC(n,j) is characterized by the value of j that is when j = 1 or ⌈ n/2 ⌉ < j ≤ n and 2 ≤ j ≤ ⌈ n/2 ⌉. When j = 1 or ⌈ n/2 ⌉ < j ≤ n, the new graph only produced an empty graph. Hence, the proponents only considered the value when/2 ≤ j ≤ ⌈ n/2 ⌉ in determining the order and size of EC(n,j) . Moreover, this paper discusses necessary and sufficient conditions where the j-edge intersection graph of Cn is isomorphic to the cycle graph. Furthermore, the researchers determined a lower bound for the independence number, and an upper bound for the domination number of EC(n,j) when j = 2. [ABSTRACT FROM AUTHOR]

This paper is motivated by the work of Taekyun Kim and Dae San Kim on the degenerate Laplace transform and degenerate gamma function, as published in the Russian Journal of Mathematical Physics. We introduce the degenerate Laplace-type integral transform and delve into its properties and interrelations. This paper focuses on the degenerate Laplace-type integral transforms of several fundamental functions, including the degenerate sine, degenerate cosine, degenerate hyperbolic sine, and degenerate hyperbolic cosine functions. Furthermore, we establish crucial connections between the degenerate Laplace-type integral transform and existing degenerate integral transforms. Specifically, we explore its relationships with the degenerate Laplace transform, the degenerate Elzaki transform, and the degenerate Sumudu transforms. [ABSTRACT FROM AUTHOR]

This paper establishes the linear algebra of the (r, β)-Stirling matrix. Along the way, this paper derives various identities, such as its factorization and relationship to the Pascal matrix and the Stirling matrix of the second kind. Additionally, this paper develops a natural extension of the Vandermonde matrix, which can be used to study and evaluate successive power sums of arithmetic progressions. [ABSTRACT FROM AUTHOR]

This paper presents a way to represent a group using a graph, which involves the concept of a generator element of a group. The graph representing a group is called the generator graph. In the generator graph, the vertices correspond to the elements of the group, and two vertices, x and y, are connected by an edge if either x or y serves as a generator for the group. The paper investigates some properties of these generator graphs and obtains the generator graphs for specific groups. Additionally, it explores the relationship between the generator graph of a group and the generating graph introduced by Lucchini et al. in their work [7]. [ABSTRACT FROM AUTHOR]

This paper investigates the companion of Ostrowski’s inequality in the framework of fractal sets. First, a new identity related to local fractional integrals is introduced, serving as the foundation for establishing a set of inequalities applicable to functions with generalized sconvex and s-concave derivatives. An illustrative example is presented to validate the obtained results, demonstrating their accuracy. Additionally, the paper discusses several practical applications, highlighting the significance of the established inequalities. The research presented in this paper contributes to the growing field of studying functions on fractal sets, which has attracted considerable interest from scientists and engineers. [ABSTRACT FROM AUTHOR]

This paper is devoted to introducing and studying two concepts, σ-skew strongly Mreflexive and σ-skew strongly M-nil-reflexive on monoid rings, which are generalizations of strongly M-reflexive and M-compatible. The paper covers the basic properties of skew monoid rings of the form R∗M. It is shown that if R is a left AP P (quasi Armendariz, semiprime rings, respectively), then R is σ-skew strongly M-reflexive. Moreover, if R is a NI-ring and M is a u.p-monoid, then R is σ-skew strongly M-nil-reflexive. Additionally, under some necessary and sufficient conditions, a skew monoid ring R ∗ M is proven to be σ-skew strongly M-nil-reflexive when σ : M → Aut(R) is a monoid homomorphism. Furthermore, if R is a left AP P, then the upper triangular matrix ring Tn(R) is σ¯-skew strongly M-nil-reflexive, where n is a positive integer. Finally, the paper provides some examples and discusses related results from the subject. [ABSTRACT FROM AUTHOR]

Let G be a graph. Then Q ⊆ V (G) is called a certified vertex cover of G if Q is a vertex cover of G and every x ∈ Q, x has either zero or at least two neighbors in V (G) \ Q. The certified vertex cover number of G, denoted by βcer(G), is the minimum cardinality of a certified vertex cover of G. In this paper, we investigate this newly defined concept on some special graphs and on the join of two graphs. We characterize certified vertex cover in these graphs and subsequently derive the simplified formulas for calculating the certified vertex cover number. Moreover, we present some bounds and properties of certified vertex cover. [ABSTRACT FROM AUTHOR]

Let G be a connected graph. A function f : V (G) → {0, 1, 2} is a convex Roman dominating function (or CvRDF) if every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2 and V1 ∪ V2 is convex. The weight of a convex Roman dominating function f, denoted by ωG CvR(f), is given byωG CvR (f) =∑ v∈V(G) f(v). The minimum weight of a CvRDF on G, denoted by γCvR(G), is called the convex Roman domination number of G. In this paper, we specifically study the concept of convex Roman domination in the corona and edge corona of graphs, complementary prism, lexicographic product, and Cartesian product of graphs. [ABSTRACT FROM AUTHOR]

This paper is concerned with the concepts of upper and lower almost (τ1, τ2)-continuous multifunctions. Furthermore, some characterizations of upper and lower almost (τ1, τ2)-continuous multifunctions are investigated. [ABSTRACT FROM AUTHOR]

This paper introduces and investigates several new classes of sets called P-α-open sets, P-semiopen sets, P-preopen sets, and P-β-open sets within the framework of primal topological spaces. Their properties and relationships with other open set generalizations are studied through examples. Additionally, the concepts of PR-sets and PRα -sets are defined and their characteristics examined. Also, the notions of P-α-continuous, P-semicontinuous, P-precontinuous and P-β-continuous mappings are initiated and their features and main characterizations determined. A new class of sets called Ψ P -sets is also introduced in primal topological spaces using the ΨP -operator. Their properties and relationships between Ψ P -sets, α-open, semi-open, and pre-open are investigated. Theorems on arbitrary unions and finite intersections of Ψe P are discussed. [ABSTRACT FROM AUTHOR]

This paper combines the McShane-Stieltjes integral and Pettis approaches by utilizing Pettis’ definition, which coincides with the Dunford integral rather than the version applicable to weakly measurable functions with Lebesgue integrable images. In this way, another integration process without measure theoretic standpoint will be introduced. To this end, we will define the McShane-Dunford-Stieltjes integral and McShane-Pettis-Stieltjes integral in Banach Space and provide its simple properties such as the uniqueness, linearity property of both the integrand and integrator, additivity and formulate the Cauchy criterion of these integrals. In addition, the existence theorem of McShane-Dunford Stieltjes integral will also be presented. [ABSTRACT FROM AUTHOR]

In this paper we establish the existence of a common fixed point from a pair of setvalued mappings. By utilizing the concept of convergence of set-valued mappings’ sequences, both ordinary and pointwise convergence, we establish a common fixed point theorem. This our newly result is a generalization of common fixed point theorem of set-valued mappings on partial metric spaces. Further, we establish newly common fixed point theorem under ϕ-contraction on partial metric spaces. [ABSTRACT FROM AUTHOR]

The paper develops a novel analysis of mutual interactions between topology and soft topology. It is known that each soft topology produces a system of crisp (parameterized) topologies. The other way round is also possible. Namely, one can generate a soft topology from a system of crisp topologies. Different methods of producing soft topologies are discussed by implementing two formulas. Then, the relationships between the resulting soft topologies are obtained. With the help of an example, it is demonstrated that one formula is more constructible than the other. Now, it is reasonable to ask which (topological) properties of a soft topology can be transferred to the set of crisp topologies, or the opposite. To address this question, we consider the standard separation axioms and show how well these axioms can be preserved when moving from a system of crisp topologies to the soft topology generated by it and contrariwise. Additionally, our findings extend and disprove some results from the literature. [ABSTRACT FROM AUTHOR]

The paper focuses on the Lotka-Volterra function in its discrete form. The purpose of the study was to determine the fixed points of the function. The study employs the Banach Fixed Point Theorem and Contraction Mapping in Metric Space on the function to demonstrate the uniqueness of the fixed points and its continuous stability after several iterations, using the fixed points as the initial conditions. The study has shown that (0, 0), (0, α−1/β), (1+γ /δ, 0 ) and (1+γ/ δ, α−1/ β)are the fixed points of the function, with the initial pair serving as a trivial one and the other three solely depending on the parameter values for the behavior of the function. The outcome of the limit points of the function as the fixed points after several iterations forms a fixed orbit structure of the function, irrespective of the value of the parameter. The study also showed the uniqueness of the fixed points, demonstrating the stability and continuity of the function in its steady state. [ABSTRACT FROM AUTHOR]

This paper derives the bivariate version of the extended standard U-quadratic (eSU) distribution using the compounding method or pseudo-family of distributions. The joint probability and cumulative distribution functions of the derived distribution are obtained and it is observed that the said distribution can generate bivariate shape distributions with the following properties: (i) X and Y have bathtub shapes; (ii) X has a constant distribution and Y has a bathtub shape; and (iii) X has an inverted bathtub and Y has a bathtub shape. Moreover, some properties of this proposed distribution are derived such as the marginal distribution, conditional distribution, conditional moments, product and ratio moments, Pearson correlation coefficient, joint moment generating function, and the stress - strength parameter. Further, the maximum likelihood estimation is performed to estimate the parameters of the derived distribution. A simulation study is carried out to evaluate the behavior of the estimates of the parameters. A new bivariate Kumaraswamy distribution is derived and used to simulate bivariate data with X and Y having bathtub shapes. A new bivariate version of the Cubic Transmuted Uniform (CTU) distribution is also derived. Finally, the proposed Bivariate eSU distribution is applied to simulated data and compared with the Bivarite Cubic Transmuted Uniform distribution. The results show that the proposed Bivariate eSU distribution provides a better fit on the simulated dataset compared with the Bivariate CTU distribution. [ABSTRACT FROM AUTHOR]

This paper seeks to showcase how a new approach can breathe new life into research within the traditional domain of Pythagorean triples, introducing innovative applications to invigorate the field. This serves not only as an exemplar but also as a wellspring of inspiration for students at both school and university levels. The demonstrations will underscore that, with fundamental mathematical concepts and unencumbered by intricate calculations, one can unveil novel results and applications with ease. The new results and applications, along with those found in the Preliminary Results section, show how the field of Pythagorean triples is still interesting and stimulating to study, despite the centuries that have elapsed. [ABSTRACT FROM AUTHOR]

Let G = (V (G), E(G)) be a simple undirected graph. A block of G is a maximal connected subgraph of G that contains no cut-vertices [11]. The family of vertex sets of blocks of G generates a unique topology. In this paper, we formally define the topology generated by the family of blocks in a graph called the block topological space. Moreover, we characterize and describe some special attributes of the block topological space. Finally, we associate a corresponding graph from a given block topological space by defining the block topological graph. [ABSTRACT FROM AUTHOR]

In this paper, the concept of Paradistributive Latticoid (PDL) as a generalization of a distributive lattice is introduced and investigated its properties. A set of equivalent conditions for a PDL to become a distributive lattice are given. The notions of an ideal and a filter in a PDL are introduced and studied their properties. Subdirect representation of a PDL is obtained [ABSTRACT FROM AUTHOR]

Let G be a finite group. A subgroup H of G is said to be weakly S-permutable in G if G has a subnormal subgroup T such that G = HT and T ∩ H ≤ HsG, where HsG is the subgroup of H generated by all those subgroups of H which are S-permutable in G. In this paper, we prove the following: For a Sylow p-subgroup P of G (p > 2), suppose that P has a subgroup D such that 1 < |D| < |P| holds and all subgroups H of P with |H| = |D| are weakly S-permutable in G. Then, the commutator subgroup G′ is p-nilpotent. We certainly believe that this result will improve and extend a current and classical theories in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study the isomorphism problem for central extensions. More precisely, in some new situations, we provide necessary and sufficient conditions for two central extensions to be isomorphic. We investigate the case when the quotient group is simple or purely nonabelian. Furthermore, we characterize isomorphisms leaving the quotient group invariant. Finally, we deal with isomorphisms of central extensions where the kernel group and the quotient group are isomorphic. [ABSTRACT FROM AUTHOR]

Privacy is given top priority as far as communication involving transfer of confidential document or data is concerned. There is therefore a need to keep confidential data secretive from being invaded by unauthorised access, and this is done through cryptography algorithms, where the Advanced Encryption Standard algorithm has been the widely accepted symmetric block encryption algorithm for such purposes. This paper therefore focuses on developing a new AES-512 bits symmetric encryption algorithm through modification of the conventional AES-128 algorithm to be used purposefully in the classroom for document transfer. The development comes with increasing the plaintext bits of the conventional AES-128 algorithm to 512 bits plaintext which undergoes five operational transformations: STATE, SKGF, SRL, SCL and AARC through key size in the Galois field, GF(29 ). A numerical example is then given to explain the use of the algorithm, and finally, we provide a comparative study of this algorithm and other existing symmetric encryption models, such as the AES-128 and DES algorithms. [ABSTRACT FROM AUTHOR]

Let φ be an analytic selfmap of the open unit disk D and g be an analytic function on D. The Volterra-type composition operators induced by the maps g and φ are defined as (Ig φ f) (z) = ∫0 z f' (φ(ζ))g(ζ)dζ and (Tg φf) (z) = ∫0 z f(φ(ζ))g′ (ζ)dζ. For 1 ≤ p < ∞, S p (D) is the space of all analytic functions on D whose first derivative f ′ lies in the Hardy space Hp (D), endowed with the norm ∥f∥Sp = |f(0)| + ∥f′ ∥Hp . Let µ : (0, 1] → (0, ∞) be a positive continuous function on D such that for z ∈ D we define µ(z) = µ(|z|). The weighted Zygmund space Zµ(D) is the space of all analytic functions f on D such that supz∈D µ(z)|f ′′(z)| is finite. In this paper, we characterize the boundedness and compactness of the Volterra-type composition operators that act between S p spaces and weighted Zygmund space [ABSTRACT FROM AUTHOR]

A topological index is a numerical character associated with a graph that is invariant under graph isomorphism and describe the graphs topology. There are several graph operations that may be used to change it into a new structure, such as constructing a stellation, bounded dual, complement, subdivided, line graph, minor, dual, and medial. In this paper, we construct transformed networks from the concealled non-kekulean benzenoid hydrocarbon structure by applying stellation and bounded dual operations, further we study their degree based topological properties by appropriately labeling the graph. Degree based topological indices are playing significant role among other types of indices in chemical, pharmaceutical and bio-informatics industry, since they corelate the structure with its physicochemical properties. [ABSTRACT FROM AUTHOR]

In this paper, we succeed in proving that a connected Artin algebra whose Jacobson cubic radical is zero with each simple module and each Gorenstein projective module having the square radical of the cover projective of its first syzygy to be zero and verifying the coincidence covers, is either self-injective or CM-free. [ABSTRACT FROM AUTHOR]

Let G be a graph with vertex and edge-sets V (G) and E(G), respectively. Then O ⊆ V (G) is called a J-open independent set of G if O is a singleton set or O is an independent set of G and for every a, b ∈ V (G), NG(a)\NG(b) ̸= ∅ and NG(b)\NG(a) ̸= ∅. The maximum cardinality of a J-open independent set of G, denoted by αJ (G), is called the J-open independence number of G. In this paper, we introduce this parameter and we show that it is always less than or equal to the standard independence (resp. J-total domination) parameter of a graph. In fact, their differences can be made arbitrarily large. In addition, we show that J-open independence parameter is incomparable with hop independence parameter. Moreover, we derive some formulas and bounds of the parameter for some classes of graphs and the join of two graphs. [ABSTRACT FROM AUTHOR]

In this paper, we examine the nonlinear Weakly Singular Volterra Integral Equation (WSVIE), u(x) = f(x)+∫0x t μ-1/xµ [u(t)]βdt. Al-Jawary and Shehan used the Daftardar-Jafari Method (DJM) and solved the above integral equation for the investigation parameter µ > 1 using specific force functions with µ and β values and obtained unique solutions. We have discovered a force function f(x) = x k1-x γk1/γk1+µ that allows the introduction of noise terms phenomena discovered by Wazwaz that cancel out the terms of the power series in the successive solution terms um, m = 0, 1, 2, ..., n: we thus obtain a maximum finite power series terms for each solution term called truncation point and denoted by x g(n) . Such that the integral solution can be written as u(x) = u0+∑ m n=1 um, where n is finite. Simplifying the solution terms, we get the unique solution u(x) = x k1, irrespective of the n−value in the truncation point. We have discovered a formula relation between the last solution term un and the truncation point as un = anx g(n) . Our results confirm the results of the two solution examples of AL-Jawary and Shehan for the investigation parameter µ > 1. We extend the parameter range to include µ > 1 and 0 < µ ≤ 1 for our solution. In addition, for any chosen rational parameter k1, the solution u(x) = x k1 is extrapolated to be valid for all integer parameter values β ≥ 2 and positive rational parameter values µ > 0 and for any finite value of n ≥ 2. [ABSTRACT FROM AUTHOR]

Let G be any graph. A sequence L = (w1, . . ., wk) of distinct vertices of G is called a legal hop independent sequence if k = 1 or L is a hop independent and NG[wi ] \ Uj=1i-1 j=1 NG[wj ] ̸= ∅ for every i ∈ {2, · · ·, k}. The maximum length of a legal hop independent sequence in G, denoted by αℓh(G), is called the legal hop independence number of G. In this paper, we investigate its relationships with the hop independence and grundy domination parameter of a graph, respectively. In fact, the legal hop independence parameter is at most equal to the grundy domination (resp. hop independence) parameter on any graph G. Moreover, we derive some formulas and bounds of this parameter on some families of graphs, join, and corona of two graphs. [ABSTRACT FROM AUTHOR]

In this paper, 2-hop domination parameter is introduced and investigated on some special graphs and on the join of two graphs. Characterizations of 2-hop dominating sets in some special graphs are formulated to derive bounds or formulas of the parameter of these graphs. Moreover, new variant of pointwise non-domination is introduced to characterize 2-hop dominating sets in the join of two graphs. This characterization is used to calculate the exact value of 2-hop domination number of the join of two graphs. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of new separation axioms named fuzzy SSPOseparation axioms by using the fuzzy strong semi preo-pen sets and we also introduce and investigate properties of α-SSPO compactness. We define and investigate the relation between fuzzy separation axioms, fuzzy pre-separation axioms, and different forms of fuzzy continuous mappings. We also investigate the existence of a countable base of fuzzy strong semi pre-open sets, we define the concept of SSPO separability, the concept of α−SSP O Lindelof sets and examine their properties. With the concepts of fuzzy strong semi pre-continuity, SSPO-irresolute continuous mappings, and other forms of fuzzy continuity, we investigate the new concept of fuzzy compactness and its properties in regard to the mentioned mappings. [ABSTRACT FROM AUTHOR]

The second largest eigenvalue of a graph is an important algebraic parameter which is related with the expansion, connectivity and randomness properties of a graph. Expanders are highly connected sparse graphs. In coding theory, Expander codes are Error Correcting codes made up of bipartite expander graphs. In this paper, first we prove the interlacing of the eigenvalues of the adjacency matrix of the bipartite graph with the eigenvalues of the bipartite quotient matrices of the corresponding graph matrices. Then we obtain bounds for the second largest and second smallest eigenvalues. Since the graph is bipartite, the results for Laplacian will also hold for Signless Laplacian matrix. We then introduce a new method called vertex-split of a bipartite graph to construct asymptotically good expander codes with expansion factor D/2 < α < D and ϵ < 1/2 and prove a condition for the vertex-split of a bipartite graph to be k-connected with respect to λ2. Further, we prove that the vertex-split of G is a bipartite expander. Finally, we construct an asymptotically good expander code whose factor graph is a graph obtained by the vertex-split of a bipartite graph. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new concept in graph theory called 2-distance zero forcing. We give some properties of this new parameter and investigate its connections with other parameters such as zero forcing and hop domination. We show that 2-distance zero forcing and hop domination(respectively, zero forcing parameter) are incomparable. Moreover, we characterize 2-distance zero forcing sets in some special graphs, and finally derive the exact values or bounds of the parameter using these results. [ABSTRACT FROM AUTHOR]

This paper focuses on the power graph of dihedral groups of order 2n, D2n, where n ≥ 3. We show the characteristic polynomial of the power graph corresponding to the adjacency, Laplacian, signless Laplacian, and normalized form of these matrices. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of a parapseudo-complementation in a paradistributive latticoid(PDL) and investigate its elementary properties. We demonstrate the independence of the axioms related to its definition, highlighting the flexibility of this concept. Additionally, we establish necessary conditions for a PDL with a minimal element to be parapseudocomplemented and explore the properties required for parapseudo-complementation to be equationally definable. Moreover, we establish a one-to-one correspondence between the set of all minimal elements and the set of all parapseudo-complementations. [ABSTRACT FROM AUTHOR]

A number of iterative schemes with high convergence order to solve nonlinear equations are presented in the literature. In this paper, a sixth-order multiple-zero finder has been developed and the dynamics of selected iterative schemes with uniparametric polynomial weight function are investigated using Möbius conjugacy map applied to the form ((z-A)(z-B))m. The complex dynamics on the Riemann sphere by analyzing the parameter spaces associated with the free critical points are studied, and the numerical experiments are carried out. [ABSTRACT FROM AUTHOR]

A set D of vertices in a graph G is a dominating set if every vertex not in D is adjacent to at least one vertex in D. The minimum cardinality of a dominating set in G is called the domination number and is denoted by γ(G). Let D be a minimum dominating set of G. If V-D contains a dominating set say D ′ of G, then D ′ is called an inverse dominating set with respect to D. The inverse domination number γ ′ (G) is the cardinality of a minimum inverse dominating set of G. A dominating set D is called a connected dominating set or an independent dominating set of G according as the induced subgraph ⟨D⟩ is connected or independent in G. The minimum of the cardinalities of the connected dominating sets of G or the independent dominating sets of G is called the connected domination number γc(G) or the independent domination number γi(G) respectively. In this paper, we determine the inverse domination numbers in X-Trees and Sibling Trees. We have also determined the independent domination numbers of both the trees and the connected domination number of Sibling Trees. A result on inverse domination number of some classes of Hypertrees is also included. [ABSTRACT FROM AUTHOR]

The object of this paper is to prove that existence global and asymptotic behavior of solutions for anomalous coupled reaction diffusion system (Gray-Scott model) with homogeneous Neumann boundary conditions. The existence and uniqueness of the local solution are given by the Banach fixed point theorem. Further, the asymptotic behavior is investigated by technique semi group estimates and the Sobolev embedding theorem . [ABSTRACT FROM AUTHOR]

In this paper, we have introduced a graph GEC generated by type-(k1, k2) E-codes which is (k1, k2) E-torsion graph. The binary codewords of the torsion code of C are the set of vertices, and the edges are defined using the construction of E-codes. Moreover, we characterized the graph obtained when k1 = 0 and k2 = 0 and calculated the degrees of every vertex and the number of edges of GEC . Moreover, we presented necessary and sufficient conditions for a vertex to be in the center of a graph given the property of the codeword corresponding to the vertex. Finally, we represent every quasi self-dual codes of short length by defining the vertex-weighted (k1, k2) E-torsion graph, where the weight of every vertex is the weight of the codeword corresponding to the vertex. [ABSTRACT FROM AUTHOR]

A subset S ⊆ V (G) is said to be a perfect equitable isolate dominating set of a graph G if it is both perfect equitable dominating set of G and isolate dominating set of G. The minimum cardinality of a perfect equitable isolate dominating set is called perfect equitable isolate domination number of G and is denoted by γpe0(G). A perfect equitable isolate dominating set S of G is called γpe0-set of G. In this paper, the authors give characterizations of a perfect equitable isolate dominating set of some graphs and graphs obtained from the join and corona of two graphs. Furthermore, the perfect equitable isolate domination numbers of these graphs is determined, and the graphs with no perfect equitable isolate dominating sets are investigated. [ABSTRACT FROM AUTHOR]

This work takes an interesting diversion, revealing the extraordinary capacity to determine the precise number of primes in a space tripled over another. Exploring the domain of K-almost prime numbers, this paper provides a clear explanation of the complex idea. In addition to outlining the conditions under which odd K-almost prime numbers must exist, it presents a novel method for figuring out how often odd numbers are as 2-almost prime, 3-almost prime, 4-almost prime, and so on, up to a specified limit n. The work goes one step further and offers useful advice on how to use these approaches to precisely calculate the prime counting function, π(n). Essentially, it offers a comprehensive exploration of the mathematical fabric, where primes reveal their mysteries in both large and small spaces. [ABSTRACT FROM AUTHOR]