In this paper, we show that the fiber cones of rational normal scrolls are Cohen–Macaulay. As an application, compute their Castelnuovo–Mumford regularities and $$\varvec{a}$$ -invariants, as well reduction number defining ideals scrolls. We also characterize Gorensteinness cone.

We determine the Castelnuovo-Mumford regularity of binomial edge ideals complement reducible graphs (cographs). For cographs with $n$ vertices maximum grows as $2n/3$. also bound by graph theoretic invariants and construct a family counterexamples to conjecture Hibi Matsuda.