Regularity of the vanishing ideal over a parallel composition of paths

Abstract

Let $G$ be a graph obtained by taking $r\ge 2$ paths and identifying all first vertices and identifying all last vertices. We compute the Castelnuovo–Mumford regularity of the quotient $S∕I\left(X\right)$, where $S$ is the polynomial ring on the edges of $G$ and $I\left(X\right)$ is the vanishing ideal of the projective toric subset parameterized by $G$. This invariant is known for several special families of graphs such as trees, cycles, complete graphs and complete bipartite graphs. For bipartite graphs, it is also known that the computation of the regularity can be reduced to the $2$-connected case. Thus, we focused on the first case of a bipartite graph where the regularity was unknown. We also prove new inequalities relating the Castelnuovo–Mumford regularity of $S∕I\left(X\right)$ with the combinatorial structure of $G$, for a general graph.