Asian-European Journal of Mathematics

The power graph of a finite group G is a special type of undirected simple graph whose vertex set is set of elements of G, in which two distinct vertices of G are adjacent if one is the power of other. Let $G=Z_2^{k-1}\times Z_4$ be a finite abelian 2-group of order $2^{k+1}$ where $k\ge 1$. In this paper, we establish that the power graph of finite abelian group G always has graceful labeling without any condition on $k$.

The Young type inequality asserts that if $a,b$ are two positive real numbers, then for each $0\le \nu \le 1$, we have

Our aim is to introduce an updated and real generalization of Kannan’s fixed point theorem with the help of $F$-contraction introduced by Wardowski for single-valued mappings. Our result can be useful to ascertain the existence of fixed point for a family of mappings for which neither the Wardowski’s result nor that of Kannan can be applied directly. Our result has been applied to solve a particular type of integral equation. Finally, we establish a Reich-type extended version of the main result.

In this paper, we define a module $M$ to be ${\mathrm{\text{SIP}}}^z$ if and only if intersection of each pair of $z$-closed direct summands is also a direct summand of $M$. We investigate structural properties of ${\mathrm{\text{SIP}}}^z$-modules and locate the implications between the other module properties which are essentially based on direct summands. We deal with decomposition theory as well as direct summands of ${\mathrm{\text{SIP}}}^z$-modules. We apply our results to matrix rings. To this end, it is obtained that the ${\mathrm{\text{SIP}}}^z$ property is not Morita invariant.

For a graph $G=(V,E)$, a double Roman dominating function is a function $f:V(G)\to \{0,1,2,3\}$ having the property that if $f(v)=0$, then vertex $v$ must have at least two neighbors assigned $2$ under $f$ or one neighbor with $f(w)=3$, and if $f(v)=1$, then vertex $v$ must have at least one neighbor with $f(w)\ge 2$. The weight of a double Roman dominating function $f$ is the value $w(f)={\sum }_{v\in V(G)}f(v)$. The double Roman domination number of a graph $G$, denoted by ${\gamma }_{dR}(G)$, equals the minimum weight of a double Roman dominating function on $G$. The double Roman domination subdivision number ${\mathrm{\text{sd}}}_{{\gamma }_{dR}}(G)$ of a graph $G$ is the minimum number of edges that must be subdivided (each edge in $G$ can be subdivided at most once) in order to increase the double Roman domination number. In this paper, we first show that the decision problem associated with sd${}_{{\gamma }_{dR}}(G)$ is NP-hard and then establish upper bounds on the double Roman domination subdivision number for arbitrary graphs.

In computability theory, the standard tool to classify preorders is provided by the computable reducibility. If $R$ and $S$ are preorders with domain $\omega$, then $R$ is computably reducible to $S$ if and only if there is a computable function $f$ such that for all $x$ and $y$, $xRy$ $\iff$$f(x)Sf(y)$. We study the complexity of preorders which arise in a natural way in computable structure theory. We prove that the relation of computable isomorphic embeddability among computable torsion abelian groups is a ${\mathrm{\Sigma }}_3^0$ complete preorder. A similar result is obtained for computable distributive lattices. We show that the relation of primitive recursive embeddability among punctual structures (in the setting of Kalimullin et al.) is a ${\mathrm{\Sigma }}_2^0$ complete preorder.

Let $B(G)$ denote the set of all blocks of a graph $G$. Two vertices are said to vv-dominate each other if they are vertices of the same block. A set $D\subseteq V$ is said to be vertex–vertex dominating set (vv-dominating set) if every vertex in $G$ is vv-dominated by some vertex in $D$. The vv-domination number ${\gamma }_{\mathrm{\text{vv}}}={\gamma }_{\mathrm{\text{vv}}}(G)$ is the cardinality of the minimum vv-dominating set of $G$. In this paper, we introduce new kind of graph energy, the minimum vv-dominating Laplacian energy of a graph denoting it as LE${}_{\mathrm{\text{vv}}}(G)$. It depends both on the underlying graph of $G$ and the particular minimum vv-dominating set of $G$. Upper and lower bounds for LE${}_{\mathrm{\text{vv}}}(G)$ are established and we also obtain the minimum vv-dominating Laplacian energy of some family of graphs.

The concept of Sombor indices (SO) of a graph was recently introduced by Gutman and the reduced Sombor index ${\mathrm{\text{SO}}}_{\mathrm{\text{red}}}$ of a graph $G$ is defined by ${\mathrm{\text{SO}}}_{\mathrm{\text{red}}}(G)={\sum }_{uv\in E(G)}\sqrt{{(d_G(u)-1)}^2+(d_G{(v)}^2-1)},$ where $d_G(v)$ is the degree of the vertex $v$. In this paper, we study the extremal properties of the reduced Sombor index and characterize the bicyclic graphs with extremal ${\mathrm{\text{SO}}}_{\mathrm{\text{red}}}$-value.

The main purpose of this work is to study the existence of solutions for a perturbed second-order evolution inclusion involving time-dependent subdifferential operators. Under suitable conditions on the set-valued perturbation, the main result of the paper is proved in the context of a separable Hilbert space. A second-order evolution quasi-variational inequality is also investigated.

Irreducible polynomials over finite fields and their applications have been quite well studied. Here, we discuss the construction of the irreducible polynomials of degree $4n$ over the finite field $F_{2^s}$ for a given irreducible polynomial of degree $n$. Furthermore, we construct the irreducible polynomials of degree $p^{2}n$ over the finite field $F_{p^s}$ for a given irreducible polynomial of degree $n$ by using the method of composition of polynomials with some conditions on coefficients and degree of a given irreducible polynomial.

This paper’s purpose is to study the nonlinear Fredholm implicit integro-differential equation in the complex plane, where the term implicit integro-differential means that the derivative of unknown function is founded inside of the integral operator. Initially, according to Banach fixed point theory, we ensure that the equation has a unique solution under particular conditions. However, we exhibit a numerical process based on the conjunction between Nyström and Picard methods, for the sake of approximating solutions of this equation. In addition to that, the convergence analysis of this numerical process is demonstrated, and some illustrated numerical examples are presented.

In this paper, the inverse problem of reconstructing the right-hand side from integral observation is studied by using variational method to minimizing objective functional combining with Tikhonov regularization. The L-curve method is suggested for choosing the regularization parameter.

The nonlinear Conjugate gradient method (CGM) is a very effective way in solving large-scale optimization problems. Zhang et al. proposed a new CG coefficient which is defined by $B_k^{\mathrm{\text{NPRP}}}$. They proved the sufficient descent condition and the global convergence for nonconvex minimization in strong Wolfe line search. In this paper, we prove that this CG coefficient possesses sufficient descent conditions and global convergence properties under the exact line search.

Several upper estimates for the numerical radius of Hilbert space operators are given. Among many other inequalities, it is shown that if $A,B\in \mathcal{ℬ}(\mathcal{\mathscr{H}})$, then

The irregularity of a simple undirected graph $G$ is defined as $\mathrm{\text{irr}}(G)={\sum }_{uv\in E(G)}|d_G(u)-d_G(v)|$, where $d_G(u)$ denotes the degree of a vertex $u\in V(G)$. The total irregularity of a graph as a new measure of graph irregularity is defined as ${\mathrm{\text{irr}}}_t(G)=\frac{1}{2}{\sum }_{u,v\in V(G)}|d_G(u)-d_G(v)|$. In this paper, the irregularity and the total irregularity of fractal graphs and the derived graphs from a class of fractal graphs are investigated. Exact formulae are presented for the computation of the irregularity and total irregularity of fractal-type graphs in terms of the parameters of the underlying graphs.

In this paper, we consider mixed approximation operators based on second-kind beta transform using Szász–Mirakjan operators. For the proposed operators, we establish some direct results, Voronovskaya-type theorem, quantitative Voronovskaya-type theorem, Grüss–Voronovskaya-type theorem, weighted approximation and functions of bounded variation.

In this paper, one-dimensional time fractional telegraph equation is deduced using fractional Ohm’s law in transmission lines. The nonlocal behavior, arising from the time fractality, is discussed via fractional calculus. Here, the time fractional differential equation related to electrical GLRC circuit is proposed in terms of the Liouville–Caputo fractional derivative. To keep the dimensionality of the physical parameters $L,C$ the new parameter $\sigma$ is introduced. This parameter ($\sigma$) characterizes the existence of fractional structure in the system. The analytical solution of the proposed equation is obtained by using Laplace transform method in terms of Mittag-Leffler function which describes physical system with memory. The order of the time derivative being considered is $0<\alpha \le 1$. The result of an example shows that the behavior of voltage depends on the value of the fractional order of the proposed equation.

All groups under consideration are finite. Let $\sigma =\{{\sigma }_i|i\in I\}$ be some partition of the set of all primes $\mathbb{P}$, $G$ be a group, and $𝔉$ be a class of groups. Then, $\sigma (G)=\{{\sigma }_i|{\sigma }_i\cap \pi (G)\ne \varnothing \}$ and $\sigma (𝔉)={\bigcup }_{G\in 𝔉}\sigma (G).$

A function $f$ of the form $f:\sigma \to \{\text{formations of groups}\}$ is called a formation $\sigma$-function. For any formation $\sigma$-function $f$ the class $L{F}_{\sigma }(f)$ is defined as follows:

Let $\tau (G)$ be a set of subgroups of $G$ such that $G\in \tau (G)$. Then, $\tau$ is called a subgroup functor if for every epimorphism $\phi$ : $A\to B$ and any groups $H\in \tau (A)$ and $T\in \tau (B)$ we have $H^{\phi }\in \tau (B)$ and $T^{{\phi }^{-1}}\in \tau (A)$. A class of groups $𝔉$ is called $\tau$-closed if $\tau (G)\subseteq 𝔉$ for all $G\in 𝔉$.

In this paper, we study the properties of $\tau$-closed $n$-multiply $\sigma$-local formations generated by some set of groups. In particular, we obtained a description of the smallest $\sigma$-local definition of such formation, we also prove that $\tau$-closed $n$-multiply $\sigma$-local formation $𝔉$ is one-generated if and only if $\sigma (𝔉)$ is finite and $𝔉$ have a integrated $\sigma$-local definition $f$ such that $f({\sigma }_i)$ is a one-generated $\tau$-closed $(n-1)$-multiply $\sigma$-local formation for all ${\sigma }_i\in \sigma (𝔉)$. In addition, we prove that the class of all $\tau$-closed one-generated $n$-multiply $\sigma$-local formations coincides with the class of all one-generated $\tau$-closed $n$-multiply $\sigma$-local formations.

In this paper, we propose and study a new algorithm for finding a common element of the sets of solutions of split equality pseudomonotone equilibrium, split equality monotone variational inclusion and fixed point problems for Bregman relatively nonexpansive mappings in $p$-uniformly convex and uniformly smooth Banach spaces. Our iterative method uses stepsizes which do not require prior knowledge of the operator norm. We apply our result to solve split equality variational inequality and split equality convex minimization problems. The result present in this paper unifies and extends several existing results in the literature.

Roopaei [J. Inequal. Appl. 2020 (2020) 120] introduced the sequence space ${{𝒞}}_p^{\alpha }$ which is a matrix domain of Copson matrix in the space ${\ell }_p$ $(1\le p<\infty ).$ In this paper, we characterize the matrix classes $({{𝒞}}_p^{\alpha },{\ell }_{\infty })$, $({{𝒞}}_p^{\alpha },c)$ and $({{𝒞}}_p^{\alpha },c_0)$ $(1\le p<\infty )$. Moreover, we characterize compact operators associated with matrices belonging to these classes via the techniques of measures of noncompactness.

This is an overview of the life and contributions of Leonid Kantorovich, a renowned Russian mathematician and economist.