The $\circ$ operation and $*$ operation of fan graphs

TL;DR

This paper investigates algebraic properties such as depth and regularity of edge ideals of fan graphs and their variants formed by specific graph operations, providing explicit computations for these invariants.

Contribution

It computes the depth and Castelnuovo--Mumford regularity of edge ideals for fan graphs and their $ ext{circ}$ and $*$ operations, extending understanding of these algebraic invariants.

Findings

Explicit formulas for depth of $S/I_G$ for fan graphs and their operations.

Explicit formulas for Castelnuovo--Mumford regularity of $S/I_G$ for these graphs.

Extension of algebraic invariant computations to graph operations $ ext{circ}$ and $*$.

Abstract

Let $G$ be a finite simple graph on the vertex set $V$ and let $I_G$ denote 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 $G=F_{k}^{W}(K_n)$ is a $k$-fan graph, or $G=G_1\circ G_2$ or $G=G_1* G_2$ is the graph obtained from fan graphs $G_1$, $G_2$ by $\circ$ operation or $*$ operation, respectively.