In this paper, we introduce a new resolvability parameter named as the local edge partition dimension \((LEPD)\) of graphs. The local edge partition dimension \((LEPD)\) makes a specialty of partitioning the vertex set of a graph into awesome instructions based totally on localized resolving properties. Our findings offer a fresh angle on graph resolvability, offering capability insights for optimizing network overall performance and structural analysis. Let \(G=(V, E)\) be a connected graph with vertex set \(V\) and edge set \(E\). A partition set \({R}_{p}=\{{R}_{p1},{R}_{p2},{R}_{p3}\dots,{R}_{pn}\}\) contain subsets of vertices of \(G\). If for every pair of adjacent edges \(p\) and \(q\) in \(G\), then \(d(p,{R}_{p})\neq d(q,{R}_{p})\) and if \(p\) and \(q\) are non-adjacent then not necessary \(d(p,{R}_{p})\neq d(q,{R}_{p})\) then \({R}_{p}\) is called a local edge resolving partition set and minimum cardinality of such set is called local edge partition dimension. We discussed local metric, local edge metric, metric, edge metric dimension, local partition, local edge partition, partition dimension, and edge partition dimension of the Petersen graph.

In this paper we construct families of bit sequences using combinatorial methods. Each sequence is derived by converting a collection of numbers encoding certain combinatorial numerics from objects exhibiting symmetry in various dimensions. Using the algorithms first described in [1] we show that the NIST testing suite described in publication 800-22 does not detect these symmetries hidden within these sequences.

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.