We present a framework for robust electric network frequency (ENF) extraction from real-world audio recordings, featuring multi-tone ENF harmonic enhancement and graph-based optimal harmonic selection. Specifically, We first extend the recently developed single-tone ENF signal enhancement method to the multi-tone scenario and propose a harmonic robust filtering algorithm (HRFA). It can respectively enhance each harmonic component without cross-component interference, thus further alleviating the effects of unwanted noise and audio content on the much weaker ENF signal. In addition, considering the fact that some harmonic components could be severely corrupted even after enhancement, disturbing rather than facilitating ENF estimation, we propose a graph-based harmonic selection algorithm (GHSA), which finds the optimal combination of harmonic components for more accurate ENF estimation. Noticeably, the harmonic selection problem is equivalently formulated as a maximum weight clique (MWC) problem in graph theory, and the Bron-Kerbosch algorithm (BKA) is adopted in the GHSA. With the enhanced and optimally selected harmonic components, both the existing maximum likelihood estimator (MLE) and weighted MLE (WMLE) are incorporated to yield the final ENF estimation results. The proposed framework is extensively evaluated using both synthetic signals and our ENF-WHU dataset consisting of $130$ real-world audio recordings, demonstrating substantially improved capability of extracting the ENF from realistically noisy observations over the existing single- and multi-tone competitors. This work further improves the applicability of the ENF as a forensic criterion in real-world situations.