In this paper, we establish sharp thresholds on the independence number of the comaximal subgroup graph Formula: see text that guarantee solvability, supersolvability, and nilpotency of the underlying group Formula: see text. Specifically: For solvability, we prove that any group Formula: see text with independence number Formula: see text must be solvable, and show that the alternating group Formula: see text is uniquely determined by its graph. For supersolvability, we show that Formula: see text implies Formula: see text is supersolvable, except for three explicit exceptions. For nilpotency, we prove that Formula: see text ensures nilpotency, except for five groups. Finally, we conclude with some open issues involving domination parameters.
Das et al. (Fri,) studied this question.