In this research article, the authors introduce the refinements of some special inequalities, like Lah-Ribarič type, Giaccardi, and Petrović’s inequalities. Also, the authors define Fejér, Giaccardi, and Petrović’s types of inequalities for different classes of convex functions.

In this study, an approximate solution of the Sitnikov problem was investigated using fourth-order Runge – Kutta method. We confirmed the periodicity and the symmetric nature of the orbits. The various values of eccentricities were obtained which showed that at eccentricity e = 0, the orbit moves in a circular shape and otherwise when e < 0. Also at every values of e, we found the numerical results which we demonstrated by simulations using MATCAD which showed that the range for the search of eccentricities can be narrowed down at different values of e, different sinusoidal frequencies were obtained.

Predation models have come close to modelling dynamic and complex economic factors despite its simplistic criticism. Based on Goodwin’s predator-prey framework, this study modelled the dynamics between employment rate and wage share of Ghana’s national output. Empirical data simulations revealed clear cyclical patterns in employment rates and wage shares, reflecting the dynamics in Goodwin’s class struggle theory. The employment rate and wage share exhibited a symbiotic relationship, where changes in one variable significantly influenced the other. The analysis further revealed that although both employment rate and wage share periodically declined, these variables were never annihilated indicative that the economy was resilient. Sensitivity analysis also demonstrated the robustness of the model, showing consistent patterns despite variations in initial conditions. After subjecting the model to stability test, the study showed that despite the economic fluctuations during the study period, the economy was generally stable mathematically, with a projected economic growth assured.

A novel topological index, the Sombor index, has been proposed by Ivan Gutman in a recent paper [1]. Motivated by this novel index, we study the new variants of Sombor index and to examine the correlation of newly introduced topological indices we have computed the values of these indices by taking all possible trees on 10 vertices. Here in this paper, we derive explicit formulae for the Sombor index of various nanostructures. These include hexagonal parallelogram \( P(\alpha, \beta) \)-nanotubes, triangular benzenoid \( G_{\alpha} \), and zigzag-edge coronoid fused with starphene nanotubes \( ZCS(k,\alpha,\beta) \), where \( k, \alpha, \beta \) are natural numbers. We also compute the Sombor index for dominating derived networks \( D_{1}, D_{2}, D_{3} \), as well as for various dendrimers such as Porphyrin Dendrimer, Ninc-Porphyrin Dendrimer, Propyl Ether Imine Dendrimers, and Polyamidoamin (PAMAM) Dendrimer. Additionally, we examine Polyamidoamin dendrimers (\( PD_{1}, PD_{2}, DS_{1} \)) and linear polyomino chains like \( L_{\alpha} \), \( Z_{\alpha} \), \( B^{1}_{\alpha}(\alpha \geq 3) \), \( B^{2}_{\alpha}(\alpha \geq 4) \). Finally, we consider benzenoid systems with different shapes, including triangular, hourglass, and jagged-rectangle configurations. By computing the Sombor index for these nanostructures, we provide a comprehensive analysis of their topological properties.

The hub set measures the connectivity of any nodes in graphs and the determination of it is found to be NP-complete. This paper deduces several properties and characterize of one such hub parameter, the doubly connected hub number for its value equal to 1 and 2. Moreover, a few bounds and Nordhaus-Gaddum type inequalities are discussed.

Let \( V(G) = \{v_1, v_2, \ldots, v_n\} \) be the vertex set and \( E(G) = \{e_1, e_2, \ldots, e_m\} \) be the edge set of a graph \( G \). The Seidel adjacency matrix of a graph \( G \) is defined as \( S(G) = [s_{ij}] \) of order \( n \times n \), in which \( s_{ij} = -1 \) if \( v_i \) is adjacent to \( v_j \), \( s_{ij} = 1 \) if \( v_i \) is not adjacent to \( v_j \) and \( s_{ii} = 0 \). We introduce here the \( (-1,1) \)-incidence matrix of \( G \) as \( B_S(G) = [c_{ij}] \) of order \( n \times m \), in which \( c_{ij} = -1 \) if \( v_i \) is incident to \( e_j \) and \( c_{ij} = 1 \) if \( v_i \) is not incident to \( e_j \). Further we explore properties of \( B_S(G) \) and of its transpose.

It is well known that positive Green’s operators are not necessarily positivity preserving. This result is important, because many physical problems require positivity in their solutions in order to make sense. In this paper we investigate the matter of just how far from being positivity preserving a positive Green’s operator can be. In particular, we will see that there exists positive Green’s operators that takes some positive functions to functions with negative mean values. We will also identify a broad class of Green’s operators that are not necessarily positivity preserving but have properties related to positivity preservation that one expects from positivity preserving Green’s operators. Finally, we will compare the results contained in this paper with those that already exist in the literature on the subject.