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

Yulong Yang; Guangjun Zhu; Yijun Cui; Shiya Duan — 2024

Czechoslovak Mathematical Journal

Let $G$ be a finite simple graph with the vertex set $V$ and let $I_{G}$ be its edge ideal in the polynomial ring $S = 𝕂 [V]$ . 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.

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The ∘ operation and * operation of Cohen-Macaulay bipartite graphs

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