We show that the Castelnuovo–Mumford regularity of the binomial edge ideal of a graph is bounded below by the length of its longest induced path and bounded above by the number of its vertices.

The author continues the study of Frobenius amplitude, introduced in an earlier paper, and compares this to various other positivity notions. These are applied to deduce vanishing theorems. A refinement of Castelnuovo-Mumford regularity is also given.

We prove that for every integer \(k\ge 1\), there exists a connected graph \(H_k\) such \(v(H_k)={\text {reg}}(H_k)+k\), where v(G) and \({\text {reg}}(G)\) denote the v-number (Castelnuovo–Mumford) regularity of G respectively.