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The closed-form expressions of the probability density function (pdf) and the moment-generating function for the exact end-to-end signal-to-noise ratio (SNR) of variable-gain relay networks are unknown for generalized fading models; thus, the exact evaluation for the ergodic capacity of these systems is cumbersome. In this paper, we develop a new unified framework for the exact ergodic capacity of multihop networks equipped with variable-gain relays, assuming that channel-state information (CSI) is available only at the receiving terminals. The resulting expression (which is based on the exact end-to-end SNR) is in the form of a single truncated infinite series and is valid for an arbitrary number of hops and for various fading models that are typically encountered in realistic scenarios. Furthermore, we show that the exact ergodic capacity of dual-hop variable-gain relay networks can be written in terms of the ergodic capacities of two equivalent single-input-single-output (SISO) channels and one single-input-multiple-output channel (SIMO), the closed-form expressions of which are already available for many commonly used fading/shadowing models. Finally, we show that there is an exact match between the Monte Carlo simulation and theoretical results over a whole range of per-hop average SNR.