The $\circ$ operation and $\ast$ operation of Cohen-Macaulay bipartite graphs

Let $G$ be a finite simple graph with the vertex set $V$ and let $I_G$ be 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 $S\mathrm{/}{I}_G$ when $G=G_1\circ G_2$ or $G=G_1\ast G_2$ is a graph obtained from Cohen-Macaulay bipartite graphs $G_1$, $G_2$ by $\circ$ operation or $\ast$ operation, respectively.