[Paper Review] Eigenvalues, Smith normal form and determinantal ideals

Aida Abiad, Carlos A. Alfaro+2 more|arXiv (Cornell University)|Oct 28, 2019

Graph theory and applications43 references4 citations

TL;DR

This paper establishes a novel link between eigenvalues, Smith normal form (SNF), and determinantal ideals of integer matrices associated with graphs. It proves that an eigenvalue divides the k-th invariant factor of the SNF if it lies in the variety of the k-th univariate integer determinantal ideal, generalizing Rushanan's result. The key contribution is a characterization of codeterminantal graphs—graphs sharing the same univariate determinantal ideals—providing a unified framework that extends both cospectral and coinvariant graph concepts, with complete and star graphs uniquely determined by the SNF of their distance Laplacian matrix.

ABSTRACT

Determinantal ideals of graphs generalize, among others, the spectrum and the Smith normal form (SNF) of integer matrices associated to graphs. In this work we investigate the relationship of the spectrum and the SNF with the determinantal ideals. We show that an eigenvalue divides the $k$-th invariant factor of its SNF if the eigenvalue belongs to a variety of the $k$-th univariate integer determinantal ideal of the matrix. This result has as a corollary a theorem of Rushanan. We also study graphs having the same determinantal ideals with at most one indeterminate; the socalled codeterminantal graphs, which generalize the concepts of cospectral and coinvariant graphs. We establish a necessary and sufficient condition for graphs to be codeterminantal on $\mathbb{R}[x]$, and we present some computational results on codeterminantal graphs up to 9 vertices. Finally, we show that complete graphs and star graphs are determined by the SNF of its distance Laplacian matrix.

Motivation & Objective

• To investigate the relationship between eigenvalues, Smith normal form (SNF), and determinantal ideals of integer matrices associated with graphs.
• To generalize the concepts of cospectral and coinvariant graphs by introducing codeterminantal graphs—graphs sharing the same univariate determinantal ideals.
• To provide a necessary and sufficient condition for graphs to be codeterminantal over ℝ[x], and to characterize when such graphs are also cospectral or coinvariant.
• To evaluate the distinguishing power of various determinantal ideals, especially univariate ideals over ℤ[x], in identifying graphs up to 9 vertices.
• To prove that complete graphs and star graphs are uniquely determined by the SNF of their distance Laplacian matrix.

Proposed method

• The paper uses univariate determinantal ideals in ℤ[x] generated by k-minors of xI − M(G), where M(G) is an integer matrix associated with graph G.
• It applies algebraic geometry and Gröbner basis theory to analyze the structure of these ideals, particularly in ℤ[x] and ℝ[x], to study their varieties and connections to eigenvalues.
• It employs computational enumeration to analyze all connected graphs up to 9 vertices, computing SNF, spectrum, and determinantal ideals to compare distinguishing power.
• It proves that if a graph is determined by its spectrum, then it is also determined by its univariate determinantal ideals in ℤ[x], leveraging the fact that Inℤ(Mx) is generated by det(Mx).
• It uses the Smith normal form of the distance Laplacian matrix to characterize complete and star graphs, showing that their SNF uniquely identifies them.
• It applies results from commutative algebra, including the theory of determinantal ideals and invariant factors, to derive divisibility relations between eigenvalues and invariant factors.

Experimental results

Research questions

• RQ1 Under what conditions does an eigenvalue of a matrix divide the k-th invariant factor of its Smith normal form?
• RQ2 What is the necessary and sufficient condition for two graphs to be codeterminantal over ℝ[x]?
• RQ3 How do codeterminantal graphs relate to cospectral and coinvariant graphs?
• RQ4 Which determinantal ideals are most effective in distinguishing non-isomorphic graphs, particularly among small graphs?
• RQ5 Are complete graphs and star graphs uniquely determined by the Smith normal form of their distance Laplacian matrix?

Key findings

• An eigenvalue divides the k-th invariant factor of the SNF if it lies in the variety of the k-th univariate integer determinantal ideal, generalizing Rushanan's theorem.
• Codeterminantal graphs over ℝ[x] are characterized by a necessary and sufficient condition based on the structure of their univariate determinantal ideals.
• The univariate determinantal ideals in ℤ[x] are the most effective for distinguishing graphs, as they unify spectral and SNF information.
• Among all matrices studied (adjacency, Laplacian, distance, distance Laplacian), the SNF of the distance Laplacian matrix performs best in distinguishing graphs up to 9 vertices.
• Complete graphs and star graphs are uniquely determined by the SNF of their distance Laplacian matrix, as shown by analyzing the Gröbner bases of 2-minor ideals in ℤ[n,m].
• If a graph is determined by its spectrum, then it is also determined by its univariate determinantal ideals in ℤ[x], since the last ideal is generated by the determinant.