Publications de l'Institut Mathematique 2009 Volume 85, Issue 99, Pages: 19-33
doi:10.2298/PIM0999019C
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Towards a spectral theory of graphs based on the signless Laplacian, I

Cvetković Dragoš, Simić Slobodan K.

A spectral graph theory is a theory in which graphs are studied by means of eigenvalues of a matrix M which is in a prescribed way defined for any graph. This theory is called M-theory. We outline a spectral theory of graphs based on the signless Laplacians Q and compare it with other spectral theories, in particular with those based on the adjacency matrix A and the Laplacian L. The Q-theory can be composed using various connections to other theories: equivalency with A-theory and L-theory for regular graphs, or with L-theory for bipartite graphs, general analogies with A-theory and analogies with A-theory via line graphs and subdivision graphs. We present results on graph operations, inequalities for eigenvalues and reconstruction problems.

Keywords: graph theory, graph spectra, adjacency matrix, signless Laplacian

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