Hall conditions for edge-weighted bipartite graphs

A weighted variant of Hall's condition for the existence of matchings is shown to be equivalent to the existence of a matching in a lexicographic product. This is used to introduce characterizations of those bipartite graphs whose edges may be replicated so as to yield semiregular multigraphs or, equivalently, semiregular edge-weightings. Such bipartite graphs will be called semiregularizable. Some infinite families of semiregularizable trees are described and all semiregularizable trees on at most 11 vertices are listed. Matrix analogues of some of the results are mentioned and are shown to imply some of the known characterizations of regularizable graphs.

Higher brain neurons succumb to acute stroke-like insult while lower brain neurons strongly resist

Pyramidal neurons (PyNs) in ‘higher’ brain are highly susceptible to acute stroke injury yet ‘lower’ brain regions better survive global ischemia, presumably because of better residual blood flow. Here we show that projection neurons in ‘lower’ brain regions of hypothalamus and brainstem intrinsically resist acute stroke-like injury independent of blood flow in the brain slice. In contrast `higher` projection neurons in neocortex, hippocampus, striatum and thalamus are highly susceptible. In live brain slices from rat deprived of oxygen and glucose (OGD), we imaged anoxic depolarization (AD) as it propagates through these regions. AD, the initial electrophysiological event of stroke, is a depolarizing front that drains residual energy in compromised gray matter. The extent of AD reliably determines ensuing damage in higher brain, but using whole-cell recordings we found that all CNS neurons do not generate a robust AD. Higher neurons generate strong AD and show no functional recovery in contrast to neurons in hypothalamus and brainstem that generate a weak and gradual AD. Most dramatically, lower neurons recover their membrane potential, input resistance and spike amplitude when oxygen and glucose is restored, while higher neurons do not. Following OGD, new recordings could be acquired in all lower (but not higher) brain regions, with some neurons even withstanding multiple OGD exposure. Two-photon laser scanning microscopy confirmed neuroprotection in lower, but not higher gray matter. Specifically pyramidal neurons swell and lose their dendritic spines post-OGD, whereas neurons in hypothalamus and brainstem display no such injury. Exposure to the Na+/K+ ATPase inhibitor ouabain (100 μM), induces depolarization similar to OGD in all cell types tested. Moreover, elevated [K+]o evokes spreading depression (SD), a milder version of AD, in higher brain but not hypothalamus or brainstem so weak AD correlates with the inability to generate SD. In summary, overriding the Na+/K+ pump using OGD, ouabain or elevated [K+]o evokes steep and robust depolarization of higher gray matter. We show that this important regional difference can be largely accounted for by the intrinsic properties of the resident neurons and that Na+/K+ ATPase pump efficiency is a major determining factor generating strong or weak spreading depolarizations.