Article

# The Erdős-Sós conjecture for graphs of girth 5

FB Mathematik, Freie Universität Berlin, Graduiertenkolleg ‘Alg. Diskr. Mathematik’, Arnimallee 2–6, 14195 Berlin, Germany; Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA

Discrete Mathematics (Impact Factor: 0.58). 01/1996; DOI: 10.1016/0012-365X(95)00207-D -
##### Article: Vascular effects of microbubble-enhanced, pulsed, focused ultrasound on liver blood perfusion.

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**ABSTRACT:**The purpose of this study was to investigate the vascular effects of microbubble-enhanced pulsed high-pressure ultrasound on liver blood perfusion. In the presence of circulating lipid-shell microbubbles, a focused ultrasound transducer was used to transcutaneously treat eight livers of healthy rabbits for perfusion analysis and to treat three livers with the abdomen open for histologic analysis. Twenty-two livers treated with the ultrasound only (n = 11) or microbubbles only (n = 11) served as the controls. The focused ultrasound was operated at a frequency of 1.22 MHz with a peak negative pressure of 4.6 MPa. The liver blood perfusion was estimated by performing contrast-enhanced ultrasound and gray-scale quantification on the livers before and after treatment. A temporary, nonenhanced region occurred in all of the experimental livers. The regional contrast gray-scale values of the experimental group dropped significantly from 88.4 before treatment to 2.7 after treatment. The liver perfusion also demonstrated a gradual recovery over a 60-min period. The liver perfusion of the control groups remained the same after treatment. We found microvascular rupture, hemorrhage and swelling hepatocytes upon histologic examination of the experimental group. Regional liver blood perfusion can be temporarily blocked by microbubble-enhanced focused ultrasound with high-pressure amplitude. These vascular effects can be explained as acute microvascular injury of the liver and may have clinical implications.Ultrasound in medicine & biology 11/2011; 38(1):91-8. · 2.46 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The Turán number of a graph H, ex(n, H), is the maximum number of edges in any graph on n vertices which does not contain H as a subgraph. Let Pl denote a path on l vertices, and let k ⋅ Pl denote k vertex-disjoint copies of Pl. We determine ex(n, k ⋅ P3) for n appropriately large, answering in the positive a conjecture of Gorgol. Further, we determine ex(n, k ⋅ Pl) for arbitrary l, and n appropriately large relative to k and l. We provide some background on the famous Erdős–Sós conjecture, and conditional on its truth we determine ex(n, H) when H is an equibipartite forest, for appropriately large n.Combinatorics Probability and Computing 01/2011; 20:837-853. · 0.61 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The Erdős-Sós conjecture says that a graph G on n vertices and number of edges e(G) > n(k− 1)/2 contains all trees of size k. In this paper we prove a sufficient condition for a graph to contain every tree of size k formulated in terms of the minimum edge degree ζ(G) of a graph G defined as ζ(G) = min{d(u) + d(v) − 2: uv ∈ E(G)}. More precisely, we show that a connected graph G with maximum degree Δ(G) ≥ k and minimum edge degree ζ(G) ≥ 2k − 4 contains every tree of k edges if d G (x) + d G (y) ≥ 2k − 4 for all pairs x, y of nonadjacent neighbors of a vertex u of d G (u) ≥ k.Acta Mathematica Sinica English Series 01/2011; 27(1-27 -1):135-140.

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