The signless Laplacian spectrum of a graph is a spectrum of the matrix , where is its adjacency matrix, while is the diagonal matrix of vertex degrees. Usually the signless Laplacian matrix is called -matrix, while its spectrum and eigenvalues are known as the -spectrum and the -eigenvalues, respectively. At the begin, the authors extend their previous survey of properties of -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 -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 -index is a nested split graph. Using this result two conjectures are confirmed. Finally some other conjectures are resolved.