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Let $ d_u $ be the degree of a vertex $ u $ of a graph $ G $. The atom-bond sum-connectivity (ABS) index of a graph $ G $ is the sum of the numbers $ (1-2(d_v+d_w)^{-1})^{1/2} $ over all edges $ vw $ of $ G $. This paper gives the characterization of the graph possessing the minimum ABS index in the class of all trees of a fixed number of pendent vertices; the star is the unique extremal graph in the mentioned class of graphs. The problem of determining graphs possessing the minimum ABS index in the class of all trees with $ n $ vertices and $ p $ pendent vertices is also addressed; such extremal trees have the maximum degree $ 3 $ when $ n\ge 3p-2\ge7 $, and the balanced double star is the unique such extremal tree for the case $ p = n-2 $.

In this manuscript, we considered a system of difference equations with delays and we established sufficient conditions to guarantee stability, asymptotic stability and exponential stability. In each type of stability, we created an appropriate space that guarantees us the existence of a fixed point that achieves the required stability.

A fractal–fractional calculus is presented in term of a generalized gamma function (ℓ−gamma function: Γℓ(.)). The suggested operators are given in the symmetric complex domain (the open unit disk). A novel arrangement of the operators shows the normalization associated with every operator. We investigate a number of significant geometric features thanks to this. Additionally, some integrals, such the Alexander and Libra integral operators, are associated with these operators. Simple power functions are among the illustrations that are provided. Additionally, the formulation of the discrete ℓ−fractal–fractional operators is conducted. We demonstrate that well-known examples are involved in the extended operators.

The ABS (atom-bond sum-connectivity) index of a graph G is denoted by ABS(G) and is defined as ∑xy∈E(G)(dx+dy)−1(dx+dy−2), where dx represents the degree of the vertex x in G. In this paper, we derive the best possible upper bounds on the ABS index for fixed-order trees possessing a given maximum degree, which provides a solution to the open problem proposed quite recently by Hussain, Liu and Hua.

Let Apn be the class of functions fz given by fz=zp+ap+nzp+n+ap+n+1zp+n+1+⋯ which are analytic in the open unit disk U. For fz∈Apn, new integral operators O−jfz and Ojfzj=0,1,2,⋯. are considered. The operators O−jfz and Ojfz satisfy OjO−jfz=O−jOjfz=fz and O−j∗Ojfz=Oj∗O−jfz=f∗fz for the convolution ∗ of O−jfz and Ojfz. In the present paper, the dominants for both operators O−jfz and Ojfz and subordinations for O−jfz and Ojfz are discussed. Also, new subclass Tpγm,δ,ρ;m,j concerning with m different boundary points is defined and discussed. Moreover, some interesting problems of Tpγm,δ,ρ;m,j associated with Ojfz are obtained. Furthermore, some interesting examples for our results are considered.

In this paper, we present a novel class of analytic functions in the form h(z)=zp+∑k=p+1∞akzk in the unit disk. These functions establish a connection between the extended Mittag–Leffler function and the quantum operator presented in this paper, which is denoted by ℵq,pn(L,a,b) and is also an extension of the Raina function that combines with the Jackson derivative. Through the application of differential subordination methods, essential properties like bounds of coefficients and the Fekete–Szegő problem for this class are derived. Additionally, some results of special cases to this study that were previously studied were also highlighted.

In this paper, we use the Faber polynomial expansion to obtain bounds for the general coefficients an of bi-univalent functions in the family of analytic functions in the open unit disk. Estimation of the bound value for the initial coefficients of the functions in these classes is also established.

This article concentrates on peristalsis of Maxwell fluid in a symmetric channel with heat and mass convective conditions. Analysis is given in presence of viscous dissipation. The relevant problems statements are constructed and solved for the series solutions of stream function, temperature and concentration. Regular perturbation technique is adopted for small wave number. Heat transfer coefficient is also derived. Results of temperature, concentration and heat transfer coefficient are graphically analyzed. Trapping phenomenon is studied. It is observed that heat and mass transfer Biot numbers have opposite effects on fluid temperature and concentration. Concentration distribution decays with the increase in Schmidt number and Soret number. Heat transfer coefficient has oscillatory behavior and magnitude of heat transfer coefficient increases when heat transfer Biot number enhances. In addition, similar trapping pattern is observed for relaxation parameter and wave number.

Quantum calculus plays a significant role in many different branches such as quantum physics, hypergeometric series theory, and other physical phenomena. In our paper and using quantitative calculus, we introduce a new family of normalized analytic functions in the open unit disk, which relates to both the generalized Mittag–Leffler function and the Jackson differential operator. By using a differential subordination virtue, we obtain some important properties such as coefficient bounds and the Fekete–Szegő problem. Some results that represent special cases of this family that have been studied before are also highlighted.

The purpose of this work is to derive sufficient conditions for the oscillation of all solutions of the third-order functional dynamic equation p2ξϕγ2p1ξϕγ1yΔξΔΔ+pξϕβygξ=0, on a time scale T. In addition, we present some Hille-type conditions for generalized third-order dynamic equations that improve and extend significant contributions reported in the literature without imposing time-scale restrictions. An example is given to demonstrate the essential results.

In the present paper, we investigate and introduce several properties of certain families of analytic functions in the open unit disc, which are defined by q-analogue of Mittag-Leffler function associated with conic domain. A number of coefficient estimates of the functions in these classes have been obtained. Sufficient conditions for the functions belong to these classes are also considered.

In this article, we consider two nonlinear neutral systems with multiple delays. Our main tool here is to use dichotomy theory to construct an implicit solution for these two systems. Utilizing Krasnoselskii’s fixed point theorem, we obtain sufficient criteria for the existence of periodic solutions, as well as for the uniqueness of solutions. The main results expand and generalize certain previously published findings.

Over the past century, private tutoring (PT) in many countries has increased dramatically. Moreover, the main disadvantage of PT is that has a byproduct and a characteristic on the educational system in developing countries in terms of contributing to conditions such as large class sizes, low public expenditures, and an inadequate number of universities. In Saudi Arabia, the spread of PT at school and university levels has yet to be addressed by researchers. One goal of this examination was to research the spread of PT in relation to mathematics being taught at school and university levels. This investigation used quantitative data of the questionnaire to determine the reasons for the spread. A total of 1000 students responded from University of Ha’il. The edge plans given results that are associated and like the assessment that is performed with the whole data. The current study found that a good family financial situation and lack of parental supervision of the children were significant factors underlying the spread of private tutoring at school and university levels.

This work aims to study analytically the nonlinear model for deoxyribonucleic acid (DNA). Based on the complete discrimination and direct method, some new wave solutions are introduced. These solutions are sorted into solitary, periodic, kink (antikink), and singular solutions. Moreover, a part of them is illustrated graphically. Based on Hamilton concepts, we study the bifurcation and phase portrait for the Hamilton system corresponding to the model under consideration.

Using the results of third-order differential subordination, we introduce certain families of admissible functions and discuss some applications of third-order differential subordination for meromorphic functions associated with a linear operator containing a generalized Mittag-Leffler function.

In this paper, we investigate a new family of normalized analytic functions and bi-univalent functions associated with the Srivastava–Attiya operator. We use the Faber polynomial expansion to estimate the bounds for the general coefficients |an| of this family. The bounds values for the initial Taylor–Maclaurin coefficients of the functions in this family are also established.

The topics studied in the geometric function theory of one variable functions are connected with the concept of Symmetry because for some special cases the analytic functions map the open unit disk onto a symmetric domain. Thus, if all the coefficients of the Taylor expansion at the origin are real numbers, then the image of the open unit disk is a symmetric domain with respect to the real axis. In this paper, we formulate the q-differential operator associated with the q-Raina function using quantum calculus, that is the so-called Jacksons’ calculus. We establish a new subclass of analytic functions in the unit disk by using this newly developed operator. The theory of differential subordination inspired our approach; therefore, we geometrically explore the most popular properties of this new operator: subordination properties, coefficient bounds, and the Fekete-Szegő problem. As special cases, we highlight certain well-known corollaries of our primary findings.

In this paper, we introduce and investigate several inclusion relationships of new k-uniformly classes of analytic functions defined by the Mittag-Leffler function. Also, integral-preserving properties of these classes associated with the certain integral operator are also obtained.

In this paper, we intorduce a family of analytic functions in the open unit disk which is bi-univalent. By the virtue of the Faber polynomial expansions, the estimation of n−th(n≥3) Taylor–Maclaurin coefficients an is obtained. Furthermore, the bounds value of the first two coefficients of such functions is established.

In this paper, we derive a number of interesting results concerning subordination and superordination relations for certain analytic functions associated with an extension of the Mittag–Leffler function.

We obtain new outcomes of analytic functions linked with operator Hα,βη,k(f) defined by Mittag–Leffler function. Moreover, new theorems of differential sandwich-type are obtained.

This article deals with Sisko fluid flow exhibiting peristaltic mechanism in an asymmetric channel with sinusoidal wave propagating down its walls. The channel walls in heat transfer process satisfy the convective conditions. The flow and heat transfer equations are modeled and non-dimensionalized. Analysis has been carried out subject to low Reynolds number and long wavelength considerations. Analytical solution is obtained by using the regular perturbation method by taking Sisko fluid parameter as a perturbed parameter. The shear-thickening and shear-thinning properties of Sisko fluid in the present nonlinear analysis are examined. Comparison is provided between Sisko fluid outcomes and viscous fluids. Velocity and temperature distributions, pressure gradient and streamline pattern are addressed with respect to different parameters of interest. Trapping and pumping processes have also been studied. As a result, the thermal analysis indicates that the implementation of a rise in a non-Newtonian parameter, the Biot numbers and Brinkman number increases the thermal stability of the liquid.