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Let $R$ be a commutative semiring with non-zero identity. In this paper, we introduce and study the graph $\Omega \left(R\right)$ whose vertices are all elements of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if the product of the co-ideals generated by $x$ and $y$ is $R$. Also, we study the interplay between the graph-theoretic properties of this graph and some algebraic properties of semirings. Finally, we present some relationships between the zero-divisor graph $\Gamma \left(R\right)$ and $\Omega \left(R\right)$.