Algebraic properties of product of graphs

TL;DR

This paper investigates algebraic invariants of the graph product, such as Betti numbers and regularity, providing explicit formulas and structural properties related to the product of graphs.

Contribution

It introduces explicit formulas for algebraic invariants of graph products and proves the closure of certain graph families under this operation.

Findings

Explicit formulas for Betti numbers, $h$-vector, and Hilbert series of graph products.

The family of graphs with regularity equal to the maximum number of pairwise 3-disjoint edges is closed under graph product.

Structural insights into algebraic properties of graph products.

Abstract

Let $G$ and $H$ be two simple graphs and let $G*H$ denotes the graph theoretical product of $G$ by $H$. In this paper we provide some results on graded Betti numbers, Castelnuovo-Mumford regularity, projective dimension, $h$-vector, and Hilbert series of $G*H$ in terms of that information of $G$ and $H$. To do this, we will provide explicit formulae to compute graded Betti numbers, $h$-vector, and Hilbert series of disjoint union of complexes. Also we will prove that the family of graphs whose regularity equal the maximum number of pairwise $3$-disjoint edges, is closed under product of graphs.