Research Statement

My research focuses on the Inverse Eigenvalue Problem for Graphs (IEPG) and the broader class of Inverse Eigenvalue Problems for Symmetric matrices (IEPS). I investigate which spectra can be realized by real symmetric matrices whose zero-nonzero patterns are constrained by an underlying graph structure. This includes studying spectral properties, multiplicities, and the interplay between combinatorial structures and algebraic invariants of graphs. A major goal of my work is to provide constructive methods to realize prescribed spectra. I combine combinatorial techniques (zero forcing, strong spectral property (SSP), strong multiplicity property (SMP), strong Arnold property (SAP)) with algebraic and matrix-theoretic methods. All constructions and validations are carefully verified using Python and SageMath. Code and notebooks are available on my GitHub. I am also interested on extremal spectral graph theory, matrix inertia, eigenvalue multiplicity obstructions, and algorithmic approaches for spectral invariants, with applications to network analysis and combinatorial optimization.

Publications

[1] A. Laikhuram, J. C.-H. Lin. Inverse eigenvalue problem for discrete Schrödinger operators of a graph, Linear Algebra Appl. (2025).
doi:10.1016/j.laa.2025.10.030

Ongoing Work

  • Matrix construction for inverse eigenvalue realizations using Python and SageMath
  • Multiplicity obstructions via SSP / SMP / SAP
  • Algorithms for spectral graph invariants and extremal problems
  • Codes and notebooks: GitHub

Thesis

Ph.D. Dissertation: Spectral characterization and matrix construction in the inverse eigenvalue problem for discrete Schrödinger operators on a graph
(Link will be available soon)