The $\circ$ operation and $\ast$ operation of fan graphs

Abstract

Let $G$ be a finite simple graph on the vertex set $V$ and let $I_G$ denote its edge ideal in the polynomial ring $S=\mathbb{K}[x_V]$. In this paper, we compute the depth and the Castelnuovo--Mumford regularity of $S/I_G$ when $G=F_k^W(K_n)$ is a $k$-fan graph, or $G=G_1\circ G_2$ or $G=G_1\ast G_2$ is the graph obtained from fan graphs $G_1$, $G_2$ by $\circ$ operation or $\ast$ operation, respectively.