Graphs and Ideals generated by some 2-minors

TL;DR

This paper investigates ideals generated by 2-minors associated with a graph's edges, providing a Groebner basis construction and primary decomposition, thus linking graph theory with algebraic geometry.

Contribution

It introduces a method to construct Groebner bases of edge ideals using paths in the graph and computes their primary decompositions.

Findings

Groebner basis constructed from graph paths

Primary decomposition of the ideals obtained

Connection established between graph structure and algebraic properties

Abstract

Let G be a finite graph on [n] = {1,2,3,...,n}, X a 2 times n matrix of indeterminates over a field K, and S = K[X] a polynomial ring over K. In this paper, we study about ideals I_G of S generated by 2-minors [i,j] of X which correspond to edges {i,j} of G. In particular, we construct a Groebner basis of I_G as a set of paths of G and compute a primary decomposition.