The ◦ operation and * 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 \(S=\mathbb{K}[V]\). We compute the depth and the Castelnuovo-Mumford regularity of S/I_G when G = G₁ ◦ G₂ or G = G₁ * G₂ is a graph obtained from Cohen-Macaulay bipartite graphs G₁, G₂ by the ◦ operation or * operation, respectively.