Asian-European Journal of Mathematics

The forgotten topological index of a graph $G$, denoted by $F(G)$, is defined as the sum of weights $d{(u)}^2+d{(v)}^2$ over all edges $uv$ of $G$, where $d(u)$ denotes the degree of the vertex $u$. In this paper, we establish upper bounds on the forgotten topological index of trees, unicyclic graphs and bicyclic graphs with a given domination number.

Let ${ℍ}_n$ be the $(2n+1)$-dimensional Heisenberg group and let ${𝕋}^n$ be the $n$-dimensional torus acting on ${ℍ}_n$ by automorphisms. We consider the semidirect product group $G_n:={𝕋}^n⋉{ℍ}_n.$ The cortex, $\mathrm{\text{cor}}(G_n),$ of $G_n$ is the set of all unitary irreducible representations $\pi$ in the unitary dual $\widehat{G_n}$ of $G_n$ that cannot be Hausdorff separated from the identity representation $1_{G_n}$ of $G_n.$ In this paper, we describe explicitly the cortex ($\mathrm{\text{cor}}(G_n)$) of $G_n$ using the coadjoint orbits of the group.

In this paper, we establish several new integral inequalities involving Caputo $k$-fractional derivatives for $\eta$-quasi-convex and $s$-Godunova–Levin convex. In order to obtain our results, we use some classical inequalities such as Hölder’s inequality and its power mean and weighted versions.

In this paper, a modified iteration method (MIM) has been proposed to solve nonlinear second-order ODEs. Convergence analysis and error estimate of the proposed method are also discussed. Computational efficiency of this method is illustrated through numerical examples.

The paper introduces the $k$th-order slant Toeplitz operator on the Lebesgue space of $n$-torus, where $k=(k_1,k_2,\dots ,k_n)$ such that $k_t\ge 2$ for all $t$. It investigates certain properties of $k$th-order slant Toeplitz operators on the Lebesgue space $L^2({𝕋}^n)$. The paper deals with a system of operator equations, characterizing the $k$th-order slant Toeplitz operators. At the end, we discuss certain spectral properties of the considered operator.

In this paper, we investigate the flow of curve and its equiform geometry in 4-dimensional Galilean space. We obtain that the Frenet equations and curvatures of inextensible flows of curves and its equiformly invariant vector fields and intrinsic quantities are independent of time. We find that the motions of curves and its equiform geometry can be defined by the inviscid and stochastic Burgers’ equations in 4-dimensional Galilean space.

In this paper, we obtain strong convergence results for asymptotically demicontractive and asymptotically hemicontractive mappings in hyperbolic spaces. We present our results in hyperbolic spaces. This class of spaces contains both linear and nonlinear spaces like CAT(0) spaces, $ℝ$-trees, Banach spaces and Hilbert spaces. Thus our results are not only novel but also much more general.

Let $G$ be a graph with $n$ vertices, and let $d_u$ be the degree of the vertex $u$ in the graph $G$. The Randić matrix of $G$ is the square matrix of order $n$ whose $(i,j)$-entry is equal to ${(d_u{d}_v)}^{\frac{-1}{2}}$ if the vertex $u$ and the vertex $v$ of $G$ are adjacent, and $0$ otherwise. The Randić eigenvalues of $G$ are the eigenvalues of its Randić matrix and the Randić energy of $G$ is the sum of the absolute values of its Randić eigenvalues. In this paper, we obtain some new results for the Randić eigenvalues and the Randić energy of a graph.

This paper considers a non-Markovian priority retrial queue which serves two types of customers. Customers in the regular queue have priority over the customers in the orbit. This means that the customer in orbit can only start retrying when the regular queue becomes empty. If another customer arrives during a retrial time, this customer is served and the retrial has to start over when the regular queue becomes empty again. In this study, a particular interest is devoted to the stochastic monotonicity approach based on the general theory of stochastic orders. Particularly, we derive insensitive bounds for the stationary joint distribution of the embedded Markov chain of the considered system.

The aim of this paper is to construct a relation between tribonacci numbers and generalized tribonacci numbers. Besides, certain conditions are obtained to generalize the representation of a positive integer $N$ which is determined in [S. Badidja and A. Boudaoud, Representation of positive integers as a sum of distinct tribonacci numbers, J. Math. Statistic. 13 (2017) 57–61] for a $k$-generalized Fibonacci numbers ${(F_n^{(k)})}_{n\in {\mathbb{N}}^{\ast }}$. Lastly, some applications to cryptography are given by using ${(F_n^{(k)})}_{n\in {\mathbb{N}}^{\ast }}$.

An Interesting result on a curious convergent series is proved by A. J. Kempner. In this short note we study and prove a result on another modification of Harmonic series.

An edge dominating set in a graph $G=(V,E)$ is a subset $F$ of $E$ such that each edge in $G$ is either in $F$ or adjacent to at least an edge in $F$. The edge domination number ${\gamma }^{\prime }(G)$ is the minimum cardinality of an edge dominating set in $G$. A subset $S$ of vertices in $G$ is $2$-independent if every vertex of $S$ has at most one neighbor in $S.$ The $2$-independence number ${\beta }_2(G)$ is the maximum cardinality of a $2$-independent set of $G.$ We show that if $T$ is a nontrivial tree, then ${\beta }_2(T)\ge 2{\gamma }^{\prime }(T)$ and we point out that this inequality is not valid for all bipartite graphs. Moreover, we characterize all trees that achieve equality in this bound.

Whereas a characterization of Weyl–Heisenberg frame operators is known, the important class of wavelet frame operators is yet to be characterized. We fill this lacuna in this paper. We first look at the problem in an abstract setting, by looking at a special class of Hilbert frames which we call admissible frames. A simple characterization of frame operators of this class of frames is obtained in terms of “local commutativity” of operators and, specializing to wavelet frames, this leads to a characterization of wavelet frame operators on $L^2(ℝ)$. Necessary and sufficient conditions for two admissible frames to be dual to each other are also presented.

In this work, by using the E. Study map, we construct timelike ruled surfaces with constant Disteli-axis based upon the curvature theory of dual hyperbolic (respectively, Lorentzian) spherical curves which match to timelike ruled surfaces in Minkowski 3-space ${𝔼}_1^3$. Meanwhile, some well-known formulae of surfaces theory into line space and their geometrical explanations are investigated.

Given a conjugation $C$ on a separable complex Hilbert space $H$, a bounded linear operator $T$ on $H$ is said to be $C$-symmetric if $CTC=T^{\ast }$. In this paper, we study some nonlinear preserving problems concerning $C$-symmetric operators and diagonal operators, as a result, we also describe those nonlinear preservers of the lattice of invariant subspaces.

We establish the existence of fixed point for F-contractive multivalued mappings of Hardy–Rogers type in metric-like spaces. Then, we introduce a new notion called bilateral approximate fixed points for multivalued mappings and establish a common bilateral approximate fixed point result for two $\alpha$-dominated multivalued contractive mappings in the sense of [M. Arshad, Z. Kadelburgb, S. Radenovic, A. Shoaibe and S. Shukla, Filomat. 31(11) (2017) 3041–3056] on a closed ball of a K-sequentially complete quasi metric-like space. Finally, one can observe that the common bilateral approximate fixed point results coincide with common fixed point results, whenever we reduce the quasi metric-like spaces to metric-like spaces.

Let $G_2$ be a group acting on an abelian group $G_1$ via a homomorphism $\alpha \in \mathrm{\text{Hom}}(G_2;\mathrm{\text{Aut}}(G_1))$ and let a 2-cocycle $𝜀\in Z^2(G_2,G_1,\alpha )$. By Schreier’s theorem, the pair $(\alpha ,𝜀)$ determines a group $G_{(\alpha ,𝜀)}$ which can arise as a non-split extension of $G_1$ by $G_2$, denoted by $G_1\underset{(\alpha ,𝜀)}{\times }{G}_2$ and called the perturbed semidirect product of $G_1$ by $G_2$ under $(\alpha ,𝜀)$. In this paper, we classify the perturbed semidirect products and give some of their properties. Furthermore, we find necessary and sufficient conditions for two perturbed semidirect products to be isomorphic.

Let $S$ be a set of vertices of a connected graph $G$. The Steiner distance of $S$ is the minimum size of a connected subgraph of $G$ containing all the vertices of $S$. The sum of all Steiner distances on sets of size $k$ is called the Steiner $k$-Wiener index. A graph $G$ is modular if for every three vertices $x,y,z$ there exists a vertex $w$ that lies on the shortest path between every two vertices of $x,y,z$. The Steiner 3-Wiener index of a modular graph is obtained in terms of its Wiener index. As concrete examples, we discuss the case of Fibonacci, Lucas cubes and the Cartesian product of modular graphs. The Steiner Wiener index of block graphs is also studied.

The forgotten topological index ($F$-index) of a graph $G$, denoted by $F(G)$, is defined as the sum of cubes of the degrees of all the vertices in a graph. A connected graph with equal order and size is called unicyclic graph, where order is number of vertices and size is number of edges. In this paper, we investigate the new sharp upper bounds on $F$-index in terms of the order and maximum degree.

The object of this paper is to study anti-invariant submanifolds of trans-Sasakian manifolds. We characterize such submanifolds on the basis of parallelism, semi-parallelism and pseudo parallelism of the second fundamental form of the submanifolds. We also characterize totally umbilical anti-invariant submanifolds of trans-Sasakian manifolds. Existence of Legendre curves on such submanifolds has been analyzed. Whether an anti-invariant submanifold of a trans-Sasakian manifold inherits local symmetry from ambient space is investigated here. Nature of Ricci soliton on anti-invariant submanifolds of trans-Sasakian manifolds has been characterized.

In this paper, we continue our study of H${}_v$MV-algebras. We apply the notion of softness to H${}_v$MV-algebras and give some related results. We introduce the notions of soft H${}_v$MV-subalgebras and soft (weak) H${}_v$MV-ideals and investigate their properties. Also, we study the lattice-theoretic properties of soft H${}_v$MV-algebras and soft H${}_v$MV-ideals over an H${}_v$MV-algebra.