Asian-European Journal of Mathematics

A proper coloring on $V(G)$ is called a $b$-coloring if every color class has a vertex $v$ such that $v$ has at least one neighbor from all other color classes. The maximum integer $k$ for which $G$ admits a $b$-coloring with $k$ colors is called the $b$-chromatic number of $G$. Another interesting coloring parameter is the neighborhood pseudo chromatic number, denoted by ${\psi }_{nhd}(G)$. It is the maximum number of colors used for a pseudo coloring of $V(G)$ such that each vertex $v$ has at least two vertices in $N[v]$ which receive the same color. Motivated by the extensive research and applications in the area of $b$-coloring and pseudo coloring, we introduce a new parameter called the neighborhood $b$-pseudo chromatic number and obtain an interesting characterization to find the neighborhood $b$-pseudo chromatic number of an arbitrary graph.

We consider the class of $\lambda$-pseudo starlike functions $f(z)=z+a_2{z}^2+a_3{z}^3+\cdots$ such that $z{(f^{\prime }(z))}^{\lambda }/f(z)$ maps the open unit disk $\left|z\right|<1$ onto a strip domain $w$ with $\frac{\alpha -\pi }{2sin\alpha }<\Re (w(z))<\frac{\alpha }{2sin\alpha }$ for some $\alpha$, $\pi /2\le \alpha <\pi$. We estimate $\left|a_2\right|$, $\left|a_3\right|$ and solve the Fekete–Szegö problem for functions in this class.

This paper examines almost Kenmotsu manifolds (briefly, AKMs) endowed with the almost Ricci–Yamabe solitons (ARYSs) and gradient ARYSs. The condition for an AKM with ARYS to be $\eta$-Einstein is established. We also show that an ARYS on Kenmotsu manifold becomes a Ricci–Yamabe soliton under certain restrictions. In this series, it is proven that a $(2n+1)$-dimensional ${(\kappa ,\mu )}^{\prime }$-AKM equipped with a gradient ARYS is either locally isometric to ${ℍ}^{n+1}(-4)\times {ℝ}^n$ or the Reeb vector field and the soliton vector field are codirectional. The properties of three-dimensional non-Kenmotsu AKMs endowed with a gradient ARYS are studied.

Let $G$ be a simple, un-directed and connected graph. The graph $G$ has a pair of sets $(V(G),E(G))$, where $V(G)$ is nonempty vertex set and $E(G)$ is an unordered pair of sets with two distinct vertices $u,v$ in $V(G)$. A total $k$-labeling is defined as a function $f_e$ from the edge set to a set $\{1,2,3,\dots ,k_e\}$ and a function $f_v$ from the vertex set to a set $\{0,2,4,\dots ,2k_v\}$, where $k=max\{k_e,2k_v\}$. The total $k$-labeling is a vertex irregular reflexive$k$-labeling of the graph $G$, if for every two different vertices $u$ and $u^{\prime }$ of $G$, $\mathrm{\text{wt}}(u)\ne \mathrm{\text{wt}}(u^{\prime })$, where $\mathrm{\text{wt}}(u)=f_v(u)+{\sum }_{uv\in E(G)}{f}_e(uv)$. The reflexive vertex strength of the graph $G$, denoted by $\mathrm{\text{rvs}}(G)$ is the minimum $k$ for graph $G$ which has a vertex irregular reflexive $k$-labeling. In this paper, we determined the exact value of the reflexive vertex strength of cycle and generalized friendship graph.

In a connected graph $G$, the distance between two vertices of $G$ is the length of a shortest path between these vertices. The eccentricity of a vertex $u$ in $G$ is the largest distance between $u$ and any other vertex of $G$. The total-eccentricity index $\tau (G)$ is the sum of eccentricities of all vertices of $G$. In this paper, we find extremal trees, unicyclic and bicyclic graphs with respect to total-eccentricity index. Moreover, we find extremal conjugated trees with respect to total-eccentricity index.

In this paper, we introduce a new class of operators called the $n$-quasi-$m$-contractive and $n$-quasi-$m$-expansive operators which are generalizations of $n$-quasi-$m$-isometric operators on Hilbert space introduced by Mecheri and Prasad, in [S. Mecheri and T. Prasad, On n-quasi-m-isometric operators, Asian-Eur. J. Math. 9(4) (2016) 1650073]. In addition, we apply those classes to the weighted composition operator. We discuss some examples concerning those classes and the weighted composition operator.

The notion of demi-complementation is introduced on distributive lattices. Certain important properties of demi-complemented distributive lattices are investigated. Some equivalent conditions are given for every demi-complemented lattice to become a pseudo-complemented lattice. The notion of demi-implication algebras is introduced and their properties are investigated. Some applications of demi-complementation to implication like algebras are studied.

Although a lot of work has been done by algebraist on soft set theory and hyperstructure theory, several fundamental results of soft intersection hyperideals in LA-semihypergroups remained untouched. This paper aims at studying some structural properties of LA-semihypergroups by applying soft set theory. In this paper, we define right regular LA-semihypergroups and introduce the concept of soft generalized bi-hyperideals in LA-semihypergroups. Further, we characterize right regular LA-semihypergroups in terms of different kinds of soft right hyperideals, soft left hyperideals and soft generalized bi-hyperideals. Idempotent soft hyperideals in LA-semihypergroups are introduced and some related results on them are obtained. It is proved that in an LA-semihypergroup with left identity, the collection of all soft left hyperideals, which are idempotent forms a commutative monoid. Examples are provided to illustrate the results.

The structure spaces, endowed with the hull kernel topology, of maximal regular congruences, which are prime on a $\mathrm{\Gamma }$-semiring as well as the structure space of prime congruences on a $\mathrm{\Gamma }$-semiring, have been studied via operator semirings. This study has been used to obtain some important results of the structure space of $C_{-}(X)$ of nonpositive real valued continuous functions over a topological space $X$. It has been found that the structure spaces of the semiring $C_{+}(X)$ of nonnegative real valued continuous functions and the $\mathrm{\Gamma }$-semiring $C_{-}(X)$ are homeomorphic. Moreover it has been shown that the structure space of $C_{-}(X)$ is the Stone–Čech compactification of $X$, where $X$ is a Tychonoff space. Furthermore, the $\mathrm{\Gamma }$-semiring analogue of the ‘Banach–Stone Theorem’ has been obtained.

Let $G$ be a nontrivial connected graph and let $c:V(G)\to \mathbb{N}$ be a vertex coloring of $G,$ where adjacent vertices may have the same color. For a vertex $v$ of $G,$ the color sum $\sigma (v)$ of $v$ is the sum of the colors of the vertices adjacent to $v.$ The coloring $c$ is said to be a sigma coloring of $G$ if $\sigma (u)\ne \sigma (v)$ whenever $u$ and $v$ are adjacent vertices in $G.$ The minimum number of colors that can be used in a sigma coloring of $G$ is called the sigma chromatic number of $G$ and is denoted by $\sigma (G).$

In this study, we show that the sigma chromatic number of the middle graph of full binary trees of height $h$ is 2. We also determine a lower bound for the sigma chromatic number of graphs containing an induced subgraph isomorphic to a complete graph joined by pendant vertices. With this lower bound, we obtain the sigma chromatic number of the middle graph of the graphs $G_{n,t},$ $\mathrm{\text{cor}}(K_n),$ stars, bistars, and the middle graph of some families of trees of height 2.

In this paper, the study of linear canonical Hankel pseudo-differential operators has been extended to Sobolev-type spaces. Two versions of pseudo-differential operators involving linear canonical Hankel transformation are defined. We have shown that the pseudo-differential operators and composition of pseudo-differential operators are bounded in certain Sobolev-type spaces associated with linear canonical Hankel transformations.

In this paper, we classify the family of entire functions $\{\rho ,\sigma \}$ such that the induced weighted composition map $A_{\rho ,\sigma }$, is a conjugation namely $J_{b,\alpha }$, where $b$ is an arbitrary complex number and $\alpha$ is a real number such that $0<\alpha <1$. In the last section, we characterize the $J_{b,\alpha }$-symmetric maximal differential operators on the Fock space ${𝔽}_{\alpha }^2$.

In this paper, we study a wide class of $H_v$-modules obtained from ordinary $R$-modules by using $P$-hyperoperations. We define a new $P$-$H_v$-module over $P$-$H_v$-rings and study several properties of it starting from the isomorphism theorem to the fundamental relation.

This paper deals with two new subclasses of holomorphic and bi-univalent functions in the open unit disk defined by generalized bivariate Fibonacci polynomials. In this paper the coefficient bounds are estimated for $\left|a_2\right|$ and $\left|a_3\right|$ which $a_2$ and $a_3$ are the Taylor–Maclaurin coefficients of the functions belonging to these new subclasses. Then, the Fekete–Szegö problem is handled for the functions in these subclasses. Also, several remarks are presented. The results of this paper generalize certain earlier results in the literature.

A class of ring-like structures of events (RLSEs) is studied that generalizes Boolean rings. Quantum logics represented by orthomodular lattices are characterized within this class and the correspondence between Boolean algebras and Boolean rings is enlarged to orthomodular lattices. The structure of RLSEs and various subclasses is analyzed and classical logics are especially identified. Moreover, sets of numerical events within different contexts of physical problems are described. A numerical event is defined as a function $p$ from a set $S$ of states of a physical system to $[0,1]$ such that $p(s)$ is the probability of the occurrence of an event when the system is in state $s\in S$. In particular, the question is answered whether a given (small) set of numerical events will give rise to the assumption that one deals with a classical physical system or a quantum-mechanical one.

In this paper, we establish the concept of $\alpha g^{\#}\psi {\mu }^{*}$-closed sets in Hereditary generalized topological spaces and study some of their basic properties. Further we define $\alpha g^{\#}\psi {\mu }^{*}$-continuity and $\alpha g^{\#}\psi {\mu }^{*}$-irresoluteness and investigate their properties in hereditary generalized topological spaces.

Let $X$ be an arbitrary infinite set, and let ${𝒫}=\{X_i:i\in I\}$ be a partition of $X$ indexed by $I$. Denote by $T(X)$ the full transformation semigroup on $X$, and let $T(X,{𝒫})=\{f\in T(X):(\forall X_i\in {𝒫})(\exists X_j\in {𝒫}),X_{i}f\subseteq X_j\}$. The character map of an element $f$ of $T(X,{𝒫})$ is the self-map ${\chi }^{(f)}:I\to I$ defined as $i{\chi }^{(f)}=j$ whenever $X_{i}f\subseteq X_j$. In this paper, we describe magnifiers in $T(X,{𝒫})$ in terms of their character and restriction maps. We further describe magnifiers in the subsemigroups $\mathrm{\Theta }(X,{𝒫})$, $\mathrm{\Sigma }(X,{𝒫})$, and ${ℬ}(X,{𝒫})$ of $T(X,{𝒫})$ that consist of all transformations whose character maps are injective, surjective, and bijective, respectively.

The main purpose of this paper is to study certain central-valued identities on a factor ring with respect to a prime ideal ${𝒫}$ of a ring ${ℛ},$ involving some specific mappings of ${ℛ}.$ In addition, examples are given to demonstrate that the constraints imposed in the assumptions of our theorems are not superfluous.

In this paper, we present a coherent and rigorous overview of the Laplace–Stieltjes transform on $H_p$ spaces.

Over the $(1,n)$-dimensional real superspace, $n\ge 2$, we study non-trivial deformations of the natural action of the affine Lie superalgebra $𝔞𝔣𝔣(n|1)$ on the direct sum of the superspaces of weighted densities. We compute the necessary and sufficient integrability conditions of a given infinitesimal deformation of this action and we prove that any formal deformation is equivalent to its infinitesimal part. This work is the simplest generalization of a result by Basdouri and Omri [Cohomology and deformation of $𝔞𝔣𝔣(1|1)$ acting on differential operators, Int. J. Geom. Methods Mod. Phys. 15 (2018) 1850072].