The Hamiltonian index of a graph and its branch-bonds

Let $G$ be an undirected and loopless finite graph that is not a path. The minimum $m$ such that the iterated line graph $L^m(G)$ is hamiltonian is called the hamiltonian index of $G,$ denoted by $h(G).$ A reduction method to determine the hamiltonian index of a graph $G$ with $h(G)\geq 2$ is given here. With it we will establish a sharp lower bound and a sharp upper bound for $h(G)$, respectively, which improves some known results of P.A. Catlin et al. [J. Graph Theory 14 (1990)] and H.-J. Lai [Discrete Mathematics 69 (1988)]. Examples show that $h(G)$ may reach all integers between the lower bound and the upper bound. \u

Spanning subgraph with Eulerian components

AbstractA graph is k-supereulerian if it has a spanning even subgraph with at most k components. We show that if G is a connected graph and G′ is the (collapsible) reduction of G, then G is k-supereulerian if and only if G′ is k-supereulerian. This extends Catlin’s reduction theorem in [P.A. Catlin, A reduction method to find spanning Eulerian subgraphs, J. Graph Theory 12 (1988) 29–44]. For a graph G, let F(G) be the minimum number of edges whose addition to G create a spanning supergraph containing two edge-disjoint spanning trees. We prove that if G is a connected graph with F(G)≤k, where k is a positive integer, then either G is k-supereulerian or G can be contracted to a tree of order k+1. This is a best possible result which extends another theorem of Catlin, in [P.A. Catlin, A reduction method to find spanning Eulerian subgraphs, J. Graph Theory 12 (1988) 29–44]. Finally, we use these results to give a sufficient condition on the minimum degree for a graph G to bek-supereulerian

Ballistic-diffusive phonon heat transport across grain boundaries

The propagation of a heat pulse in a single crystal and across grain boundaries (GBs) is simulated using a concurrent atomistic-continuum method furnished with a coherent phonon pulse model. With a heat pulse constructed based on a Bose-Einstein distribution of phonons, this work has reproduced the phenomenon of phonon focusing in single and polycrystalline materials. Simulation results provide visual evidence that the propagation of a heat pulse in crystalline solids with or without GBs is partially ballistic and partially diffusive, i.e., there is a co-existence of ballistic and diffusive thermal transport, with the long-wavelength phonons traveling ballistically while the short-wavelength phonons scatter with each other and travel diffusively. To gain a quantitative understanding of GB thermal resistance, the kinetic energy transmitted across GBs is monitored on the fly and the time-dependent energy transmission for each specimen is measured; the contributions of coherent and incoherent phonon transport to the energy transmission are estimated. Simulation results reveal that the presence of GBs modifies the nature of thermal transport, with the coherent long-wavelength phonons dominating the heat conduction in materials with GBs. In addition, it is found that phonon-GB interactions can result in reconstruction of GBs

Degree sums and subpancyclicity in line graphs

A graph is called subpancyclic if it contains a cycle of length k for each k between 3 and the circumference of the graph. In this paper, we show that if the degree sum of the vertices along each 2-path of a graph G exceeds (n+6)/2, or if the degree sum of the vertices along each 3-path of G exceeds (2n+16)/3, then its line graph L(G) is subpancyclic. Simple examples show that these bounds are best possible. Our results shed some light on the content of a famous Metaconjecture of Bondy

In vitro and in vivo antitumor effects of acetylshikonin isolated from Arnebia euchroma (Royle) Johnst (Ruanzicao) cell suspension cultures

<p>Abstract</p> <p>Background</p> <p>Shikonin derivatives have cytotoxic and antitumor effects. This study aims to investigate the antitumor effects of acetylshikonin isolated from a Chinese medicinal herb <it>Arnebia euchroma (Royle) Johnst</it>.</p> <p>Methods</p> <p>The 3-(4,5-dimethylthiazol- 2-yl)-2,5-diphenyltetrazolium bromide (MTT) assay was used to determine the <it>in vitro </it>antitumor effects of acetylshikonin on human lung adenocarcinoma cell line A549, human hepatocellular carcinoma cell line Bel-7402, human breast adenocarcinoma cell line MCF-7 and mouse Lewis lung carcinoma (LLC) cell line. C₅₇BL/6 mice with LLC model were used to study the <it>in vivo </it>antitumor effects of acetylshikonin. The expression of bax, bcl-2 and caspase-3 proteins in LLC tissue was determined with immunohistochemical staining.</p> <p>Results</p> <p>In A549, Bel-7402, MCF-7 and LLC cell lines, acetylshikonin inhibited cell growth in a dose-dependent manner. IC50 (means ± SD) were 5.6 ± 0.86 μg/ml, 6.82 ± 1.5 μg/ml, 3.04 ± 0.44 μg/ml and 2.72 ± 0.38 μg/ml respectively. Acetylshikonin suppressed tumor growth in C57BL/6 mice with LLC. The inhibition rate of acetylshikonin (2 mg/kg) was 42.85%. Immunohistochemical staining revealed that in the acetylshikonin groups the expression of bax and caspase-3 increased, whereas the expression of bcl-2 decreased, suggesting that acetylshikonin induced tumor cell apoptosis through activating the pro-apoptotic bcl-2 family and caspase-3.</p> <p>Conclusion</p> <p>Acetylshikonin isolated from <it>Arnebia euchroma (Royle) Johnst </it>cell suspension cultures exhibits specific <it>in vivo </it>and <it>in vitro </it>antitumor effects.</p

Star subdivisions and connected even factors in the square of a graph

For any positive integer $s$, a $[2,2s]$-factor in a graph $G$ is a connected even factor with maximum degree at most $2s$. We prove that if every induced $S(K_{1, 2s+1})$ in a graph $G$ has at least 3 edges in a block of degree at most two, then $G^2$ has a $[2,2s]$-factor. This extends the results of Hendry and Vogler and of Abderrezzak et al.Comment: 10 page