For a vertex V of a graph , the lower connectivity, denoted by V ( ), is the smallest number of vertices that contains V and those vertices whose deletion from produces a disconnected or a trivial graph. The average lower connectivity denoted by av ( ) is the value (∑ V∈ ( ) V ( ))/| ( )|. It is shown that this parameter can be used to measure the vulnerability of networks. This paper contains results on bounds for the average lower connectivity and obtains the average lower connectivity of some graphs.
Neuroinflammation plays pivotal roles in the pathogenesis of Alzheimer's disease (AD). IL-6 is pleiotropic cytokine which plays significant pathological role in inflammatory diseases and causes prolonged inflammation. Additionally, IL-6 activates microglia cells and enhances the accumulation of amyloid-β peptides. Moreover, IL-6 signal transduction is mediated by membrane-bound and soluble IL-6 receptors. Tocilizumab which is a humanized anti-human IL-6 receptor (IL-6R) monoclonal antibody binds to both of these receptors and inhibits IL-6 signaling by this route. The objective was to investigate tocilizumab's potential effects in the treatment of AD. Male Sprague-Dawley rats were divided into three groups: sham (control), streptozotocin (STZ), and tocilizumab-STZ. We used a single dose of intracerebroventricular (ICV) tocilizumab, beginning 1 h prior to injection of STZ for 3 weeks. The rats in STZ and tocilizumab-STZ groups were given ICV-STZ (3 mg/kg). Behavioral parameters were evaluated on days 17-20 and the rats were sacrificed on day-21 to examine histopathological changes. STZ injection caused significant decrease in the mean escape latency in passive avoidance and also declined the performance improvement in Morris water maze tests. Tocilizumab-STZ group significantly improved learning and spatial memory functions by increasing RLT in the passive avoidance and by shortening escape latency in reaching the platform in the Morris water maze. Histopathological changes were examined using hematoxylin and eosin and immunohistochemical (IHC) stainings. IHC analysis revealed that while protein expressions of amyloid-ß (3.5 ± 0.2) and IL-6 (2.9 ± 0.4) showed intense immune-positivity in STZ group, amyloid-ß (1.3 ± 0.1) and IL-6 (1.5 ± 0.2) immunoreactivities were substantially decreased in tocilizumab treatment group. We conclude that tocilizumab treatment attenuated significantly STZ-induced cognitive impairment and histopathological changes. Further studies would be desirable to investigate clinically relevant protective effects of tocilizumab in AD.
In this study, we introduce multidelay fractional differential equations with variable coefficients in a unique formula. A novel graph-operational matrix method based on the fractional Caputo, Riemann-Liouville, Caputo-Fabrizio, and Jumarie derivative types is developed to efficiently solve them. We also make use of the collocation points and matrix relations of the matching polynomial of the complete graph in the method. We determine which of the fractional derivative types is more appropriate for the method. The solutions of model problems are improved via a new residual error analysis technique. We design a general computer program module. Thus, we can explicitly monitor the usefulness of the method. All results are scrutinized in tables and figures. Finally, an illustrative algorithm is presented.
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