A Study on Edge-Set Graphs of Certain Graphs

Let $G(V,E)$ be a simple connected graph, with $\mathrm{\mid }E\mathrm{\mid }=ϵ\mathrm{.}$ 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 $ϵ\mathrm{.}$ We also discuss the characteristics and properties of the edge-set graphs corresponding to certain standard graphs.