Search

The ℱ -coindex (forgotten topological coindex) for a simple connected graph G is defined as the sum of the terms ζ G 2 y + ζ G 2 x over all nonadjacent vertex pairs x , y of G , where ζ G y and ζ G x are the degrees of the vertices y and x in G , respectively. The ℱ -index of a graph is defined as the sum of cubes of the vertex degrees of the graph. This was introduced in 1972 in the same paper where the first and second Zagreb indices were introduced to study the structure dependency of total π -electron energy. Therefore, considering the importance of the ℱ -index and ℱ -coindex, in this paper, we study these indices, and we present new bounds for the ℱ -index and ℱ -coindex.

The existing Z -test for correlated proportions under classical statistics is applied when observations in the data are exact and precise. In the current paper, Z -test for correlated proportions is proposed to deal with the data having indeterminate observations. The statistic of Z -test for correlated proportions under neutrosophy is introduced in the paper. The testing procedure of the proposed test is given. The application of the proposed test is given using the price of oil data. From the application, it is clear that the proposed test outperforms the existing Z -test for correlated proportions under classical statistics in terms of flexibility, adequacy, and information. The simulation study also shows the effect of indeterminacy on the statistic of the proposed test.

This paper is devoted to a study of the concept of edge-group choosability of graphs. We say that G is edge- k -group choosable if its line graph is k -group choosable. In this paper, we study an edge-group choosability version of Vizing conjecture for planar graphs without 5-cycles and for planar graphs without noninduced 5-cycles (2010 Mathematics Subject Classification: 05C15, 05C20).

The main purpose of this paper is to study almost α -cosymplectic pseudo metric manifold satisfying certain η -parallel tensor fields. We first focus on the concept of almost α -cosymplectic pseudo metric manifold and its curvature properties. Then, we obtain some results related to the η -parallelity of h , φ h , and τ . Moreover, the deformation of almost α -Kenmotsu pseudo metric structure is given. We conclude the paper with an illustrative example of almost α -cosymplectic pseudo metric manifold.

The number of conjugate classes of derangements of order $n$ is the same as the number $h(n)$ of the restricted partitions with every portion greater than $1$. It is also equal to the number of isotopy classes of $2\times n$ Latin rectangles. Sometimes the exact value is necessary, while sometimes we need the approximation value. In this paper, a recursion formula of $h(n)$ will be obtained, also will some elementary approximation formulae with high accuracy for $h(n)$ be presented. Although we may obtain the value of $h(n)$ in some computer algebra system, it is still meaningful to find an efficient way to calculate the approximate value, especially in engineering, since most people are familiar with neither programming nor CAS software. This paper is mainly for the readers who need a simple and practical formula to obtain the approximate value (without writing a program) with more accuracy, such as to compute the value in an pocket science calculator without programming function. Some methods used here can also be applied to find the fitting functions for some types of data obtained in experiments., Comment: 19 pages, 25 figures, 15 tables

Suppose that p and q are two distinct odd prime numbers with n = p q . In this paper, the uniform representation of general two-prime generalized cyclotomy with order two over ℤ n was demonstrated. Based on this general generalized cyclotomy, a type of binary sequences defined over F l was presented and their minimal polynomials and linear complexities were derived, where l = r s with a prime number r and gcd l , n = 1 . The results have indicated that the linear complexities of these sequences are high without any special requirements on the prime numbers. Furthermore, we employed these sequences to obtain a few cyclic codes over F l with length n and developed the lower bounds of the minimum distances of many cyclic codes. It is important to stress that some cyclic codes in this paper are optimal.

In this paper, the authors investigated the concept of s , m -exponential-type convex functions and their algebraic properties. New generalizations of Hermite–Hadamard-type inequality for the s , m -exponential-type convex function ψ and for the products of two s , m -exponential-type convex functions ψ and ϕ are proved. Many refinements of the (H–H) inequality via s , m -exponential-type convex are obtained. Finally, several new bounds for special means and new error estimates for the trapezoidal and midpoint formula are provided as well. The ideas and techniques of this paper may stimulate further research in different areas of pure and applied sciences.

In this paper, we introduce a new derivative operator involving q -Al-Oboudi differential operator for meromorphic functions. By using this new operator, we define a new subclass of meromorphic functions and obtain the Fekete–Szegő inequalities.

Multiplicative hyperrings are an important class of algebraic hyperstructures which generalize rings further to allow multiple output values for the multiplication operation. Let R be a commutative multiplicative hyperring. A proper hyperideal I of R is called 2-prime if x ∘ y ⊆ I for some x , y ∈ R , then, x 2 ⊆ I or y 2 ⊆ I . The 2-prime hyperideals are a generalization of prime hyperideals. In this paper, we aim to study 2-prime hyperideals and give some results. Moreover, we investigate δ -2-primary hyperideals which are an expansion of 2-prime hyperideals.

The notion of Δ -metric spaces has been proposed in this study as a generalization of b -metric spaces, extended b -metric spaces, and p -metric spaces. A number of topological characteristics of such spaces have been investigated in this paper. On such spaces, a noncompactness measure has been established, and some results in the framework of noncompactness measure have been achieved. We prove an analogous of the Banach contraction principle in such spaces based on this approach. In order to investigate the validity of the underlying space and our proven fixed-point theorems, supporting examples have been presented. Furthermore, the well-posedness of the fixed-point problem has been tested using our fixed-point result.

The graphical representation of finite groups is studied in this paper. For each finite group, a simple graph is associated for which the vertex set contains elements of group such that two distinct vertices x and y are adjacent iff x 2 = y 2 . We call this graph an equal-square graph of the finite group G , symbolized by E S G . Some interesting properties of E S G are studied. Moreover, examples of equal-square graphs of finite cyclic groups, groups of plane symmetries of regular polygons, group of units U n , and the finite abelian groups are constructed.

Let F be the forgotten topological index of a graph G . The exponential of the forgotten topological index is defined as e F G = ∑ x , y ∈ S t x , y G e x 2 + y 2 , where t x , y G is the number of edges joining vertices of degree x and y . Let T n be the set of trees with n vertices; then, in this paper, we will show that the path P n has the minimum value for e F over T n .

The Jordan–Hölder theorem is proved by using Zassenhaus lemma which is a generalization of the Second Isomorphism Theorem for groups. Goursat’s lemma is a generalization of Zassenhaus lemma, it is an algebraic theorem for characterizing subgroups of the direct product of two groups G 1 × G 2 , and it involves isomorphisms between quotient groups of subgroups of G 1 and G 2 . In this paper, we first extend Goursat’s lemma to R -algebras, i.e., give the version of Goursat’s lemma for algebras, and then generalize Zassenhaus lemma to rings, R -modules, and R -algebras by using the corresponding Goursat’s lemma, i.e., give the versions of Zassenhaus lemma for rings, R -modules, and R -algebras, respectively.

In this paper, we study the notion of approximate biprojectivity and left φ -biprojectivity of some Banach algebras, where φ is a character. Indeed, we show that approximate biprojectivity of the hypergroup algebra L 1 K implies that K is compact. Moreover, we investigate left φ -biprojectivity of certain hypergroup algebras, namely, abstract Segal algebras. As a main result, we conclude that (with some mild conditions) the abstract Segal algebra B is left φ -biprojective if and only if K is compact, where K is a hypergroup. We also study the approximate biflatness and left φ -biflatness of hypergroup algebras in terms of amenability of their related hypergroups.

A modular irregular graph is a graph that admits a modular irregular labeling. A modular irregular labeling of a graph G of order n is a mapping of the set of edges of the graph to 1,2 , … , k such that the weights of all vertices are different. The vertex weight is the sum of its incident edge labels, and all vertex weights are calculated with the sum modulo n . The modular irregularity strength is the minimum largest edge label such that a modular irregular labeling can be done. In this paper, we construct a modular irregular labeling of two classes of graphs that are biregular; in this case, the regular double-star graph and friendship graph classes are chosen. Since the modular irregularity strength of the friendship graph also holds the minimal irregularity strength, then the labeling is also an irregular labeling with the same strength as the modular case.

The physical and structural properties of molecular structure or graph such as boiling point, melting point, surface tension, or solubility are studied using topological index (TI). Topological index is a mathematical formula that can be applied to any graph which models some molecular structures. The various operations play an important role in graph theory such as joining, union, intersection, products, and subdivision. In this paper, we computed the bounds for general Randic coindex of F -sum graphs such as ( S -sum, R -sum, Q -sum, and T -sum) in the form of their factor graphs. At the end, results are illustrated by numerical table for the particular F -sum graphs.

In this paper, we study the largest component of the near-critical random intersection graph G n , m , p with n nodes and m elements, where m = Θ n which leads to the fact that the clustering is tunable. We prove that with high probability the size of the largest component in the weakly supercritical random intersection graph with tunable clustering on n vertices is of order n ϵ n , and it is of order ϵ − 2 n log n ϵ 3 n in the weakly subcritical one, where ϵ n ⟶ 0 and n 1 / 3 ϵ n ⟶ ∞ as n ⟶ ∞ .

In this paper, we define α -admissible and α - ϕ -fuzzy cone contraction in fuzzy cone metric space to prove some fixed point theorems. Some related sequences with contraction mappings have been discussed. Ultimately, our theoretical results have been utilized to show the existence of the solution to a nonlinear integral equation. This application is also illustrative of how fuzzy metric spaces can be used in other integral type operators.

The aim of this paper is to define fuzzy contraction in the context of complex valued extended b -metric space and prove fuzzy fixed-point results. Our results improve and extend certain recent results in literature. Moreover, we discuss an illustrative example to highlight the realized improvements. As application, we derive fixed-point results for multivalued mappings in the setting of complex valued extended b -metric space.

Let G be a graph with n vertices, and let L G and Q G denote the Laplacian matrix and signless Laplacian matrix, respectively. The Laplacian (respectively, signless Laplacian) permanental polynomial of G is defined as the permanent of the characteristic matrix of L G (respectively, Q G ). In this paper, we show that almost complete graphs are determined by their (signless) Laplacian permanental polynomials.

In this paper, a finite-dimensional Lie superalgebra K n , m over a field of prime characteristic is constructed. Then, we study some properties of K n , m . Moreover, we prove that K n , m is an extension of a simple Lie superalgebra, and if m = n − 1 , then it is isomorphic to a subalgebra of a restricted Lie superalgebra.

In this paper, we firstly introduce the generalized Reich‐Ćirić‐Rus-type and Kannan-type contractions in cone b -metric spaces over Banach algebras and then obtain some fixed point theorems satisfying these generalized contractive conditions, without appealing to the compactness of X . Secondly, we prove the existence and uniqueness results for fixed points of asymptotically regular mappings with generalized Lipschitz constants. The continuity of the mappings is deleted or relaxed. At last, we prove that the completeness of cone b -metric spaces over Banach algebras is necessary if the generalized Kannan-type contraction has a fixed point in X . Our results greatly extend several important results in the literature. Moreover, we present some nontrivial examples to support the new concepts and our fixed point theorems.

Let G be a simple, connected, and finite graph. For every vertex v ∈ V G , we denote by N G v the set of neighbours of v in G . The locating-dominating number of a graph G is defined as the minimum cardinality of W ⊆ V G such that every two distinct vertices u , v ∈ V G \ W satisfies ∅ ≠ N G u ∩ W ≠ N G v ∩ W ≠ ∅ . A graph G is called k -regular graph if every vertex of G is adjacent to k other vertices of G . In this paper, we determine the locating-dominating number of k -regular graph of order n , where k = n − 2 or k = n − 3 .

In this paper, we focus on the degree of the greatest common divisor ( gcd ) of random polynomials over F q . Here, F q is the finite field with q elements. Firstly, we compute the probability distribution of the degree of the gcd of random and monic polynomials with fixed degree over F q . Then, we consider the waiting time of the sequence of the degree of gcd functions. We compute its probability distribution, expectation, and variance. Finally, by considering the degree of a certain type gcd , we investigate the probability distribution of the number of rational (i.e., in F q ) roots (counted with multiplicity) of random and monic polynomials with fixed degree over F q .

In the present paper, we focused our attention to study pseudo-Ricci symmetric spacetimes in Gray’s decomposition subspaces. It is proved that PRS n spacetimes are Ricci flat in trivial, A , and B subspaces, whereas perfect fluid in subspaces I , I ⊕ A , and I ⊕ B , and have zero scalar curvature in subspace A ⊕ B . Finally, it is proved that pseudo-Ricci symmetric GRW spacetimes are vacuum, and as a consequence of this result, we address several corollaries.

Convexity theory becomes a hot area of research due to its applications in pure and applied mathematics, especially in optimization theory. The aim of this paper is to introduce a broader class of convex functions by unifying geometrically strong convex function with h convex functions. This new class of functions is called as generalized geometrically strongly modified h -convex functions. We established Hermite–Hadamard-type inequalities for the generalized geometrically strongly modified h -convex functions. Our results can be considered as generalization and extension of literature.

The Randić index of a graph G is defined as the sum of weights 1 / d u d v over all edges u v of G , where d u and d v are the degrees of the vertices u and v in G , respectively. In this paper, we will obtain lower and upper bounds for the Randić index in terms of size, maximum degree, and minimum degree. Moreover, we obtain a generally lower and a general upper bound for the Randić index.

Metric dimension is the extraction of the affine dimension (obtained from Euclidean space E d ) to the arbitrary metric space. A family ℱ = G n of connected graphs with n ≥ 3 is a family of constant metric dimension if dim G = k (some constant) for all graphs in the family. Family ℱ has bounded metric dimension if dim G n ≤ M , for all graphs in ℱ . Metric dimension is used to locate the position in the Global Positioning System (GPS), optimization, network theory, and image processing. It is also used for the location of hospitals and other places in big cities to trace these places. In this paper, we analyzed the features and metric dimension of generalized convex polytopes and showed that this family belongs to the family of bounded metric dimension.

LetGbe a graph andH⊆Gbe subgraph ofG. The graphGis said to bea,d-Hantimagic total graph if there exists a bijective functionf:VH∪EH⟶1,2,3,…,VH+EHsuch that, for all subgraphs isomorphic toH, the totalHweightsWH=WH=∑x∈VHfx+∑y∈EHfyforms an arithmetic sequencea,a+d,a+2d,…,a+n−1d, whereaanddare positive integers andnis the number of subgraphs isomorphic toH. Ana,d-Hantimagic total labelingfis said to be super if the vertex labels are from the set1,2,…,|VG. In this paper, we discuss supera,d-C3-antimagic total labeling for generalized antiprism and a supera,d-C8-antimagic total labeling for toroidal octagonal map.

Cyclic triangle-free process (CTFP) is the cyclic analog of the triangle-free process. It begins with an empty graph of order n and generates a cyclic graph of order n by iteratively adding parameters, chosen uniformly at random, subject to the constraint that no triangle is formed in the cyclic graph obtained, until no more parameters can be added. The structure of a cyclic triangle-free graph of the prime order is different from that of composite integer order. Cyclic graphs of prime order have better properties than those of composite number order, which enables generating cyclic triangle-free graphs more efficiently. In this paper, a novel approach to generating cyclic triangle-free graphs of prime order is proposed. Based on the cyclic graphs of prime order, obtained by the CTFP and its variant, many new lower bounds on R 3 , t are computed, including R 3,34 ≥ 230 , R 3,35 ≥ 242 , R 3,36 ≥ 252 , R 3,37 ≥ 264 , R 3,38 ≥ 272 . Our experimental results demonstrate that all those related best known lower bounds, except the bound on R 3,34 , are improved by 5 or more.

In this paper, we use the elementary methods and the estimates for character sums to prove the following conclusion. Let p be a prime large enough. Then, for any positive integer n with p 1 / 2 + ɛ ≤ n < p , there must exist two primitive roots α and β modulo p with 1 < α , β ≤ n − 1 such that the equation n = α + β holds, where 0 < ɛ < 1 / 2 is a fixed positive number. In other words, n can be expressed as the exact sum of two primitive roots modulo p .

In this paper, we use the elementary methods and the estimates for character sums to study a problem related to primitive roots and the Pythagorean triples and prove the following result: let p be an odd prime large enough. Then, there must exist three primitive roots x , y , and z modulo p such that x 2 + y 2 = z 2 .

Algebraic structures play a prominent role in mathematics with wide ranging applications in many disciplines such as theoretical physics, computer sciences, control engineering, information sciences, coding theory, and topological spaces. This provides sufficient motivation to researchers to review various concepts and results from the realm of abstract algebra in the broader framework of fuzzy setting. In this paper, we introduce the notions of int-softm,n-ideals, int-softm,0-ideals, and int-soft0,n-ideals of semigroups by generalizing the concept of int-soft bi-ideals, int-soft right ideals, and int-soft left ideals in semigroups. In addition, some of the properties of int-softm,n-ideal, int-softm,0-ideal, and int-soft0,n-ideal are studied. Also, characterizations of various types of semigroups such asm,n-regular semigroups,m,0-regular semigroups, and0,n-regular semigroups in terms of their int-softm,n-ideals, int-softm,0-ideals, and int-soft0,n-ideals are provided.

In this paper, we study fuzzy congruence relations and kernel fuzzy ideals of an Ockham algebra A , f , whose truth values are in a complete lattice satisfying the infinite meet distributive law. Some equivalent conditions are derived for a fuzzy ideal of an Ockham algebra A to become a fuzzy kernel ideal. We also obtain the smallest (respectively, the largest) fuzzy congruence on A having a given fuzzy ideal as its kernel.

In this paper, we initiate the concept of α -multiplier on almost distributive lattices. We prove some useful results by using the notion of α -multiplier and generalize the idea of multiplier on almost distributive lattices.

This paper is concerned with the periodicity of entire functions with finite growth order, and some sufficient conditions are given. Let f is a transcendental entire function with finite growth order, zero is a Picard exceptional value of f , and a given differential monomial Q f of f is periodic, then f is also periodic. We are also interested in finding the following: let f is a transcendental entire function with finite growth order, d is a Picard exceptional value of f and f z Δ c n f z is a periodic function, then f is also a periodic function. These results extend Yang’s conjecture.

In this paper, we discuss one-dimensional differential equation with ω -period. By using the fixed point theory, the existence of a periodic solution is obtained; by using the second Lyapunov method, the uniqueness and stability of the periodic solution are obtained.

Edge even graceful labeling of a graph G with p vertices and q edges is a bijective f from the set of edge E G to the set of positive integers 2,4 , … , 2 q such that all the vertex labels f ∗ V G , given by f ∗ u = ∑ u v ∈ E G f u v mod 2 k , where k = max p , q , are pairwise distinct. There are many graphs that do not have edge even graceful labeling, so in this paper, we have extended the definition of edge even graceful labeling to r -edge even graceful labeling and strong r -edge even graceful labeling. We have obtained the necessary conditions for more path-related graphs and cycle-related graphs to be an r -edge even graceful graph. Furthermore, the minimum number r for which the graphs: tortoise graph, double star graph, ladder and diagonal ladder graphs, helm graph, crown graph, sunflower graph, and sunflower planar graph, have an r -edge even graceful labeling was found. Finally, we proved that the even cycle C 2 n has a strong 2 -edge even graceful labeling when n is even.

In this paper, we use the analytic methods, the properties of the sixth-order characters, and the classical Gauss sums to study the computational problems of a certain special sixth residues’ modulo p and give two exact calculating formulas for them.

A set S ⊆ V of a graph G = V , E is called a co-independent liar’s dominating set of G if (i) for all v ∈ V , N G v ∩ S ≥ 2 , (ii) for every pair u , v ∈ V of distinct vertices, N G u ∪ N G v ∩ S ≥ 3 , and (iii) the induced subgraph of G on V − S has no edge. The minimum cardinality of vertices in such a set is called the co-independent liar’s domination number of G , and it is denoted by γ coi L R G . In this paper, we introduce the concept of co-independent liar’s domination number of the middle graph of some standard graphs such as path and cycle graphs, and we propose some bounds on this new parameter.

In this paper, we establish a Hausdorff metric over the family of nonempty closed subsets of an extended b -metric space. Thereafter, we introduce the concept of multivalued fuzzy contraction mappings and prove related α -fuzzy fixed point theorems in the context of extended b -metric spaces that generalize Nadler’s fixed point theorem as well as many preexisting results in the literature. Further, we establish α -fuzzy fixed point theorems for Ćirić type fuzzy contraction mappings as a generalization of previous results. Moreover, we give some examples to support the obtained results.

According to the length-preserving projection and Euler discretization method, biorthogonal wavelet function on a smooth curve C is constructed in this paper, such as a logarithm curve. The properties of biorthogonal wavelet filters on a smooth curve C are discussed, such as induced refinable equation and symmetry. Moreover, an example is given for discussing the biorthogonal scaling function and its dual on a logarithm curve C . Finally, a numerical application is given for dealing with financial data.

In the present paper, we establish a Chen–Ricci inequality for a C-totally real warped product submanifold M n of Sasakian space forms M 2 m + 1 ε . As Chen–Ricci inequality applications, we found the characterization of the base of the warped product M n via the first eigenvalue of Laplace–Beltrami operator defined on the warping function and a second-order ordinary differential equation. We find the necessary conditions for a base B of a C-totally real-warped product submanifold to be an isometric to the Euclidean sphere S p .

In this paper, we initiate the concept of orthogonal partial b -metric spaces. We ensure the existence of a unique fixed point for some orthogonal contractive type mappings. Some useful examples are given, and an application is also provided in support of the obtained results.

The 1-crossing index of a graph G is the smallest integer k such that the k th iterated line graph of G has crossing number greater than 1. In this paper, we show that the 1-crossing index of a graph is either infinite or it is at most 5. Moreover, we give a full characterization of all graphs with respect to their 1-crossing index.

Let M be a set of positive integers. A set S of nonnegative integers is called an M ‐ set if a and b ∈ S , then a − b ∉ M . If S ⊆ 0,1 , … , n is an M − set with the maximal cardinality, then S is called a maximal M − set of 0,1 , … , n . If S ∩ 0,1 , … , n is a maximal M − set of 0,1 , … , n for all integers n ≥ 0 , then we call S an optimal M − set. In this paper, we study the existence of an optimal M − set.

In this paper, some characterizations are given in terms of the boundary value and Poisson extension for the Dirichlet-type space D μ . The multipliers of D μ and Hankel-type operators from D μ to L 2 P μ d A are also investigated.

LetGbe a graph withnvertices. For every realα∈0,1, writeAαGfor the matrixAαG=αDG+1−αAG, whereAGandDGdenote the adjacency matrix and the degree matrix ofG, respectively. The collection of eigenvalues ofAαGtogether with multiplicities are called theAα-spectrum ofG. A graphGis said to be determined by itsAα-spectrum if all graphs having the sameAα-spectrum asGare isomorphic toG. In this paper, we show that some joins are determined by theirAα-spectra forα∈0,1/2 or 1/2,1.

In this paper, we study a rational type common fixed-point theorem (CFP theorem) in complex-valued generalized b -metric spaces ( G b -metric spaces) by using three self-mappings under the generalized contraction conditions. We find CFP and prove its uniqueness. To justify our result, we provide an illustrative example. Furthermore, we present a supportive application of the three Urysohn type integral equations (UTIEs) for the validity of our result. The UTIEs are

In this paper, we define a new family of separation axioms in the classical topology called functionally T i spaces for i = 0,1,2 . With the assistant of illustrative examples, we reveal the relationships between them as well as their relationship with T i spaces for i = 0,1,2 . We demonstrate that functionally T i spaces are preserved under product spaces, and they are topological and hereditary properties. Moreover, we show that the class of each one of them represents a transitive relation and obtain some interesting results under some conditions such as discrete and Sierpinski spaces.