Integral closure and normality of edge ideals of some edge-weighted graphs

Let $G_\omega$ be an edge-weighted simple graph. In this paper, we give a complete characterization of the graph $G_\omega$ whose edge ideal $I(G_\omega)$ is integrally closed. We also show that if $G_\omega$ is an edge-weighted star graph, a path or a cycle, and $I(G_\omega)$ is integrally closed, then $I(G_\omega)$ is normal.