*(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 |