# Estimation of Laplacian spectra of direct and strong product graphs

arXiv: https://arxiv.org/abs/1507.03030 · HTML: https://mlsanity.com/paper/1507.03030/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jul 2015.

## Built on by (most cited first, 4 of 4)

- [Scalable Multi-agent Covering Option Discovery based on Kronecker Graphs](https://mlsanity.com/paper/2307.11629/) (2023)
- [Learning Multi-agent Skills for Tabular Reinforcement Learning using Factor Graphs](https://mlsanity.com/paper/2201.08227/) (2022)
- [Matrix-weighted networks for modeling multidimensional dynamics](https://mlsanity.com/paper/2410.05188/) (2024)
- [Learning Kronecker-Structured Graphs from Smooth Signals](https://mlsanity.com/paper/2505.09822/) (2025)

## Abstract (arXiv)

Calculating a product of multiple graphs has been studied in mathematics, engineering, computer science, and more recently in network science, particularly in the context of multilayer networks. One of the important questions to be addressed in this area is how to characterize spectral properties of a product graph using those of its factor graphs. While several such characterizations have already been obtained analytically (mostly for adjacency spectra), characterization of Laplacian spectra of direct product and strong product graphs has remained an open problem. Here we develop practical methods to estimate Laplacian spectra of direct and strong product graphs from spectral properties of their factor graphs using a few heuristic assumptions. Numerical experiments showed that the proposed methods produced reasonable estimation with percentage errors confined within a +/-10% range for most eigenvalues.