Depth and regularity of powers of edge ideals of edge-weighted trees

Abstract

For an increasing weighted tree $G_\omega$, we obtain an asymptotic value and a sharp bound on the index stability of the depth function of its edge ideal $I(G_\omega)$. Moreover, if $G_\omega$ is a strictly increasing weighted tree, we provide the minimal free resolution of $I(G_\omega)$ and an exact formula for the regularity of all powers of $I(G_\omega)$.

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