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Spectrum of Laplacians for Graphs with Self-Loops

This note introduces a result on the location of eigenvalues, i.e., the spectrum, of the Laplacian for a family of undirected graphs with self-loops. We extend on the known results for the spectrum of undirected graphs without self-loops or multiple edges. For this purpose, we introduce a new concept of pseudo-connected graphs and apply a lifting of the graph with self-loops to a graph without self-loops, which is then used to quantify the spectrum of the Laplacian for the graph with self-loops.