Prime order element graph of a group

In this paper, we introduce a new graph called prime order element graph \(\Gamma (G)\) $\mathrm{\Gamma }(G)$ defined on a finite group G. Its set of vertices is the group G and two distinct vertices \(x,y \in G\) $x,y\in G$ are adjacent if order of xy is prime. We study the connectivity, degree, girth, domination number, bipartiteness etc. of the proposed graph. In many cases, we find useful information about the group G from its associated graph \(\Gamma (G)\) $\mathrm{\Gamma }(G)$ . Finally, we conclude with some open issues and direction of further studies.