Abstract:
The domination number of a graph is the smallest number of vertices which dominate all remaining vertices by edges of . The bondage number of a nonempty graph is the smallest number of edges whose removal from results in a graph with domination number greater than the domination number of . The concept of the bondage number was formally introduced by Fink et al. in 1990. Since then, this topic has received considerable research attention and made some progress, variations, and generalizations. This paper gives a survey on the bondage number, including known results, conjectures, problems, and some comments, also selectively summarizes other types of bondage numbers. 1. Introduction For terminology and notation on graph theory not given here, the reader is referred to Xu [1]. Let be a finite, undirected, and simple graph. We call and the order and size of and denote them by and , respectively, unless otherwise specified. Through this paper, the notations , , and always denote a path, a cycle, and a complete graph of order , respectively, the notation denotes a complete -partite graph with and , with , and is a star. For two vertices and in a connected graph , we use to denote the distance between and in . For a vertex in , let be the open set of neighbors of and the closed set of neighbors of . For a subset , , and , where . Let be the set of edges incident with in ; that is, . We denote the degree of by . The maximum and the minimum degrees of are denoted by and , respectively. A vertex of degree zero is called an isolated vertex. An edge incident with a vertex of degree one is called a pendant edge. The bondage number is an important parameter of graphs which is based upon the well-known domination number. A subset is called a dominating set of if ; that is, every vertex in has at least one neighbor in . The domination number of , denoted by , is the minimum cardinality among all dominating sets; that is, A dominating set is called a -set of if . The domination is such an important and classic conception that it has become one of the most widely studied topics in graph theory and also is frequently used to study property of networks. The domination, with many variations and generalizations, is now well studied in graph and networks theory. The early vast literature on domination includes the bibliography compiled by Hedetniemi and Laskar [2] and a thorough study of domination appears in the books by Haynes et al. [3, 4]. However, the problem determining the domination number for general graphs was early proved to be NP-complete (see GT2 in Appendix in

Abstract:
The bondage number of a nonempty graph $G$ is the cardinality of a smallest edge set whose removal from $G$ results in a graph with domination number greater than the domination number of $G$. This lecture gives a survey on the bondage number, including the known results, problems and conjectures. We also summarize other types of bondage numbers.

Abstract:
Using a multiphase transport model that includes both initial partonic and final hadronic scatterings, we have studied higher-order anisotropic flows as well as dihadron correlations as functions of pseudorapidity and azimuthal angular differences between trigger and associated particles in Pb-Pb collisions at $\sqrt{s_{NN}}=2.76$ TeV. With parameters in the model determined previously from fitting the measured multiplicity density of mid-pseudorapidity charged particles in central collisions and their elliptic flow in mid-central collisions, the calculated higher-order anisotropic flows from the two-particle cumulant method reproduce approximately those measured by the ALICE Collaboration, except at small centralities where they are slightly overestimated. Similar to experimental results, the two-dimensional dihadron correlations at most central collisions show a ridge structure at the near side and a broad structure at the away side. The short- and long-range dihadron azimuthal correlations, corresponding to small and large pseudorapidity differences, respectively, are studied for triggering particles with different transverse momenta and are found to be qualitatively consistent with experimental results from the CMS Collaboration. The relation between the short-range and long-range dihadron correlations with that induced by back-to-back jet pairs produced from initial hard collisions is also discussed.

Abstract:
By assuming that the isospin- and momentum-dependent MDI interaction has a form similar to the Gogny-like effective two-body interaction with a Yukawa finite-range term and the momentum dependence only originates from the finite-range exchange interaction, we determine its parameters by comparing the predicted potential energy density functional in uniform nuclear matter with what has been usually given and used extensively in transport models for studying isospin effects in intermediate-energy heavy-ion collisions as well as in investigating the properties of hot asymmetric nuclear matter and neutron star matter. We then use the density matrix expansion to derive from the resulting finite-range exchange interaction an effective Skyrme-like zero-range interaction with density-dependent parameters. As an application, we study the transition density and pressure at the inner edge of neutron star crusts using the stability conditions derived from the linearized Vlasov equation for the neutron star matter.

Abstract:
We have obtained a new set of parameters in a multiphase transport (AMPT) model that are able to describe both the charged particle multiplicity density and elliptic flow measured in Au+Au collisions at center of mass energy $\sqrt{s_{NN}}=200$ GeV at the Relativistic Heavy Ion Collider (RHIC), although they still give somewhat softer transverse momentum spectra. We then use the model to predict the triangular flow due to fluctuations in the initial collision geometry and study its effect relative to those from other harmonic components of anisotropic flows on the di-hadron azimuthal correlations in both central and mid-central collisions.

Abstract:
Using the AMPT model for relativistic heavy ion collisions, we have studied the di-hadron azimuthal angular correlations triggered by emitted jets in Au+Au collisions at center of mass energy $\sqrt{s_{NN}}=200$ GeV and impact parameter $b=8$ fm. A double-peak structure for the associated particles at the away side of trigger particles is obtained after subtracting background correlations due to the elliptic flow. Both the near-side peak and the away-side double peaks in the azimuthal angular correlations are, however, significantly suppressed (enhanced) in events of small (large) triangular flow, which are present as a result of fluctuations in the initial collision geometry. After subtraction of background correlations due to the triangular flow, the away-side double peaks change into a single peak with broad shoulders on both sides. The away side of the di-hadron correlations becomes essentially a single peak after further subtraction of higher-order flows.

Abstract:
The multiplicity and elliptic flow of charged particles produced in Pb-Pb collisions at center of mass energy $\sqrt{s_{NN}}=2.76$ TeV from the Large Hadron Collider are studied in a multiphase transport (AMPT) model. With the standard parameters in the HIJING model, which is used as initial conditions for subsequent partonic and hadronic scatterings in the AMPT model, the resulting multiplicity of final charged particles at mid-pseudorapidity is consistent with the experimental data measured by the ALICE Collaboration. This value is, however, increased by about 25% if the final-state partonic and hadronic scatterings are turned off. Because of final-state scatterings, particular those among partons, the final elliptic flow of charged hadrons is also consistent with the ALICE data if a smaller but more isotropic parton scattering cross section than previously used in the AMPT model for describing the charged hadron elliptic flow in heavy ion collisions at the Relativistic Heavy Ion Collider is used. The resulting transverse momentum spectra of charged particles as well as the centrality dependence of their multiplicity density and the elliptic flow are also in reasonable agreement with the ALICE data. Furthermore, the multiplicities, transverse momentum spectra and elliptic flows of identified hadrons such as protons, kaons and pions are predicted.

Abstract:
A routing $R$ of a given connected graph $G$ of order $n$ is a collection of $n(n-1)$ simple paths connecting every ordered pair of vertices of $G$. The vertex-forwarding index $\xi(G,R)$ of $G$ with respect to $R$ is defined as the maximum number of paths in $R$ passing through any vertex of $G$. The vertex-forwarding index $\xi(G)$ of $G$ is defined as the minimum $\xi(G,R)$ over all routing $R$'s of $G$. Similarly, the edge-forwarding index $ \pi(G,R)$ of $G$ with respect to $R$ is the maximum number of paths in $R$ passing through any edge of $G$. The edge-forwarding index $\pi(G)$ of $G$ is the minimum $\pi(G,R)$ over all routing $R$'s of $G$. The vertex-forwarding index or the edge-forwarding index corresponds to the maximum load of the graph. Therefore, it is important to find routings minimizing these indices and thus has received much research attention in the past ten years and more. In this paper we survey some known results on these forwarding indices, further research problems and several conjectures.

Abstract:
In the last two decades, 5-fluorouracil (5-FU) is widely used in clinical practice to inhibit the fibroblasts to proliferate and improve the success rate of glaucoma-filtering surgery, but 5-FU has many toxic effects to normal ocular tissues. The self-assembled peptide hydrogels may serve as a new class of biomaterials for applications including tissue engineering and drug delivery. How to deliver 5-FU quickly and precisely to the target sites of ocular tissue by a self-assembled peptide hydrogel remains unexplored. RGD (arginine-glycine-aspartic acid) sequence is cell attachment site in extracellular matrix (ECM). Thus, If the self-assembled peptide hydrogel containing the RGD sequence that act as a specific attachment site for the proliferated fibroblasts adhesion could be designed, after integrated 5-FU, a novel targeting drug delivery system will be put into practice in the future.