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In this paper, we introduce some explicit and recursive formulas for the n$$ n $$th q$$ q $$‐derivative of a quotient of two functions. We establish the connections between linear algebra, the homomorphisms between commutative algebras and the properties of q$$ q $$‐differential operator. One of the formulas involves the determinant of the functions and their q$$ q $$‐derivatives. Then, we show that the formulas in the classical mathematical analysis are their special cases. Finally, we used formulas to obtain some interesting q$$ q $$‐identities. [ABSTRACT FROM AUTHOR]

We call a Krasner right S‐hypermodule A regular if each cyclic subhypermodule of A is a direct summand of A, and we also call A semiregular if every finitely generated subhypermodule of A lies above a direct summand of A. In this study, some properties of such hypermodules are achieved. This paper consists of concentrating over the radical, as well as their connections with (semi)regularity of hypermodules. Finally, the relationship between f‐supplement submodule and semiregularity is given. [ABSTRACT FROM AUTHOR]

In this study, we define the binomial transforms of third-order Jacobsthal and modified third-order Jacobsthal polynomials. Further, the generating functions, Binet formulas and summation of these binomial transforms are found by recurrence relations. Also, we establish the relations between these transforms by deriving new formulas. Finally, the Vajda, d'Ocagne, Catalan and Cassini formulas for these transforms are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of the intrinsic product of neutrosophic sets, and some related properties are investigated. Characterizations of neutrosophic subrings, neutrosophic ideals, neutrosophic quasi-ideals, and neutrosophic bi-ideals are given. [ABSTRACT FROM AUTHOR]

This paper is dedicated to study the Invertibility properties of all plithogenic square matrices of high orders between 20 and 24, where we present the mathematical conditions and formulas for computing the inverses of all plithogenic square matrices with real plithogenic entries. This goal will be completed by proving many theorems that describe the Invertibility of all classic matrix parts of the corresponding symbolic m-plithogenic matrix. [ABSTRACT FROM AUTHOR]

We derive one specific example of a Machin-like formula for approximating π using the unit circle and the reciproval of the golden section. The efficiency of our derived formula is compared to other, similar formulas using Lehmer's measure. [ABSTRACT FROM AUTHOR]

We compute a formula for the statistical average of numbers whose base two expression has no more than a given number of digits and contains a given number of ones. We call the collection of such numbers a Hamming circle centered at 0 and conjecture a formula for the mean of Hamming circles centers at any positive integer. [ABSTRACT FROM AUTHOR]

Summary: A largely forgotten result proved by Amèpre roughly states that the magnetic field generated by a circuit can be simulated by a magnetic sheet whose border coincides with the circuit. We show that this can be seen as a neat, nonobvious application of Stokes' theorem. [ABSTRACT FROM AUTHOR]

In this paper, we demonstrate the local well-posedness and blow up of solutions for a class of time- and space-fractional diffusion wave equation in a fractional power space associated with the Laplace operator. First, we give the definition of the solution operator which is a noteworthy extension of the solution operator of the corresponding time-fractional diffusion wave equation. We have analyzed the properties of the solution operator in the fractional power space and Lebesgue space. Next, based on some estimates of the solution operator and source term, we prove the well-posedness of mild solutions by using the contraction mapping principle. We have also investigated the blow up of solutions by using the test function method. The last result describes the properties of mild solutions when α → 1 − . The main feature of the proof is the reasonable use of continuous embedding between fractional space and Lebesgue space. [ABSTRACT FROM AUTHOR]

This paper mainly studies the least-squares solutions of matrix equation A X B = C under a semi-tensor product. According to the definition of the semi-tensor product, the equation is transformed into an ordinary matrix equation. Then, the least-squares solutions of matrix-vector and matrix equations respectively investigated by applying the derivation of matrix operations. Finally, the specific form of the least-squares solutions is given. [ABSTRACT FROM AUTHOR]

The main identity of midpoint type inequalities is generalized and using this identity some composite midpoint type inequalities are estimated. Moreover, applications to some special means and some error estimates for the composite midpoint formula are discussed. [ABSTRACT FROM AUTHOR]

Elliptic curve cryptography (ECC) is widely acknowledged as a method for implementing public key cryptography on devices with limited resources thanks to its use of small keys. A crucial and complex operation in ECC calculations is scalar point multiplication. To improve its execution time and computational complexity in low-power devices, such as embedded systems, several algorithms have been suggested for scalar point multiplication, with each featuring different techniques and mathematical formulas. In this research, we focused on combining some techniques to produce a scalar point multiplication algorithm for elliptic curves over finite fields. The employed methodology involved mathematical analysis to investigate commonly used point multiplication methods. The aim was to propose an efficient algorithm that combined the best computational techniques, resulting in lower computational requirements. The findings show that the proposed method can overcome certain implementation issues found in other multiplication algorithms. In certain scenarios, the proposed method offers a more efficient approach by reducing the number of point doubling and point addition operations on elliptic curves using the inverse of the targeted point. [ABSTRACT FROM AUTHOR]

From the viewpoint of "extra dimension detecting," the phenomenon of the transition of the free point particle into 3d space is investigated. In this way, we formulate the problem using the second-class constrained system. To investigate it using a gauge theoretical approach, we use two methods to convert its two second-class constraints to first-class ones. In symplectic embedding, we construct a pair of scaler and vector gauge potentials, which can be interpreted as interactions for detecting extra dimensions. A Wess-Zumino variable appears as a new coordinate in potentials, and the particle's mass plays the role of a globally conserved charge related to the constructed gauge theory for extra dimensions. [ABSTRACT FROM AUTHOR]

We prove a sharp stability result for the Brunn–Minkowski inequality for A,B⊂R2. Assuming that the Brunn–Minkowski deficit δ=∣A+B∣1/2/(∣A∣1/2/(∣A∣1/2/(∣A∣1/2+∣B∣1/2/(∣A∣1/2)−1 is sufficiently small in terms of t=∣A∣1/2/(∣A∣1/2/(∣A∣1/2/(∣A∣1/2+∣B∣1/2/(∣A∣1/2), there exist homothetic convex sets KA​⊃A and KB​⊃B such that ∣((KA​)/A)∣​+((∣BKB​))/B∣​≤Ct−1/2/(∣A∣1/2δ1/2/(∣A∣1/2. The key ingredient is to show for every, if δ is sufficiently small then ∣co(A+B)/(A+B)∣≤(1+ϵ)(∣co(A)/A∣+∣co(B)/B∣). [ABSTRACT FROM AUTHOR]

We introduce a polynomial invariant Vτ​∈Z[H1​(M)/torsion] associated to a veering triangulation τ of a 3-manifold M. In the special case where the triangulation is layered, i.e. comes from a fibration, Vτ​ recovers the Teichmüller polynomial of the fibered face canonically associated to τ. Via Dehn filling, this gives a combinatorial description of the Teichmüller polynomial for any hyperbolic fibered 3-manifold. For a general veering triangulation τ, we show that the surfaces carried by τ determine a cone in homology that is dual to its cone of positive closed transversals. Moreover, we prove that this is equal to the cone over a (generally non-fibered) face of the Thurston norm ball, and that τ computes the norm on this cone in a precise sense. We also give a combinatorial description of Vτ​ in terms of the flow graph for τ and its Perron polynomial. This perspective allows us to characterize when a veering triangulation comes from a fibration, and more generally to compute the face of the Thurston norm ball determined by τ. [ABSTRACT FROM AUTHOR]

In this paper, we find the necessary and sufficient conditions under which two classes of (α, β)-metrics, i.e. generalized Randers change (G.R.C.) exponential metric F = α exp(β/α) + ϵβ and Randers metric ˜F = α + β are projectively related. Note that α and α are Riemannian metrics, β and ˜β are 1-forms on a manifold M with dimension n > 2 and ϵ ̸= 0 is a constant. Furthermore, we study this projective relation when F has special curvature properties. [ABSTRACT FROM AUTHOR]

In the present paper we introduce and investigate some types of harmonic Finsler manifolds and find out the interrelation between them. We give some characterizations of such spaces in terms of the Finsler mean curvature of geodesic spheres and the Finsler Laplacian of the distance function induced by the Finsler metric. We investigate some properties of the Finsler mean curvature of geodesic spheres of different radii. We prove that certain harmonic Finsler manifolds are of isotropic S-curvature and isotropic Ricci curvature. Moreover, we give some examples of non-Riemannian Finsler harmonic manifolds of constant flag curvature and constant S-curvature. Finally, we provide a technique to construct harmonic Finsler manifolds of Randers type. [ABSTRACT FROM AUTHOR]

In this paper, we study the uniqueness problem of meromorphic functions when two differential polynomials share a set of roots of unity with finite weight. Our results not only generalize a result of Lahiri and Sinha [Commun. Korean Math. Soc., 35 (2020), 773-787], at the same time improve the recent results of Khoai and An [Advanced Studies: Euro-Tbilisi Math. J., 15 (2022), 39-51]. [ABSTRACT FROM AUTHOR]

It is proved that for any decomposable perfect measure space (Z, A, µ), the space L∞ω (µ, E*) of essentially bounded weak* measurable functions on Z to E* is linearly isometric to the space C(Z, E**) of continuous functions on Z to E**, the latter space is being provided with the supremum norm ‖𝑔‖∞ = sup 𝑧∈𝑍 ‖𝑔(z)‖, where E** stands for the space E* endowed with its weak* topology. [ABSTRACT FROM AUTHOR]

In Euclidean geometry, Carnot's theorem is a direct application of the theorem Pythagoras. In [4,6] A.A. Ungar, employs the Einstein gyrovector spaces for the introduction of the gyrotrigonometry. Ungar's and other researcher's works play a major role in translating some theorems from Euclidean geometry to corresponding theorems in Einstein gyrovector spaces. In [2] Demirel and Soytürk proved that hyperbolic Carnot theorem. In this paper, we present Carnot's theorem in Einstein's gyrovector spaces in terms of gamma factors. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of a Quasi B-algebra, building upon the foundational work on Post algebras by G. Epstein and the investigation of semi B-algebras. We systematically explore the properties of Quasi B-algebras, establishing equivalent conditions for the transition from a semi B-algebra to a Quasi B-algebra. [ABSTRACT FROM AUTHOR]

In this paper, as a further generalization of ideals, we introduce the notion of tri-quasi ideals as a generalization of ideals, left ideals, right ideals, tri-ideals, bi-ideals, quasi ideals, interior ideals, bi-interior ideals, weak interior ideals, bi-quasi ideals, quasi-interior ideals and bi-quasi-interior ideals of a semiring. Some charecterizations of a regular semiring and a simple semiring using tri-quasi ideals are given and study the properties of tri-quasi ideals of semirings. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of solutions for the following Kirchhoff-Choquard type equation involving the fractional p-Laplacian on the Heisenberg group: ... where (-Δ)ps is the fractional p-Laplacian on the Heisenberg group ℍN, M is the Kirchhoff function, V (ξ) is the potential function, 0 < s < 1,1 < p < N/s, μ > 0, f (ξ, u) is the nonlinear function, 0 < λ < Q, Q = 2N + 2, and Qλ* = 2Q-λ/Q-2 is the Sobolev critical exponent. Using the Krasnoselskii genus theorem, the existence of infinitely many solutions is obtained if μ is sufficiently large. In addition, using the fractional version of the concentrated compactness principle, we prove that problem has m pairs of solutions if μ is sufficiently small. As far as we know, the results of our study are new even in the Euclidean case. [ABSTRACT FROM AUTHOR]

This paper presents a piecewise convexification method with a box classification strategy to approximate the entire globally optimal solution set of non-convex optimization problems with box constraints. First, the box classification strategy is proposed based on the convexity of the objective function on the sub-boxes, which helps to reduce the number of box divisions and improve the computational efficiency. At the same time, we construct the piecewise convexification problem of the original non-convex optimization problem by applying the α-based Branch-and-Bound (a BB) method, and we define the (approximate) solution set of the piecewise convexification problem based on the result of classifying the sub-boxes. Then, it is deduced that the globally optimal solution set can be approximated by the (approximate) solution set of the piecewise convexification problem. Finally, a piecewise convexification algorithm is proposed that includes a new subset selection technique for division and two new termination tests. The results of our experiments demonstrate the effectiveness and general superiority of our approach over the competition. [ABSTRACT FROM AUTHOR]

In this paper, we study the multiplicity of solutions to the following Kirchhoff type equation with zero mass and Hardy-Littlewood-Sobolev critical nonlinearity ... where N ≥ 3, λ > 0, μ ∈ (0, min{N, 4}), 2μ* = 2n-μ/n-2 is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, and m satisfies some local monotonicity conditions near zero. The non- linearity f is odd in u and satisfies some classical superlinear and quasi-critical growth conditions. For any given k ∈ ℕ, k pairs of nontrivial solutions are obtained for λ large enough by a version of the symmetric mountain pass theorem and a version of the second concentration compactness principle. [ABSTRACT FROM AUTHOR]

A continuous operator T between two Banach lattices E and F is called almost order-weakly compact, whenever for each almost order bounded subset A of E, T(A) is a relatively weakly compact subset of F. We show that the positive operator T from E into a Dedekind complete Banach lattice F is almost order-weakly compact iff T(xn)... 0 in F for each disjoint almost order bounded sequence {xn} in E. In this manuscript, we study some properties of this class of operators and its relationships with the others known classes of operators. [ABSTRACT FROM AUTHOR]

Mathematical series and sequences are crucial in scientific disciplines to identify patterns, make predictions, and deduce mathematical correlations between variables. Chemistry, biology and physics rely heavily on mathematical series to model complex systems, make precise predictions, and identify fundamental principles of chemical and biological processes. The study used a qualitative approach to identify mathematical series used in scientific research and evaluate their application in chemistry and biology. A comprehensive literature review was conducted to gather pertinent papers and articles from credible scientific databases, followed by a thematic analysis strategy to examine the content. The findings of the study revealed that mathematical series are widely used in various fields, including chemistry, biology, and physics. The Taylor series, power series expansion, Fibonacci series, power series and binomial series are some of the most commonly used series. They approximate functions, express reaction rates, solve linear equations, depict spiral patterns, study population growth, and analyze genetics and molecular biology. They are crucial tools in physics, quantum mechanics, and natural phenomena description. [ABSTRACT FROM AUTHOR]

The contribution of this work is to provide tables of Salem numbers with trace −3 and small degrees, namely degrees 2d = 34, 36, 38, and 40. The implemented method also generates a list of totally positive polynomials of degrees d = 17, 18, 19, and 20, with trace 2d − 3, a single root strictly greater than 4, and all other roots strictly between 0 and 4. [ABSTRACT FROM AUTHOR]

We study partial Hölder regularity for nonlinear elliptic systems in divergence form with double-phase growth, modeling double-phase non-Newtonian fluids in the stationary case. [ABSTRACT FROM AUTHOR]

We study local boundedness for subsolutions of nonlinear nonuniformly elliptic equations whose prototype is given by , where the variable coefficient and its inverse are allowed to be unbounded. Assuming certain integrability conditions on and depending on and the dimension, we show local boundedness. Moreover, we provide counterexamples to regularity showing that the integrability conditions are optimal for every . [ABSTRACT FROM AUTHOR]

We establish three Hardy-type inequalities in which the arithmetic means of a sequence of non-negative real numbers are replaced by the weighted means over nested subsets of the sequence. This work originated from some delicate calculations previously performed for evaluating the norm of infinite L-matrices. [ABSTRACT FROM AUTHOR]

In this paper, we present some further refinements of the Tan-Xie inequality for sector matrices and its applications due to Nasiri and Furuichi [J. Math. Inequal., 15 (2021), 1425-1434]. [ABSTRACT FROM AUTHOR]

In virtue of the conception of the multiplicative tempered fractional integrals, put forward by Fu et al. in the published article [AIMS Math., 6 (7): 7456-7478, 2021], we present a fractional integral identity for * differentiable functions. Based upon it, we develop several inequalities of Hermite-Hadamard type in association with * differentiable multiplicative mpreinvexity and (s,m) -preinvexity. [ABSTRACT FROM AUTHOR]

This paper aims to establish some results on the structure of fixed point sets for mappings in 2-inner product spaces. To this end, we employ some well-known techniques of 2-inner product spaces. [ABSTRACT FROM AUTHOR]

In this paper, the relationship between r -convexity and Schur-convexity is investigated first. Moreover, the r -convexity and Schur-convexity of a class of functions are studied. As applications, some new inequalities on Minkowski's inequality are established. [ABSTRACT FROM AUTHOR]

In this paper, for a function f: X → R, we introduce the following expression: Δλ (f)(x,y):=λ f (x)+(1-λ) f (y)- f (λ x+(1-λ)y), where x,y ∈X and λ ∈ R. The purpose of this article is to characterize this expression, by finding various estimates of it. We also give some characterizations of Δλ (f)(x,y) when function f is convex, which prove refinements of Young's inequality. Finally, we give several inequalities in a normed space. [ABSTRACT FROM AUTHOR]

The article explores the longest distance problem in mathematics, specifically focusing on the longest path between two points. It introduces a mathematical model and discusses the requirements for acceptable paths, such as continuity and differentiability. The article provides mathematical formulas and proofs to support its arguments. It also discusses the longest path problem with restricted slope and the optimization of path length. The findings are useful for solving optimization problems and mathematical analysis. [Extracted from the article]

In the paper, we discussed the distribution of unique range sets and its elements over the extended complex plane from a different point of view and obtained some new results regarding the structure and position of unique range sets. These new results have immense applications like classifying different subsets of C to be or not to be a unique range set, exploring the fact that every bi-linear transformation preserves unique range sets for meromorphic functions, providing simpler and shorter proofs of existence of some unique range sets, unfolding the fact that zeros or poles of any meromorphic function lie in a unique range set, in particular, identifying the Fundamental Theorem of Algebra to a more specific region and many more applications. We have also posed some open questions to unveil the mysterious arrangement of the elements of unique range sets. [ABSTRACT FROM AUTHOR]

Invesigation of bounds for Hankel determinat of analytic univalent functions is prominentintrest of many researcher from early twenth century to study geometric properties. Manyauthors obtained non sharp upper bound of third Hankel determinat for different subclasses ofanalytic univalent functions until Kwon et al. [5] obtained exact estimation of the fourth coefficeientof Carath'eodory class. Recently authors made use of an exact estimation of the fourthcoefficient, well known second and third coefficient of Carath'eodory class obtained sharp bound for the third Hankel determinant associated with subclasses of analytic univalent functions.Let w = f(z) = z + a2z² + · · · be analytic in the unit disk D = {z ∊ C: |z| < 1}, and Sbe the subclass of normalized univalent functions with f(0) = 0, and f'(0) = 1. Let z = f-1be the inverse function of f, given by f-1(w) = w + t2w² + · · · for some |w| < ro(f). LetSc ⊂ S be the subset of convex functions in D. In this paper, we estimate the best possibleupper bound for the third Hankel determinant for the inverse function z = f-1 when f ∊ Sc.Let Sc be the class of convex functions. We prove the following statements (Theorem): Iff ∊ Sc, then - H3,1(f-1)| = 1 36 and the inequality is attained for p0(z) = (1 + z³)/(1 - z³). [ABSTRACT FROM AUTHOR]

Groups of infinite unitriangular matrices over associative unitary rings are considered. These groups naturally act on infinite dimensional free modules over underlying rings. They are profinite in case underlying rings are finite. Inspired by their connection with groups defined by finite automata the problem to construct faithful representations of free products of groups by banded infinite unitriangular matrices is considered. For arbitrary prime p a sufficient conditions on a finite set of banded infinite unitriangular matrices over unitary associative rings of characteristic p under which they generate the free product of cyclic p-groups is given. The conditions are based on certain properties of the actions on finite dimensional free modules over underlying rings. It is shown that these conditions are satisfied. For arbitrary free product of finite number of cyclic p-groups constructive examples of the sets of infinite unitriangular matrices over unitary associative rings of characteristic p that generate given free product are presented. These infinite matrices are constructed from finite dimensional ones that are nilpotent Jordan blocks. A few open questions concerning properties of presented examples and other types of faithful representations are formulated. [ABSTRACT FROM AUTHOR]

We provide a new construction of rationality for cubic fourfolds via Mori's theory and the minimal model program. As an application we present the solution of the Kuznetsov conjecture for d D 42 (the first open case). Our methods also show an explicit connection between the rationality of cubic fourfolds belonging to the first four admissible families Cd with d D 14; 26; 38, and 42 and some birational models of K3 surfaces of degree d contained in well known rational Fano fourfolds. [ABSTRACT FROM AUTHOR]

The objective of this study is to incorporate topological space into the realm of n-Cylindrical Fuzzy Neutrosophic Sets (n-CyFNS), which are the most novel type of fuzzy neutrosophic sets. In this paper, we introduce n-Cylindrical Fuzzy Neutrosophic Topological Spaces (n-CyFNTS), n-Cylindrical Fuzzy Neutrosophic (n-CyFN) open sets, and n-CyFN closed sets. We also defined the n-CyFN base, n-CyFN subbase, and some related theorems here. [ABSTRACT FROM AUTHOR]

This paper aims to study Pythagorean and Fermatean Fuzzy Subgroups (FFSG) in the context of T-norm and S-conorm functions. The paper examines the extensions of fuzzy subgroups, specifically "Pythagorean Fuzzy Subgroups (PFSG)" and "FFSG", along with their properties. In the existing literature on Pythagorean and FFSG, the standard properties for membership and non-membership functions are based on the "min" and "max" operations, respectively. However, in this work, we develop a theory that utilizes the T-norm for "min" and the S-conorm for "max", providing definitions of Pythagorean and FFSG with these functions, along with relevant examples. By incorporating this approach, we introduce multiple options for selecting the minimum and maximum values. Additionally, we prove several results related to Pythagorean and FFSG using the T-norm and S-conorm, and discuss important properties associated with them. [ABSTRACT FROM AUTHOR]

In this paper, we define the term "η-fuzzy subgroup" and show that every fuzzy subgroup is a η-fuzzy subgroup. We define some of the algebraic properties of the concept of η-fuzzy cosets. Furthermore, we initiate the study of the η-fuzzy normal subgroup and the quotient group with respect to the η-fuzzy normal subgroup and demonstrate some of their various group theoretical properties. [ABSTRACT FROM AUTHOR]

In our work in this paper, we define intuitionistic fuzzy complex subgroups with respect to t-norm T and s-norm S and investigate some properties of them in detail. Next, we obtain some results about them and give some relationships between them. Later, we introduce the inverse, composition, intersection and normality of them and we prove some basic new results and present some properties of them such that the inverse and composition of two intuitionistic fuzzy complex subgroups with respect to t-norm T and s-norm S will be intuitionistic complex fuzzy subgroups with respect to t-norm T and s-norm S. Also we consider and give some characterizations of them. Finally, we discuss them under group homomorphisms and investigate some related properties such that the image and preimage of two intuitionistic fuzzy complex subgroups with respect to t-norm T and s-norm S will be intuitionistic complex fuzzy subgroups with respect to t-norm T and s-norm S. [ABSTRACT FROM AUTHOR]

An effective and flexible method for encoding ambiguous data is using cubic sets. The concept of incline algebraic substructure is considered and is interlinked with the notation of the cubic set to define cubic subincline. The sense of cubic sub incline of algebra is established with relevant results. Additionally, the results such as homomorphic image, preimage, cartesian product and level sets of cubic sub incline are worked out in this study, and several of its associated findings were looked into. [ABSTRACT FROM AUTHOR]

We consider semidefinite relaxations of Stable-Set and Coloring, which are based on quadratic 0-1 optimization. Information about the stability number and the chromatic number is hidden in the objective function. This leads to simplified relaxations which depend mostly on the number of vertices of the graph. We also propose tightenings of the relaxations which are based on the maximal cliques of the underlying graph. Computational results on graphs from the literature show the strong potential of this new approach. [ABSTRACT FROM AUTHOR]

Since discovery of fully homomorphic encryption by Gentry in 2009, interest of lattice-based cryptography has significantly increased. Several researches have improved the efficiency of homomorphic encryption (HE) schemes. Despite all those powerful results, huge computational cost of underlying operations limits feasibility of practical implementations. In order to avoid computations over large numbers, residue number system arithmetic was included into HE cryptosystems scheme. In this paper, residue number system (RNS) and its representation were introduced and applications of RNS representation in some homomorphic encryptions was presented and some of them are analyzed extensively. This paper aims to present a brief summary of RNS variants of some homomorphic encryptions. [ABSTRACT FROM AUTHOR]

Let G = (V, E) be a graph. A Roman dominating function (RDF) f : V {0,1,2} in satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of an RDF f is f(V) = ∑ue f (v). The Roman domination number of a graph G, denoted by yr(G), is the minimum weight of an RDF on G. In this paper, the Roman domination numbers of some graphs are given. [ABSTRACT FROM AUTHOR]

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