On the power graph of the direct product of two groups

The power graph P (G) of a finite group G is the graph with vertex set G and two distinct vertices are adjacent if either of them is a power of the other. Here we show that the power graph P(G1×G2) of the direct product of two groups G1 and G2 is not isomorphic to either of the direct, cartesian and normal product of their power graphs P (G1) and P (G2). A new product of graphs, namely generalized product, has been introduced and we prove that the power graph P(G1×G2) is isomorphic to a generalized product of P (G1) and P (G2).