Robust Monochromatic Edges in a Random Coloring

Each vertex of the complete graph $K_n$ is independently colored red or blue, each with probability $\tfrac{1}{2}$. An edge is monochromatic if both its endpoints have the same color. Find the expected number of monochromatic edges.

Additional robustness twist: before observation, an independent random relabeling of outcome labels is applied. Compute the same target and justify invariance.