  Maintaining cerebral blood flow is critical for adequate neuronal function.
Previous computational models of brain capillary networks have predicted that
heterogeneous cerebral capillary flow patterns result in lower brain tissue
partial oxygen pressures. It has been suggested that this may lead to number of
diseases such as Alzheimer's disease, acute ischemic stroke, traumatic brain
injury and ischemic heart disease. We have previously developed a computational
model that was used to describe in detail the effect of flow heterogeneities on
tissue oxygen levels. The main result in that paper was that, for a general
class of capillary networks, perturbations of segment diameters or conductances
always lead to decreased oxygen levels. This result was verified using both
numerical simulations and mathematical analysis. However, the analysis depended
on a novel conjecture concerning the Laplace operator of functions related to
the segment flow rates and how they depend on the conductances. The goal of
this paper is to give a mathematically rigorous proof of the conjecture for a
general class of networks. The proof depends on determining the number of trees
and forests in certain graphs arising from the capillary network.
