Given two graphs $G,H$ is there a product $\star$ such that 1. and 2. holds where $\alpha$ refers to independence number?

1. $$\alpha(G\star H)=\alpha(G)\alpha(H)=\alpha(H\star G)$$

2. $$G\star H\cong H\star G$$

Lexicographic product satisfies 1. but not 2.

Also other than Lexicographic product is there any other product that satisfies 1.?