Abstract: The star chromatic index of graph $G$ is denoted by $\chi_{s}^{\prime}(G)$. In this paper, we studied the relationship between  $\chi_{s}^{\prime}(G)$, $|V(G)|=\nu$, and $|E(G)|=\varepsilon$, and proved that $\lceil\frac{8\varepsilon}{3\nu}\rceil\leqslant\chi_{s}^{\prime}(G)$ for $\Delta(G)\geqslant2$. The star chromatic index of 2-dimensional grid was obtained. We also got the attainable bounds for the star chromatic index of hypercubes and $d$-dimensional grids.

Key words: star edge coloring, star chromatic index, hypercube, $d$-dimensional grid