Ideals of graphs: finding a set of generators

TL;DR

This paper investigates the algebraic structure of graph ideals by identifying minimal generators of their Koszul homology algebra, providing explicit generators for specific graph classes such as trees and cycles.

Contribution

It introduces a method to find minimal generators of the Koszul homology algebra for certain classes of graphs, extending understanding of graph ideals in algebraic combinatorics.

Findings

Identified a generator element star for each vertex in the homology algebra.

Established that star and circle generators suffice for trees and certain cycles.

Described generators for graphs with two cycles sharing a vertex and for connected graphs formed by known components.

Abstract

In this paper, we consider homological properties of so-called graph ideals. Consider $\Gamma$ is a graph with vertices $t_1$, ..., $t_s$, without self-loops and multiple adjacencies. We can associate with such a graph an ideal $I({\Gamma})$ of polynomial ring $$ A({\Gamma}) = k[t_1,...,t_s] $$ over k, generated by $x_{ij}=t_it_j$, $i\ne j$, corresponding to edges of $\Gamma$. The object of this paper is an algebra of Koszul homology $$ H((x_{ij}),A({\Gamma})) $$ of Koszul complex $K((x_{ij}),A({\Gamma})).$ The result of this paper is finding a minimal multiplicative system of generators of this algebra for some graphs $\Gamma$. There is an element $\bigstar$ in homology algebra corresponding each vertex in the graph, that should be included in every set of generators of each graph. This is a sufficient system for trees. Also, there is a generator element $\bigcirc$ for every cycle with length n if n mod 3=2. System of $\bigstar$-s and maybe $\bigcirc$ is sufficient for a graph with only one cycle. Also, here described a set of generators for a graph that is two cycles with exactly one common vertex. If a graph is two graphs with known algebra generators, connected by an edge, the answer for the whole graph is also described in this paper.