讲座内容：

In 1981, Cvetkovic pointed out 12 directions in further investigations of graph spectra, one of which is ''classifying and ordering graphs''. Along this classic direction, we pay our attention to the order of the largest eigenvalue of the signless Laplacian matrix of a graph, which is usually called the signless Laplacian spectral radius or Q-index of a graph. Let G(m, g) (resp. G(m, > g)) be the family of connected graphs on m edges with girth g (resp. no less than g), where g>3. In this paper, we firstly order the first ([g/2]+2) largest Q-indices of graphs in G(m, g), where m> 3g> 12. Secondly, we order the first ([g/2]+3) largest Q-indices of graphs in G(m, >g), where m> 3g>12. As a complement, we give the first five largest Q-indices of graphs in G(m, 3) with m> 9. Finally, we give the order of the first eleven largest Q-indices of all connected graphs with size m.