Spectral Perturbations of the Line Graph Laplacian

TL;DR

This paper investigates how small changes in the edge weights of line graphs affect their Laplacian eigenvalues, providing bounds and insights relevant for extending DSP-like operations to complex graph structures.

Contribution

It derives closed-form bounds on eigenvalue perturbations of line graph Laplacians under single edge weight modifications, facilitating generalized graph Fourier transforms.

Findings

Eigenvalue bounds are validated through simulations.

Single edge perturbations can significantly alter the Laplacian spectrum.

Results support extending GFT to more complex graphs with multiple perturbations.

Abstract

The graph Laplacian is an important tool in Graph Signal Processing (GSP) as its eigenvalue decomposition acts as an analogue to the Fourier transform and is known as the Graph Fourier Transform (GFT). The line graph has a GFT that is a direct analogue to the Discrete Cosine Transform Type II (DCT-II). Leveraging Fourier transform properties, one can then define processing operations on this graph structure that is loosely analogous to processing operations encountered in Digital Signal Processing (DSP) theory. This raises the question of whether well defined DSP-like operations can be derived from the GFT for more complex graph structures. One potential approach to this problem is to perturb simple graph structures and study the perturbation's impact on the graph Laplacian. This paper explores this idea by examining the eigenvalue decomposition of the Laplacian of undirected line graphs that undergo a single edge weight perturbation. This single perturbation can perturb either an existing edge weight or create new edge between distant unconnected vertices. The eigenvalue bounds are expressed in closed form and agree with simulated examples. The theory can be extended to include multiple perturbations such that the GFT can be defined for a more general graph structure.