Detection of neurological disorders such as Alzheimer, Epilepsy, etc. through electroencephalogram (EEG) signal analysis has become increasingly popular in recent years. Alcoholism is one of the severe brain disorders that not only affects the nervous system but also leads to behavioural issues. This work presents a weighted visibility graph (WVG) approach for the detection of alcoholism, which consists of three phases. The first phase maps the EEG signals to WVG. Then, the second phase extracts important network features, viz., modularity, average weighted degree, weighted clustering coefficient, and average degree. It further identifies the most significant channels and combines their discriminative features to form feature vectors. Then, these feature vectors are trained by different machine learning classifiers in the third phase, achieving 98.91% classification accuracy. The visibility graph (VG) is not only robust to noise, but it also inherits many dynamical properties of EEG time series. Moreover, preserving weight on the links in VG aids in detecting sudden changes in the EEG signal. Experimental analysis of the alcoholic EEG signals indicates that the average accuracy of the proposed approach is higher or comparable to other reported studies.