Pseudo-Gorenstein edge rings and a new family of almost Gorenstein edge rings

TL;DR

This paper investigates the properties of edge rings, focusing on when they are pseudo-Gorenstein and identifying a new family of almost Gorenstein edge rings through analysis of their $h$-polynomials.

Contribution

It introduces a characterization of pseudo-Gorenstein edge rings and constructs a new family of almost Gorenstein edge rings based on $h$-polynomial computations.

Findings

Characterization of pseudo-Gorenstein edge rings.

Identification of a new family of almost Gorenstein edge rings.

Explicit computation of $h$-polynomials for these rings.

Abstract

In this paper, we study edge rings and their $h$-polynomials. We investigate when edge rings are pseudo-Gorenstein, which means that the leading coefficients of the $h$-polynomials of edge rings are equal to $1$. Moreover, we compute the $h$-polynomials of a special family of edge rings and show that some of them are almost Gorenstein.