Generation of cubic graphs and snarks with large girth

We describe two new algorithms for the generation of all non-isomorphic cubic graphs with girth at least $k\ge 5$ which are very efficient for $5\le k \le 7$ and show how these algorithms can be efficiently restricted to generate snarks with girth at least $k$. Our implementation of these algorithms is more than 30, respectively 40 times faster than the previously fastest generator for cubic graphs with girth at least 6 and 7, respectively. Using these generators we have also generated all non-isomorphic snarks with girth at least 6 up to 38 vertices and show that there are no snarks with girth at least 7 up to 42 vertices. We present and analyse the new list of snarks with girth 6.Comment: 27 pages (including appendix

Polyhedra with few 3-cuts are hamiltonian

In 1956, Tutte showed that every planar 4-connected graph is hamiltonian. In this article, we will generalize this result and prove that polyhedra with at most three 3-cuts are hamiltonian. In 2002 Jackson and Yu have shown this result for the subclass of triangulations. We also prove that polyhedra with at most four 3-cuts have a hamiltonian path. It is well known that for each $k \ge 6$ non-hamiltonian polyhedra with $k$ 3-cuts exist. We give computational results on lower bounds on the order of a possible non-hamiltonian polyhedron for the remaining open cases of polyhedra with four or five 3-cuts.Comment: 21 pages; changed titl

A possible feature of thermal matter in relativistic jets of radio-loud quasars

It has been suggested that relativistic jets in quasars may contain a considerable amount of thermal matter. In this paper, we explore the possibility that the K-alpha line from the thermal matter may appear at tens of keV due to a high Doppler blue-shift. In the jet comoving frame, the energy density of photons originally emitted by the accretion disk and reflected off the broad line region clouds dominates over that of photons of other origin. We discuss the photoionization states of the thermal matter and find that the irons elements are neutral. The high metallicity in quasars enhances the possibility to detect the thermal matter in the relativistic jet in some radio-loud quasars. A highly Doppler blue-shifted K-alpha line may be detected. We make a rediction for 3C 273, in which the K-alpha line luminosity might be of the order $3.0\times 10^{44}$ erg/s with an equivalent width of 2.4 keV. Such a line could be detected in a future mission.Comment: 4 pages, 1 Figur

Conformally Einstein Products and Nearly K\"ahler Manifolds

In the first part of this note we study compact Riemannian manifolds (M,g) whose Riemannian product with R is conformally Einstein. We then consider compact 6--dimensional almost Hermitian manifolds of type W_1+W_4 in the Gray--Hervella classification admitting a parallel vector field and show that (under some regularity assumption) they are obtained as mapping tori of isometries of compact Sasaki-Einstein 5-dimensional manifolds. In particular, we obtain examples of inhomogeneous locally (non-globally) conformal nearly K\"ahler compact manifolds

CENTRALITY MEASURES ON BRINKMANN GRAPH: BEFORE AND AFTER NODE DELETION

Brinkmann graph is a 4-regular graph with 21 nodes and 42 edges discovered by Gunnar Brinkmann in 1992. To our knowledge, the research specifically on Brinkmann graph is still hard to find. Therefore, this research was carried out to analyze the Brinkmann graph in term of its centrality. The centrality measures used are degree, betweenness, and closeness centrality. In this paper, we presented the centrality measures not only on the Brinkmann graph but also on the Brinkmann graph after node deletion to see how the impact of node deletion to the centrality of graph. Before deletion, the results showed that according to the betweenness centrality, there exist 7 nodes who act as mediators or bridges in the Brinkmann graph. Therefore, when a node among these nodes has deleted, it affected not only any other mediator nodes and the furthest nodes from the deleted node but also the nodes that are adjacent to the deleted nodes

Active Galactic Nuclei and the Truncation of Star Formation in K+A Galaxies

We have searched for active galactic nuclei (AGNs) in K+A galaxies, using multiwavelength imaging and spectroscopy in the Bootes field of the NOAO Deep Wide-Field Survey. The K+A galaxies, which have had their star formation rapidly truncated, are selected via their strong Balmer absorption lines and weak H-alpha emission. Our sample consists of 24 K+A galaxies selected from 6594 0.10<z<0.35 galaxies brighter than I=20 with optical spectroscopy from the AGN and Galaxy Evolution Survey. Two thirds of the K+A galaxies are likely ongoing galaxy mergers, with nearby companion galaxies or tidal tails. Galaxy mergers may be responsible for the truncation of star formation, or we are observing the aftermath of merger triggered starbursts. As expected, the optical colors of K+A galaxies largely fall between blue galaxies with ongoing star formation and red passive galaxies. However, only 1% of the galaxies with colors between the red and blue populations are K+A galaxies, and we conclude that the truncation of star formation in K+A galaxies must have been unusually abrupt (<100 Myr). We examined the AGN content of K+A galaxies with both optical emission-line ratios (BPT diagrams) and Chandra X-ray imaging. At least half of all K+A galaxies display the optical emission-line ratios of AGNs, and a third of M_R<-22 K+A galaxies host AGNs with X-ray luminosities of 10^{42} erg/s. The faintest K+A galaxies do not show clear evidence for hosting AGNs, having emission-line ratios consistent with photoionization by massive stars and few X-ray detections. We speculate that two mechanisms may be responsible for the truncation of star formation in K+A galaxies, with AGN feedback only playing a role in M_R<-20.5 galaxies.Comment: Accepted for publication in the Astrophysical Journal. 13 pages, 8 figure