Regularity of symbolic powers of certain graphs

TL;DR

This paper investigates the algebraic properties of a class of graphs called $G_{n,r}$, proving Minh's conjecture that the regularity of their edge ideal's powers are equal and computing key invariants like the Waldschmidt constant.

Contribution

The paper proves Minh's conjecture for the edge ideals of $G_{n,r}$ graphs and calculates their Waldschmidt constant and resurgence, expanding understanding of symbolic and ordinary powers.

Findings

Regularity of symbolic and ordinary powers are equal for $I(G_{n,r})$.

Computed Waldschmidt constant for the class of graphs.

Determined resurgence for the class of graphs.

Abstract

Let $G_{n,r}$ denote the graph with $n$ vertices $\{x_1,\ldots,x_n\}$ in cyclic order and for each vertex $x_i$ consider the set $A_i=\{x_{i-r},\ldots,x_{i-1},x_{i+1},x_{i+2},\ldots, x_{i+r}\},$ where $x_{i-j}$ is the vertex $x_{n+i-j}$, whenever $i<j$ and $0\leq r\leq \Bigl\lfloor\dfrac{n}{2}\Bigr\rfloor -1$. In $G_{n,r}$, every vertex $x_i$ is adjacent to all the vertices of $V(G_{n,r})\backslash A_i$. Let $I=I(G_{n,r})$ be the edge ideal of $G_{n,r}$. We show that Minh's conjecture is true for $I,$ i.e. regularity of ordinary powers and symbolic powers of $I$ are equal. We compute the Waldschmidt constant and resurgence for the whole class.