Degree of $h$-polynomials of edge ideals

TL;DR

This paper explores the degree of $h$-polynomials of edge ideals in graphs, providing combinatorial formulas for specific graph classes and characterizing graphs with maximal algebraic invariants.

Contribution

It introduces the first combinatorial interpretation of the degree of $h$-polynomials and characterizes graphs where algebraic invariants reach their maximum.

Findings

Derived combinatorial formulas for paths, cycles, and bipartite graphs.

Characterized connected graphs with maximal sum of regularity and $h$-polynomial degree.

Identified the maximum value of algebraic invariants as the number of vertices.

Abstract

In this paper, we investigate the degree of $h$-polynomials of edge ideals of finite simple graphs. In particular, we provide combinatorial formulas for the degree of the $h$-polynomial for various fundamental classes of graphs such as paths, cycles, and bipartite graphs. To the best of our knowledge, this marks the first investigation into the combinatorial interpretation of this algebraic invariant. Additionally, we characterize all connected graphs in which the sum of the Castelnuovo-Mumford regularity and the degree of the $h$-polynomial of an edge ideal reaches its maximum value, which is the number of vertices in the graph.