  We present in this paper a computational approach based on molecular dynamics
simulations and graph theory to characterize the structure of liquid water
considering not only the local structural arrangement within the first (or
second) hydration shell, but also the medium- to long-range order. In
particular, a new order parameter borrowed from the graph-theory framework,
i.e. the node total communicability (NTC ), is introduced to analyze the
dynamic network of water molecules in the liquid phase. This order parameter is
able not only to accurately report on the different high-density-liquid (HDL)
and low-density-liquid (LDL) water phases postulated in the liquid-liquid phase
transition hypothesis, but also to unveil the presence of very high density
liquid (VHDL) clusters, both under pressure and at ambient conditions. To the
best of our knowledge, VHDL water patches under moderate pressures were not
observed before.
