On the power graph of the direct product of two groups

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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).