A Study on Edge-Set Graphs of Certain Graphs

TL;DR

This paper introduces the concept of edge-set graphs derived from simple connected graphs, exploring their properties and how they depend on the original graph's structure and edge count.

Contribution

It defines the edge-set graph construction and analyzes its characteristics for certain standard graphs, highlighting its dependence on the original graph's structure and edges.

Findings

Edge-set graphs are dependent on the structure and number of edges of the original graph.

Properties of edge-set graphs are characterized for specific standard graphs.

The study provides insights into the relationship between a graph and its edge-set graph.

Abstract

Let $G(V, E)$ be a simple connected graph, with $|E| = \epsilon.$ In this paper, we define an edge-set graph $\mathcal G_G$ constructed from the graph $G$ such that any vertex $v_{s,i}$ of $\mathcal G_G$ corresponds to the $i$-th $s$-element subset of $E(G)$ and any two vertices $v_{s,i}, v_{k,m}$ of $\mathcal G_G$ are adjacent if and only if there is at least one edge in the edge-subset corresponding to $v_{s,i}$ which is adjacent to at least one edge in the edge-subset corresponding to $v_{k,m}$ where $s,k$ are positive integers. It can be noted that the edge-set graph $\mathcal G_G$ of a graph $G$ id dependent on both the structure of $G$ as well as the number of edges $\epsilon.$ We also discuss the characteristics and properties of the edge-set graphs corresponding to certain standard graphs.