SOME EXAMPLES ABOUT IRREDUCIBLE DECOMPOSITION OF POWERS OF EDGE IDEALS

Abstract

In recent years, the construction of algebraic structures on graphs has attracted significant interest from many researchers, especially finding the irreducible decomposition of the powers of an edge ideal. In this paper, we consider A = Q[x₁, ..., x_k] as a polynomial ring in k variables over the field Q, G = (V, E) as a graph with the vertex set {x₁,...,x_k} and J_G as the edge ideal associated with G. Our main result is constructing a simple, connected non bipartite graph G over a polynomial ring of 9 variables and calculating the isolated and the embedded irreducible components of powers of the edge ideal , with some small number n. In the next research, we can apply this technique to study more the structure of an edge ideal, for examples calculating the indexes astab(J_G) or distab(J_G). It helps us gain a better understanding of the structure of components within graphs, thereby facilitating the development of more efficient algorithms for processing and analyzing graphs.