In islanded AC microgrids, achieving optimal power distribution across distributed generators (DGs) and restoring frequency post-primary control under limited communication resources is paramount for ensuring stability and flexibility. In this paper, we develop a distributed adaptive secondary control strategy tailored for islanded AC microgrids. This strategy facilitates effective active power sharing and frequency regulation using solely local network information. A unique adaptive edge-based event-triggered transmission mechanism is also proposed, obviating Zeno behavior, to reduce communication overhead. It requires only the degree information of a single node in the DG network topology. Contrasting most existing research that necessitates global network topology details, like the Laplacian matrix, this work establishes a fully distributed control performance criterion that operates devoid of global data. The effectiveness of the proposed control approach is validated using a modified IEEE 34-bus test system.