Ideals of graphs: finding a set of generators

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.