  Typically, graph structures are represented by one of three different
matrices: the adjacency matrix, the unnormalised and the normalised graph
Laplacian matrices. The spectral (eigenvalue) properties of these different
matrices are compared. For each pair, the comparison is made by applying an
affine transformation to one of them, which enables comparison whilst
preserving certain key properties such as normalised eigengaps. Bounds are
given on the eigenvalue differences thus found, which depend on the minimum and
maximum degree of the graph. The monotonicity of the bounds and the structure
of the graphs are related. The bounds on a real social network graph, and on
three model graphs, are illustrated and analysed. The methodology is extended
to provide bounds on normalised eigengap differences which again turn out to be
in terms of the graph's degree extremes. It is found that if the degree extreme
difference is large, different choices of representation matrix may give rise
to disparate inference drawn from graph signal processing algorithms; smaller
degree extreme differences result in consistent inference, whatever the choice
of representation matrix. The different inference drawn from signal processing
algorithms is visualised using the spectral clustering algorithm on the three
representation matrices corresponding to a model graph and a real social
network graph.
