Balanced and 1-balanced graph constructions

AbstractThere are several density functions for graphs which have found use in various applications. In this paper, we examine two of them, the first being given by b(G)=|E(G)|/|V(G)|, and the other being given by g(G)=|E(G)|/(|V(G)|−ω(G)), where ω(G) denotes the number of components of G. Graphs for which b(H)≤b(G) for all subgraphs H of G are called balanced graphs, and graphs for which g(H)≤g(G) for all subgraphs H of G are called 1-balanced graphs (also sometimes called strongly balanced or uniformly dense in the literature). Although the functions b and g are very similar, they distinguish classes of graphs sufficiently differently that b(G) is useful in studying random graphs, g(G) has been useful in designing networks with reduced vulnerability to attack and in studying the World Wide Web, and a similar function is useful in the study of rigidity. First we give a new characterization of balanced graphs. Then we introduce a graph construction which generalizes the Cartesian product of graphs to produce what we call a generalized Cartesian product. We show that generalized Cartesian product derived from a tree and 1-balanced graphs are 1-balanced, and we use this to prove that the generalized Cartesian products derived from 1-balanced graphs are 1-balanced

Finishing the euchromatic sequence of the human genome

The sequence of the human genome encodes the genetic instructions for human physiology, as well as rich information about human evolution. In 2001, the International Human Genome Sequencing Consortium reported a draft sequence of the euchromatic portion of the human genome. Since then, the international collaboration has worked to convert this draft into a genome sequence with high accuracy and nearly complete coverage. Here, we report the result of this finishing process. The current genome sequence (Build 35) contains 2.85 billion nucleotides interrupted by only 341 gaps. It covers ∼99% of the euchromatic genome and is accurate to an error rate of ∼1 event per 100,000 bases. Many of the remaining euchromatic gaps are associated with segmental duplications and will require focused work with new methods. The near-complete sequence, the first for a vertebrate, greatly improves the precision of biological analyses of the human genome including studies of gene number, birth and death. Notably, the human enome seems to encode only 20,000-25,000 protein-coding genes. The genome sequence reported here should serve as a firm foundation for biomedical research in the decades ahead

Nowhere zero flows in line graphs

AbstractCai an Corneil (Discrete Math. 102 (1992) 103–106), proved that if a graph has a cycle double cover, then its line graph also has a cycle double cover, and that the validity of the cycle double cover conjecture on line graphs would imply the truth of the conjecture in general. In this note we investigate the conditions under which a graph G has a nowhere zero k-flow would imply that L(G), the line graph of G, also has a nowhere zero k-flow. The validity of Tutte's flow conjectures on line graphs would also imply the truth of these conjectures in general

Purification and Characterization of a Novel Kazal-Type Trypsin Inhibitor from the Leech of Hirudinaria manillensis

Kazal-type serine proteinase inhibitors are found in a large number of living organisms and play crucial roles in various biological and physiological processes. Although some Kazal-type serine protease inhibitors have been identified in leeches, none has been reported from Hirudinaria manillensis, which is a medically important leech. In this study, a novel Kazal-type trypsin inhibitor was isolated from leech H. manillensis, purified and named as bdellin-HM based on the sequence similarity with bdellin-KL and bdellin B-3. Structural analysis revealed that bdellin-HM was a 17,432.8 Da protein and comprised of 149 amino acid residues with six cysteines forming three intra-molecular disulfide bonds. Bdellin-HM showed similarity with the Kazal-type domain and may belong to the group of “non-classical” Kazal inhibitors according to its CysI-CysII disulfide bridge position. Bdellin-HM had no inhibitory effect on elastase, chymotrypsin, kallikrein, Factor (F) XIIa, FXIa, FXa, thrombin and plasmin, but it showed a potent ability to inhibit trypsin with an inhibition constant (Ki) of (8.12 ± 0.18) × 10−9 M. These results suggest that bdellin-HM from the leech of H. manillensis plays a potent and specific inhibitory role towards trypsin