On Independence Number of Comaximal Subgroup Graph

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.