The strong chromatic index of complete cubic Halin graphs

A complete cubic Halin graph is a cubic Halin graph whose characteristic tree is a complete cubic tree, in which all leaves are at the same distance from the root vertex. In this work, we determine the strong chromatic index of the complete cubic Halin graph.     

Parallel domain decomposition method for finite element approximation of 3D steady state non‐Newtonian fluids

We introduce a stabilized finite element method for the 3D non‐Newtonian Navier–Stokes equations and a parallel domain decomposition method for solving the sparse system of nonlinear equations arising from the discretization. Non‐Newtonian flow problems are, generally speaking, more challenging than Newtonian flows because the nonlinearities are not only in the convection term but also in the viscosity term, which depends on the shear rate. Many good iterative methods and preconditioning techniques that work well for the Newtonian flows do not work well for the non‐Newtonian flows. We employ a Galerkin/least squares finite element method, with stabilization parameters adjusted to count the non‐Newtonian effect, to discretize the equations, and the resulting highly nonlinear system of equations is solved by a Newton–Krylov–Schwarz algorithm. In this study, we apply the proposed method to some inelastic power‐law fluid flows through the eccentric annuli with inner cylinder rotation and investigate the robustness of the method with respect to some physical parameters, including the power‐law index and the Reynolds number ratios. We then report the superlinear speedup achieved by the domain decomposition algorithm on a computer with up to 512 processors. Copyright © 2015 John Wiley & Sons, Ltd.     

The smallest values of algebraic connectivity for trees

The algebraic connectivity of a graph G is the second smallest eigenvalue of its Laplacian matrix. Let ■n be the set of all trees of order n. In this paper, we will provide the ordering of trees in ■n up to the last eight trees according to their smallest algebraic connectivities when n ≥ 13. This extends the result of Shao et al. [The ordering of trees and connected graphs by algebraic connectivity. Linear Algebra Appl., 428, 1421-1438 (2008)].     

Bioactive Luminescent Transition‐Metal Complexes for Biomedical Applications

The serendipitous discovery of the anticancer drug cisplatin cemented medicinal inorganic chemistry as an independent discipline in the 1960s. Luminescent metal complexes have subsequently been widely applied for sensing, bio‐imaging, and in organic light‐emitting diode applications. Transition‐metal complexes possess a variety of advantages that make them suitable as therapeutics and as luminescent probes for biomolecules. It is thus highly desirable to develop new luminescent metal complexes that either interact with DNA through different binding modes or target alternative cellular machinery such as proteins as well as to provide a more effective means of monitoring disease progression. In this Review, we highlight recent examples of biologically active luminescent metal complexes that can target and probe a specific biomolecule, and offer insights into the future potential of these compounds for the investigation and treatment of human diseases.     

Cell rotation graphs of strongly connected orientations of plane graphs with an application

The cell rotation graph D(G) on the strongly connected orientations of a 2-edge-connected plane graph G is defined. It is shown that D(G) is a directed forest and every component is an in-tree with one root; if T is a component of D(G), the reversions of all orientations in T induce a component of D(G), denoted by T⁻, thus (T,T⁻) is called a pair of in-trees of D(G); G is Eulerian if and only if D(G) has an odd number of components (all Eulerian orientations of G induce the same component of D(G)); the width and height of T are equal to that of T⁻, respectively. Further it is shown that the pair of directed tree structures on the perfect matchings of a plane elementary bipartite graph G coincide with a pair of in-trees of D(G). Accordingly, such a pair of in-trees on the perfect matchings of any plane bipartite graph have the same width and height.     

Enumeration of Wiener indices in random polygonal chains

The Wiener index of a connected graph (molecule graph) G is the sum of the distances between all pairs of vertices of G, which was reported by Wiener in 1947 and is the oldest topological index related to molecular branching. In this paper, simple formulae of the expected value of the Wiener index in a random polygonal chain and the asymptotic behavior of this expectation are established by solving a difference equation. Based on the results above, we obtain the average value of the Wiener index of all polygonal chains with n polygons. As applications, we use the unified formulae to obtain the expected values of the Wiener indices of some special random polygonal chains which were deeply discussed in the context of organic chemistry or statistical physics.     

Judicious Choice of N‐Heterocycles for the Realization of Sky‐Blue‐ to Green‐Emitting Carbazolylgold(III) C^C^N Complexes and Their Applications for Organic Light‐Emitting Devices

A new class of sky‐blue‐ to green‐emitting carbazolylgold(III) C^C^N complexes containing pyrazole or benzimidazole moieties has been successfully designed and synthesized. Through the judicious choice of the N‐heterocycles in the cyclometalating ligand and the tailor‐made carbazole moieties, maximum photoluminescence quantum yields of 0.52 and 0.39 have been realized in the green‐ and sky‐blue‐emitting complexes, respectively. Solution‐processed and vacuum‐deposited organic light‐emitting devices (OLEDs) based on the benzimidazole‐containing complexes have been prepared. The sky‐blue‐emitting device shows an emission peaking at 484 nm with a narrow full‐width at half‐maximum of 57 nm (2244 cm^?1), demonstrating the potential of this class of complexes in the application of OLEDs with high color purity. In addition, high maximum external quantum efficiencies of 12.3 % and a long operational half‐lifetime of over 5300 h at 100 cd m^?2 have been achieved in the vacuum‐deposited green‐emitting devices.