Connected graphs of fixed order and size with maximal Q-index: Some spectral bounds

The Q-index of a simple graph G is the largest eigenvalue of the matrix Q, the signless Laplacian of G. It is well-known that in the set of connected graphs with fixed order and size, the graphs with maximal Q-index are the nested split graphs (also known as threshold graphs). In this paper, we focus our attention on the eigenvector techniques for getting some (lower and upper) bounds on the Q-index of nested split graphs. In addition, we give some computational results in order to compare these bounds.