The $h$-vectors of the edge rings of a special family of graphs

TL;DR

This paper explicitly computes the $h$-vectors of edge rings for a specific family of graphs using initial ideals and simplicial complexes, contributing to understanding their algebraic invariants.

Contribution

It introduces a method to compute $h$-vectors for a particular class of graphs' edge rings, filling a gap in explicit examples.

Findings

Explicit $h$-vectors for the studied graph family

Application of initial ideals and simplicial complexes in computation

Enhanced understanding of algebraic invariants of edge rings

Abstract

The $h$-vectors of homogeneous rings are one of the most important invariants that often reflect ring-theoretic properties. On the other hand, there are few examples of edge rings of graphs whose $h$-vectors are explicitly computed. In this paper, we compute the $h$-vector of a special family of graphs, by using the technique of initial ideals and the associated simplicial complex.