On binomial edge ideals of corona of graphs

Abstract

For a simple graph $G$, let $J_G$ denote the corresponding binomial edge ideal. This article considers the binomial edge ideal of the corona product of two connected graphs $G$ and $H$. The corona product of $G$ and $H$, denoted by $G\circ H$, is a construction where each vertex of $G$ is connected (via the coning-off) to an entire copy of $H$. This is a direct generalization of a cone construction. Previous studies have shown that for $J_{G \circ H}$ to be Cohen-Macaulay, both $G$ and $H$ must be complete graphs. However, there are no general formulae for the dimension, depth, or Castelnuovo-Mumford regularity of $J_{G\circ H}$ for all graphs $G$ and $H$. In this article, we provide a general formula for the dimension, depth and Castelnuovo-Mumford regularity of the binomial edge ideals of certain corona and corona-type (somewhat a generalization of corona) products of special interests. Additionally, we study the Cohen-Macaulayness, unmixedness and related properties of binomial edge ideals corresponding to above class of graphs. We have also added a short note on the reduction of the Bolognini-Macchia-Strazzanti Conjecture to all graphs with a diameter of $3$.