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

TL;DR

This paper investigates how the depth and regularity of edge ideals in polynomial rings are affected when constructing new graphs from Cohen-Macaulay bipartite graphs using specific operations, providing explicit formulas for these invariants.

Contribution

It computes the depth and Castelnuovo--Mumford regularity of edge ideals for graphs formed via $ullet$ operations on Cohen-Macaulay bipartite graphs, extending understanding of algebraic invariants under graph operations.

Findings

Explicit formulas for depth and regularity after $ullet$ operations

Extension of Cohen-Macaulay bipartite graph properties

Enhanced understanding of algebraic invariants in graph theory

Abstract

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 $S/I_G$ 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 $\circ$ operation or $*$ operation, respectively.