Cacti with Extremal PI Index

The vertex PI index $PI(G) = \sum_{xy \in E(G)} [n_{xy}(x) + n_{xy}(y)]$ is a distance-based molecular structure descriptor, where $n_{xy}(x)$ denotes the number of vertices which are closer to the vertex $x$ than to the vertex $y$ and which has been the considerable research in computational chemistry dating back to Harold Wiener in 1947. A connected graph is a cactus if any two of its cycles have at most one common vertex. In this paper, we completely determine the extremal graphs with the largest and smallest vertex PI indices among all the cacti. As a consequence, we obtain the sharp bounds with corresponding extremal cacti and extend a known result.Comment: Accepted by Transactions on Combinatorics, 201

Evidence for Two-Component Jet in Sw J1644+57

The continued observations of Sw J1644+57 in X-ray and radio bands accumulated a rich data set to study the relativistic jet launched in this tidal disruption event. We find that the re-brightening feature in the radio light curve can be naturally explained by the two-component jet model. The possible origin of this structured jet are the Blandford-Znajek and Blandford-Payne mechanisms. We also show that this two-component jet model can interpret the two kinds of quasi-periodic variations in the X-ray light curve: a 200 second quasi-periodic oscillation (QPO) and a 2.7-day quasi-periodic variation. The latter is interpreted by a precessing outer jet launched near the Bardeen-Petterson radius of a warped disk. The $\sim$ 200s QPO could be associated with a second, narrower jet sweeping the observer line-of-sight periodically, which is launched from a spinning black hole in the misaligned direction with respect to the black hole's angular momentum.Comment: 6 pages, 3 figures. In proceedings of "Swift: 10 Years of Discovery" congress (Rome, 2-4 December 2014), PoS(SWIFT 10)17

Sharp bounds for the modified multiplicative zagreb indices of graphs with vertex connectivity at most k

© 2019, University of Nis. All rights reserved. Zagreb indices and their modified versions of a molecular graph originate from many practical problems such as two dimensional quantitative structure-activity (2D QSAR) and molecular chirality. Nowadays, they have become important invariants which can be used to characterize the properties of graphs from different aspects. Let Vkn (or Ekn respectively) be a set of graphs of n vertices with vertex connectivity (or edge connectivity respectively) at most k. In this paper, we explore some properties of the modified first and second multiplicative Zagreb indices of graphs in Vkn and Ekn. By using analytic and combinatorial tools, we obtain some sharp lower and upper bounds for these indices of graphs in Vkn and Ekn. In addition, the corresponding extremal graphs which attain the lower or upper bounds are characterized. Our results enrich outcomes on studying Zagreb indices and the methods developed in this paper may provide some new tools for investigating the values on modified multiplicative Zagreb indices of other classes of graphs