The spectral extremal problem is a classic problem in spectral graph theory. For a simple graph G , let q ( G ) denote the Q -spectral radius. We first characterize the graphs with maximal Q -spectral radius among all graphs of size m with maximum degree Δ ≥ m + 1 2 . For two graphs G 1 and G 2 of size m ≥ 11 , employing this result, we prove that q ( G 1 ) > q ( G 2 ) if Δ ( G 1 ) > Δ ( G 2 ) and Δ ( G 1 ) ≥ m 2 + 3 , which improves the main result of [Bull. Malays. Math. Sci. Soc. 45(2022)2165-2174]. Let r ≥ m 2 + 3 be an integer. Employing the above results, we completely characterize the graphs with maximal Q -spectral radius among all connected graphs of size m with maximum degree at most r , with covering number β and with independence number α ≥ m 3 , respectively.