# Comparing Graph Spectra of Adjacency and Laplacian Matrices

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.

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