A review of graph and complex network theory in water distribution networks: Mathematical foundation, application and prospects

•Mathematical foundations and key application areas of graph theory are reviewed.•Popular applications are state estimation, performance evaluation, optimization, and critical segment identification.•Data availability limits exploration and application of WDNs’ topological patterns.•Combination of graph theory and hydraulic information leads to better results.•Challenges include tailoring topological indicators and integrating with GNNs. Graph theory (GT) and complex network theory play an increasingly important role in the design, operation, and management of water distribution networks (WDNs) and these tasks were originally often heavily dependent on hydraulic models. Facing the general reality of the lack of high-precision hydraulic models in water utilities, GT has become a promising surrogate or assistive technology. However, there is a lack of a systematic review of how and where the GT techniques are applied to the field of WDNs, along with an examination of potential directions that GT can contribute to addressing WDNs' challenges. This paper presents such a review and first summarizes the graph construction methods and topological properties of WDNs, which are mathematical foundations for the application of GT in WDNs. Then, main application areas, including state estimation, performance evaluation, partitioning, optimal design, optimal sensor placement, critical components identification, and interdependent networks analysis, are identified and reviewed. GT techniques can provide acceptable results and valuable insights while having a low computational burden compared with hydraulic models. Combining GT with hydraulic model significantly enhances the performance of analysis methods. Four research challenges, namely reasonable abstraction, data availability, tailored topological indicators, and integration with Graph Neural Networks (GNNs), have been identified as key areas for advancing the application and implementation of GT in WDNs. This paper would have a positive impact on promoting the use of GT for optimal design and sustainable management of WDNs. [Display omitted]