This paper evaluates the performance of adaptive modulation in multi-relay networks with selective relaying, under Nakagami-m fading. In the system model, the source decides independently whether to forward the source message to the destination via the best (partial relay selection) relay path or direct path by comparing the end-to-end instantaneous signal-to-noise ratio (SNR) at the destination, which is independent of the modulation scheme. Adaptive discrete-rate M-ary quadrature amplitude modulation with fixed switching thresholds is implemented by dividing the SNR region into five modes. In particular, impact of imperfect (outdated) channel estimation due to feedback delay is quantified for relay selection. We derive the cumulative distribution function for the upper-bound of end-to-end SNR in closed-form. Further, lower-bounds of outage probability and average bit error rate, and upper-bound of spectral efficiency are derived in closed-forms. Monte Carlo simulation results validate our numerical analysis.