This note addresses impracticalities or possible absurdities with regards to the definition corresponding of some graph parameters. To remedy the impracticalities the principle of transmitting the definition is put forward. The latter principle justifies a comprehensive review of many known graph parameters, the results related thereto, as well as the methodology of applications which draw a distinction between connected versus disconnected simple graphs. To illustrate the notion of transmitting the definition, various parameters are re-examined such as, connected domination number, graph diameter, girth, vertex-cut, edge-cut, chromatic number, irregularity index and quite extensively, the hub number of a graph. Ideas around undefined viz-a-viz permissibility viz-a-viz non-permissibility are also discussed.

A bijective mapping \(\varsigma\) assigns each vertex of a graph \(G\) a unique positive integer from 1 to \(|V(G)|\), with edge weights defined as the sum of the values at its endpoints. The mapping ensures that no two adjacent edges at a common vertex have the same weight, and each \(k\)-color class is connected to every other \(k-1\) color class. A graph \(G\) possesses \(b\)-color local edge antimagic coloring if it satisfies the aforementioned criteria and it corresponds to a maximum graph coloring. This paper extensively studies the bounds, non-existence, and results of b-color local edge antimagic coloring in fundamental graph structures.

In this article, we establish fixed point outcomes for mappings that are asymptotically regular within the context of \(b\)-metric spaces. These findings broaden and enhance the familiar outcomes found in existing literature. Additionally, we present corollaries to demonstrate that our results are more encompassing compared to the established findings in the literature.

Investigating the sequence spaces \(e_{p}^{r},\) \(0\leq p<\infty ,\) and \( e_{\infty }^{r}\), is the aim of this work, which is done with some consideration to [1] and [2]. Also, we put forward some elite features of these spaces in terms of their bounded linear operators. To be more specific, we provide a response to the following: which of these spaces contain the properties of the Approximation, the Dunford-Pettis, the Radon-Riesz, and the Hahn-Banach extensions. Our study also examines the rotundity and smoothness of these spaces.

In this article, we present mathematical simulations of non-separable functions (those that would “correspond” to two entangled quantum particles) that lose this character only as a result of approaching the quantum-classical frontier. No mathematical representation of the action of deteriorating agents of quantum entanglement was included in the simulation. Such loss manifests itself both from the point of view of position space and momentum space. For certain limits, compatible with the space considered, the non-separable functions defined here transform into separable functions or cancel each other out at this boundary, thus erasing the (mathematical representation of) the quantum characteristic with no equivalent in the classical world. These simulations do not concern the loss of a physical property or characteristic, but rather the loss of a mathematical characteristic of a function for two quantum particles. The “ghostly action at a distance”, colloquially expressed by Prof. A. Einstein, has a “spatially limited and non-instantaneous action” as it’s opposite, which mathematically takes place in simulations of non-separable quantum functions, as shown here.

When mathematical models of biological phenomena deal with an unknown parameter, it is often assumed that such a parameter follows a normal distribution. This introduces a symmetry assumption into the model. The purpose of this paper is to investigate and quantify the effect of asymmetry on model prediction. We introduce an asymmetry into a model of sexual conflict and toxin allocation by replacing a normal distribution by a shifted beta distribution. This way, we can naturally consider a large family of continuously changing distributions. We isolate the effect of skewness on the model prediction and demonstrate that in most cases, increasing skewness causes a slight increase in optimal toxicity allocation. We conclude that overall, the effect of the skewness is much smaller than the effect of the mean. In fact, for the particular model we studied, skewness does not seem to affect qualitative predictions.

In this paper, the study identified existence regularity of a random attractor for the stochastic dynamical system generated by non-autonomous strongly damping wave equation with linear memory and additive noise defined on \(\mathbb{R}^{n}\). First, to prove the existence of the pullback absorbing set and the pullback asymptotic compactness of the cocycle in a certain parameter region by using tail estimates and the decomposition technique of solutions. Then it proved the existence and uniqueness of a random attractor.

Chemical graph theory, a branch of graph theory, uses molecular graphs for its representation. In QSAR/QSPR studies, topological indices are employed to evaluate the bioactivity of chemicals. Degree-based entropy, derived from Shannon’s entropy, is a functional statistic influenced by the graph and the probability distribution of its vertex set, with informational graphs forming the basis of entropy concepts. Planar octahedron networks have diverse applications in pharmacy hardware and system management. This article explores the Benzenoid Planar Octahedron Network (\(BPOH(n)\)), Benzenoid Dominating Planar Octahedron Network (BDPOH(n)), and Benzenoid Hex Planar Octahedron Network (\(BHPOH(n)\)). We compute degree-based entropies, including Randić entropy, atom bond connectivity (ABC), and geometric arithmetic (GA) entropy, for the Benzenoid planar octahedron network.

We study analytical solutions of a bi-dimensional low-mass gaseous disc slowly rotating around a central mass and submitted to small radial periodic perturbations. Hydrodynamics equations are solved for the equilibrium and perturbed configurations. A wave-like equation for the gas-perturbed specific mass is deduced and solved analytically for several cases of exponents of the power law distributions of the unperturbed specific mass and sound speed. It is found that, first, the gas perturbed specific mass displays exponentially spaced maxima, corresponding to zeros of the radial perturbed velocity; second, the distance ratio of successive maxima of the perturbed specific mass is a constant depending on disc characteristics and, following the model, also on the perturbation’s frequency; and, third, inward and outward gas flows are induced from zones of minima toward zones of maxima of perturbed specific mass, leading eventually to the possible formation of gaseous annular structures in the disc. The results presented may be applied in various astrophysical contexts to slowly rotating thin gaseous discs of negligible relative mass, submitted to small radial periodic perturbations.