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

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