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Eigenvalue bounds for the signless Laplacian. (English) Zbl 1164.05038

The signless Laplacian spectrum of a graph is a spectrum of the matrix Q=A+D, where A is its adjacency matrix, while D is the diagonal matrix of vertex degrees. Usually the signless Laplacian matrix is called Q-matrix, while its spectrum and eigenvalues are known as the Q-spectrum and the Q-eigenvalues, respectively. At the begin, the authors extend their previous survey of properties of Q-spectrum. The paper also contains a number of computer-generated conjectures. Mostly, the conjectures give some bounds for the first, the second or the least Q-eigenvalue of an arbitrary graph. Some comments on the conjectures are given.

In further, the authors prove their main result: among the connected graphs with fixed order and size, the graph with maximal Q-index is a nested split graph. Using this result two conjectures are confirmed. Finally some other conjectures are resolved.


MSC:
05C50Graphs and linear algebra