Some new approaches to neighborhoods via graphs

This paper aims to increase the accuracy measure of the subgraph of a graph and generate new nano topologies on the power set of vertices and edges of a graph. Firstly, we introduce Ej-neighborhoods and Cj-neighborhoods which depend on vertices and edges of a simple directed graph by using j-neighborhoods for j E {out, in, n, U}. Then, we apply these neighborhoods to present the concepts of Ej-approximations and Cj-approximations. We investigate their main properties and relationships among them. Besides, we define the accuracy measures of a subgraph with the help of these approximations and show that Cj-accuracy measures are the highest when we compare these accuracy measures with the previous one. Furthermore, we generate new nano topologies via obtained approximations and illustrate that these topologies may not be comparable. Finally, we give an application in physics to elucidate the current approximations are more general. Throughout the paper, we summarize all with tables and to the

Automated summary

This paper aims to improve the accuracy measure of a graph's subgraph and create new nano topologies on the power set of the graph's vertices and edges. It introduces Ej-neighborhoods and Cj-neighborhoods based on the vertices and edges of a simple directed graph, using j-neighborhoods for j E {out, in, n, U}. The paper applies these neighborhoods to describe Ej-approximations and Cj-approximations, investigates their properties and relationships, defines the accuracy measures of a subgraph using these approximations, and shows that Cj-accuracy measures are the highest. Furthermore, the paper generates new nano topologies using these approximations and demonstrates that these topologies may not be comparable. Finally, an application in physics is presented to show the wider applicability of the current approximations. Throughout the paper, all findings are summarized using tables.