[Paper Review] On the ideal of orthogonal representations of a graph in $\mathbb{R}^2$

Jürgen Herzog, Antonio Macchia+2 more|arXiv (Cornell University)|Nov 13, 2014

Commutative Algebra and Its Applications5 references3 citations

TL;DR

This paper studies the algebraic structure of orthogonal representations of graphs in ℝ² by analyzing the ideal generated by orthogonality conditions between non-adjacent vertices. It proves that for any field K with char(K) ≠ 2, the ideal is radical, and when √−1 ∉ K, provides a reduced primary decomposition, showing the variety of orthogonal embeddings is a union of varieties defined by prime ideals—applying directly to ℝ and resolving key algebraic and geometric properties of the representation space.

ABSTRACT

In this paper, we study orthogonal representations of simple graphs $G$ in $\mathbb{R}^d$ from an algebraic perspective in case $d = 2$. Orthogonal representations of graphs, introduced by Lovász, are maps from the vertex set to $\mathbb{R}^d$ where non-adjacent vertices are sent to orthogonal vectors. We exhibit algebraic properties of the ideal generated by the equations expressing this condition and deduce geometric properties of the variety of orthogonal embeddings for $d=2$ and $\mathbb{R}$ replaced by an arbitrary field. In particular, we classify when the ideal is radical and provide a reduced primary decomposition if $\sqrt{-1} ot\in K$. This leads to a description of the variety of orthogonal embeddings as a union of varieties defined by prime ideals. In particular, this applies to the motivating case $K = \mathbb{R}$.

Motivation & Objective

• To understand the algebraic and geometric structure of orthogonal representations of graphs in ℝ² from a commutative algebra perspective.
• To characterize when the ideal of orthogonality relations is radical or prime, particularly over fields where √−1 ∉ K.
• To provide a reduced primary decomposition of the Lovász-Saks-Schrijver ideal when √−1 ∉ K, enabling a geometric description of the variety of orthogonal embeddings.
• To classify graphs for which the ideal is unmixed or prime, linking algebraic properties to graph-theoretic invariants.

Proposed method

• The authors reframe orthogonal representations in ℝ² as a polynomial ideal L_G in K[x₁,…,xₙ,y₁,…,yₙ], generated by quadratic forms x_ix_j + y_iy_j for non-edges {i,j} in G.
• They reduce the problem to studying permanental edge ideals by applying a linear transformation under the assumption √−1 ∈ K, simplifying the ideal structure.
• Using the theory of initial ideals, they show that the permanental edge ideals have squarefree initial ideals under a suitable monomial order, implying the ideals are radical.
• They establish a correspondence between minimal prime ideals of L_G and certain minimal vertex subsets S ⊆ V(G) with specific connectivity properties, using the set M(G) of minimal such subsets.
• They prove that L_G is prime if and only if G is a disjoint union of edges and isolated vertices, under the condition √−1 ∉ K.
• They derive a criterion for unmixedness of L_G based on the number of bipartite components and the size of minimal vertex sets S ∈ M(G).

Experimental results

Research questions

• RQ1 When is the ideal L_G of orthogonal representations in ℝ² radical, and what are the conditions on the field K for this to hold?
• RQ2 What is the structure of the variety of orthogonal embeddings over an arbitrary field K, particularly when √−1 ∉ K?
• RQ3 When is the ideal L_G prime, and how does this relate to the graph structure of G?
• RQ4 What conditions ensure that L_G is unmixed, and how can this be characterized via graph invariants?
• RQ5 How does the minimal prime ideal decomposition of L_G relate to the combinatorics of G, especially through the set M(G) of minimal vertex subsets?

Key findings

• For any field K with char(K) ≠ 2, the ideal L_G is radical, meaning it contains all polynomials vanishing on the variety of orthogonal representations.
• When √−1 ∉ K, the ideal L_G admits a reduced primary decomposition, and the variety of orthogonal embeddings is the union of varieties defined by prime ideals.
• L_G is prime if and only if G is a disjoint union of edges and isolated vertices, under the condition √−1 ∉ K.
• The ideal L_G is unmixed if and only if for every non-empty S ∈ M(G), the number of bipartite components b(S) equals |S| + b, where b is the number of bipartite components of G.
• For the cycle graph C_n, L_{C_n} is unmixed if and only if n is odd, and for the complete graph K_n, L_{K_n} is unmixed only when n = 2 or 3.
• The paper provides a complete classification of unmixedness and primality for L_G in terms of graph-theoretic properties, particularly when √−1 ∉ K.