2019/10/28 by Aida Abiad, Abiad, Aida, Carlos A. Alfaro +5
Computer Science · Mathematics · #05C25 #05C50 #05E99 #13C40 #13P10 #Advanced Optimization Algorithms Research #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.1910.12502
openalex publication_date 2019/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.