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Let $G$ be a finite simple graph with the vertex set $V$ and let $I_G$ be its edge ideal in the polynomial ring $S=𝕂\left[V\right]$. We compute the depth and the Castelnuovo-Mumford regularity of $S/I_G$ when $G=G_1\circ G_2$ or $G=G_1*G_2$ is a graph obtained from Cohen-Macaulay bipartite graphs $G_1$, $G_2$ by the $\circ$ operation or $*$ operation, respectively.