Summary: The independence graph \(\operatorname{Ind}(G)\) of a graph \(G\) is the graph with vertices as maximum independent sets of \(G\) and two vertices are adjacent, if and only if the corresponding maximum independent sets are disjoint. In this work, we find the independence graph of Cartesian product of \(d\) copies of complete graphs \(K_q\), which is known as the Hamming graph \(H(d, q)\). \textit{D. Greenwell} and \textit{L. Lovász} [Acta Math. Acad. Sci. Hung. 25, 335--340 (1974; Zbl 0294.05108)] found that the independence number of direct product of \(d\) copies of \(K_q\) as \(q^{d-1}\). We prove that the independence number of Hamming graph \(H(d, q)\), which is cartesian product of \(d\) copies of \(K_q\), is also \(q^{d-1}\). As an application of our findings, we find answers for rook problem in higher dimensional square chess board.