Ordering Q-indices of graphs: Given size and circumference

In 1981, Cvetković identified 12 directions for further investigations of graph spectra, one of which is “classifying and ordering graphs”. Following this classic direction, we focus on the classification and ordering of Q-indices of graphs. Let H(m,c) be the family of connected graphs on m edges with circumference c. In this paper, we order the Q-indices of graphs in H(m,c) from the first to the (⌊c2⌋+1)-th, where m≥max⁡{3c−3,c+8}≥13. Additionally, we determine the first seven largest Q-indices of graphs in H(m,4) for m≥12.