Graph pebbling is a network optimization model for transporting discrete resources that are consumed in transit: the movement of 2 pebbles across an edge consumes one of the pebbles. The pebbling number of a graph is the fewest number of pebbles t so that, from any initial configuration of t pebbles on its vertices, one can place a pebble on any given target vertex via such pebbling steps. It is known that deciding whether a given configuration on a particular graph can reach a specified target is NP-complete, even for diameter 2 graphs, and that deciding whether the pebbling number has a prescribed upper bound is Π[P over 2]-complete. On the other hand, for many families of graphs there are formulas or polynomial algorithms for computing pebbling numbers; for example, complete graphs, products of paths (including cubes), trees, cycles, diameter 2 graphs, and more. Moreover, graphs having minimum pebbling number are called Class 0, and many authors have studied which graphs are Class 0 and what graph properties guarantee it, with no characterization in sight. In this paper we investigate an important family of diameter 3 chordal graphs called split graphs; graphs whose vertex set can be partitioned into a clique and an independent set. We provide a formula for the pebbling number of a split graph, along with an algorithm for calculating it that runs in O(n[superscript β]) time, where β = 2ω/(ω + 1) [= over ∼] 1.41 and ω [= over ∼] 2.376 is the exponent of matrix multiplication. Furthermore we determine that all split graphs with minimum degree at least 3 are Class 0.

Background: To determine the effects of high sucrose diets on vascular reactivity. We hypothesized that similar to high fat diets (HFD), HSD feeding would lead to increased adiposity resulting in inflammation and oxidative stress-mediated impairment of vasodilation.

Methods: Male Sprague-Dawley rats were fed control chow (Chow), HSD or HFD diets for 6 weeks. The role of inflammation and oxidative stress on impaired vasodilation were assessed in isolated mesenteric arterioles.

Results: HSD and HFD induced increased adiposity, oxidative stress and inflammation. HFD rats developed fasting hyperglycemia. Both HSD and HFD rats developed impaired glucose tolerance and hyperleptinemia. Nitric oxide (NO)-mediated vasodilation was significantly attenuated in both HSD and HFD rats but was normalized by treatment with antioxidants or anti-inflammatory drugs. Endothelial NO synthase (eNOS) protein expression was not affected by diet. Sensitivity to NO was reduced since NOS inhibition attenuated vasodilation in Chow rats but did not further impair vasodilation in HSD or HFD rats. Likewise, responsiveness to a NO donor was attenuated in both experimental groups.

Conclusions: Oxidative stress diminishes vasodilatory responsiveness in HSD and HFD rats through ROS-mediated scavenging of NO and decreased smooth muscle sensitivity to NO. Inflammation also plays a significant role in the impaired vasodilation.

Background: Rats fed high fat (HFD) or high sucrose (HSD) diets develop increased adiposity as well as impaired vasodilatory responsiveness stemming from oxidative stress. Moreover, HFD rats become hypertensive compared to either control (Chow) or HSD fed rats, suggesting elevated vascular tone. We hypothesized that rats with increased adiposity and oxidative stress demonstrate augmented pressure-induced vasoconstriction (i.e. myogenic tone) that could account for the hypertensive state.

Methods: Male Sprague-Dawley rats were fed Chow, HFD or HSD for 6 weeks. The effects of oxidative stress and endogenous nitric oxide on myogenic responses were examined in small mesenteric arteries by exposing the arteries to incremental intraluminal pressure steps in the presence of antioxidants or an inhibitor of nitric oxide synthase, LNNA (100 μM).

Results: Contrary to the hypothesis, rats fed either HSD or HFD had significantly impaired myogenic responses despite similar vascular morphology and passive diameter responses to increasing pressures. Vascular smooth muscle (VSM) calcium levels were normal in HFD arteries suggesting that diminished calcium sensitivity was responsible for the impaired myogenic response. In contrast, VSM calcium levels were reduced in HSD arteries but were increased with pre-exposure of arteries to the antioxidants tiron (10 mM) and catalase (1200 U/mL), also resulting in enhanced myogenic tone. These findings show that oxidative stress impairs myogenic tone in arteries from HSD rats by decreasing VSM calcium. Similarly, VSM calcium responses were increased in arteries from HFD rats following treatment with tiron and catalase, but this did not result in improved myogenic tone. Nitric oxide is involved in the impaired myogenic response in HFD, but not HSD, rats since inhibition with LNNA resulted in maximal myogenic responses at lower intraluminal pressures and VSM calcium levels, further implicating reduced calcium sensitivity in the impaired response.

Conclusion: The impaired myogenic responses observed in isolated arteries from HSD and HFD rats are attributed to changes in VSM calcium signaling.