## 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:

 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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