A general hydrogen-bond connectivity descriptor based on graph theory

Hydrogen bonds play a central role in determining the structure and behaviour of liquids, particularly water, whose anomalous properties arise from its extended and highly heterogeneous hydrogen-bond network. Here, we introduce a graph-theoretical framework related to the Node Total Communicability (NTC) that enables a systematic, molecule-resolved description of hydrogen-bond networks from Molecular Dynamics (MD) simulations. In contrast to earlier related approaches with the NTC, the hydrogen-bond network is explicitly mapped onto a directed graph, allowing the asymmetric nature of hydrogen bonding to be retained. The method captures both local and longer-range connectivity, providing a unified metric to probe the structural organization of molecules. As a representative test case, we apply this approach to water and aqueous salt solutions, demonstrating how the presence of ions modifies hydrogen-bond connectivity. From this perspective, the analysis shows that ionic effects are confined to the first hydration shell, while the hydrogen-bond network beyond remains essentially unperturbed.