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Signless Laplacians of finite graphs

✍ Scribed by Dragoš Cvetković; Peter Rowlinson; Slobodan K. Simić

Publisher
Elsevier Science
Year
2007
Tongue
English
Weight
191 KB
Volume
423
Category
Article
ISSN
0024-3795

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✦ Synopsis

We survey properties of spectra of signless Laplacians of graphs and discuss possibilities for developing a spectral theory of graphs based on this matrix. For regular graphs the whole existing theory of spectra of the adjacency matrix and of the Laplacian matrix transfers directly to the signless Laplacian, and so we consider arbitrary graphs with special emphasis on the non-regular case. The results which we survey (old and new) are of two types: (a) results obtained by applying to the signless Laplacian the same reasoning as for corresponding results concerning the adjacency matrix, (b) results obtained indirectly via line graphs. Among other things, we present eigenvalue bounds for several graph invariants, an interpretation of the coefficients of the characteristic polynomial, a theorem on powers of the signless Laplacian and some remarks on star complements.

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