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.