The optimal pebbling number of staircase graphs

Let G be a graph with a distribution of pebbles on its vertices. A pebbling move consists of removing two pebbles from one vertex and placing one pebble on an adjacent vertex. The optimal pebbling number of G is the smallest number of pebbles which can be placed on the vertices of G such that, for any vertex v of G, there is a sequence of pebbling moves resulting in at least one pebble on v. We determine the optimal pebbling number for several classes of induced subgraphs of the square grid, which we call staircase graphs. © 2018 Elsevier B.V

Optimal pebbling number of graphs with given minimum degree

Consider a distribution of pebbles on a connected graph $G$. A pebbling move removes two pebbles from a vertex and places one to an adjacent vertex. A vertex is reachable under a pebbling distribution if it has a pebble after the application of a sequence of pebbling moves. The optimal pebbling number $\pi^*(G)$ is the smallest number of pebbles which we can distribute in such a way that each vertex is reachable. It was known that the optimal pebbling number of any connected graph is at most $\frac{4n}{\delta+1}$, where $\delta$ is the minimum degree of the graph. We strengthen this bound by showing that equality cannot be attained and that the bound is sharp. If $\operatorname{diam}(G)\geq 3$ then we further improve the bound to $\pi^*(G)\leq\frac{3.75n}{\delta+1}$. On the other hand, we show that for arbitrary large diameter and any $\epsilon>0$ there are infinitely many graphs whose optimal pebbling number is bigger than $\left(\frac{8}{3}-\epsilon\right)\frac{n}{(\delta+1)}$

Reduced Physiological Complexity in Robust Elderly Adults with the APOE ε4 Allele

BACKGROUND:It is unclear whether the loss of physiological complexity during the aging process is due to genetic variations. The APOE gene has been studied extensively in regard to its relationship with aging-associated medical illness. We hypothesize that diminished physiological complexity, as measured by heart rate variability, is influenced by polymorphisms in the APOE allele among elderly individuals. METHODOLOGY/PRINCIPAL FINDINGS:A total of 102 robust, non-demented, elderly subjects with normal functions of daily activities participated in this study (97 males and 5 females, aged 79.2+/-4.4 years, range 72-92 years). Among these individuals, the following two APOE genotypes were represented: epsilon4 non-carriers (n = 87, 85.3%) and epsilon4 carriers (n = 15, 14.7%). Multi-scale entropy (MSE), an analysis used in quantifying complexity for nonlinear time series, was employed to analyze heart-rate dynamics. Reduced physiological complexity, as measured by MSE, was significantly associated with the presence of the APOE epsilon4 allele in healthy elderly subjects, as compared to APOE epsilon4 allele non-carriers (24.6+/-5.5 versus 28.9+/-5.2, F = 9.429, p = 0.003, respectively). CONCLUSIONS/SIGNIFICANCE:This finding suggests a role for the APOE gene in the diminished physiological complexity seen in elderly populations

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete