The Non-Coprime Graph of Finite Groups

The non-coprime graph ΠG of a finite group G is a graph with the vertexset G\{e}, where two distinct vertices u and v are adjacent if they havenon-coprime orders. In this paper, the main properties of the Cartesian andtensor product of the non-coprime graph of two finite groups are investigated.We also describe the non-coprime graph of some special groups including thedihedral and semi-dihedral groups. Some open questions are also proposed.