  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.
