Abstract:
We consider a random model for directed graphs whereby an arc is placed from one vertex to another with a prescribed probability which may vary from arc to arc. Using perturbation bounds as well as Chernoff inequalities, we show that the stationary distribution of a Markov process on a random graph is concentrated near that of the "expected" process under mild conditions. These conditions involve the ratio between the minimum and maximum in- and out-degrees, the ratio of the minimum and maximum entry in the stationary distribution, and the smallest singu- lar value of the transition matrix. Lastly, we give examples of applications of our results to well-known models such as PageRank and G(n, p).

Abstract:
The Cheeger constant, $h_G$, is a measure of expansion within a graph. The classical Cheeger Inequality states: $\lambda_{1}/2 \le h_G \le \sqrt{2 \lambda_{1}}$ where $\lambda_1$ is the first nontrivial eigenvalue of the normalized Laplacian matrix. Hence, $h_G$ is tightly controlled by $\lambda_1$ to within a quadratic factor. We give an alternative Cheeger Inequality where we consider the $\infty$-norm of the corresponding eigenvector in addition to $\lambda_1$. This inequality controls $h_G$ to within a linear factor of $\lambda_1$ thereby providing an improvement to the previous quadratic bounds. An additional advantage of our result is that while the original Cheeger constant makes it clear that $h_G \to 0$ as $\lambda_1 \to 0$, our result shows that $h_G \to 1/2$ as $\lambda_1 \to 1$.

Abstract:
Hoffman proved that for a simple graph $G$, the chromatic number $\chi(G)$ obeys $\chi(G) \le 1 - \frac{\lambda_1}{\lambda_{n}}$ where $\lambda_1$ and $\lambda_n$ are the maximal and minimal eigenvalues of the adjacency matrix of $G$ respectively. Lov\'asz later showed that $\chi(G) \le 1 - \frac{\lambda_1}{\lambda_{n}}$ for any (perhaps negatively) weighted adjacency matrix. In this paper, we give a probabilistic proof of Lov\'asz's theorem, then extend the technique to derive generalizations of Hoffman's theorem when allowed a certain proportion of edge-conflicts. Using this result, we show that if a 3-uniform hypergraph is 2-colorable, then $\bar d \le -\frac{3}{2}\lambda_{\min}$ where $\bar d$ is the average degree and $\lambda_{\min}$ is the minimal eigenvalue of the underlying graph. We generalize this further for $k$-uniform hypergraphs, for the cases $k=4$ and $5$, by considering several variants of the underlying graph.

Abstract:
A singularly perturbed linear system of second order partial differential equations of parabolic reaction-diffusion type with given initial and boundary conditions is considered. The leading term of each equation is multiplied by a small positive parameter. These singular perturbation parameters are assumed to be distinct. The components of the solution exhibit overlapping layers. Shishkin piecewise-uniform meshes are introduced, which are used in conjunction with a classical finite difference discretisation, to construct a numerical method for solving this problem. It is proved that the numerical approximations obtained with this method are first order convergent in time and essentially second order convergent in the space variable uniformly with respect to all of the parameters.

Abstract:
We present a counterexample to a lower bound for power domination number given in Liao, Power domination with bounded time constraints, J. Comb. Optim., in press 2014. We also define the power propagation time and make connections between the power domination propagation ideas in Liao and the (zero forcing) propagation time in Hogben et al, Propagation time for zero forcing on a graph, Discrete Appl. Math., 2012.

Abstract:
Objective: To describe a 2010 outbreak of nine cases of measles in Australia possibly linked to an index case who travelled on an international flight from South Africa while infectious.Methods: Three Australian state health departments, Victoria, Queensland and New South Wales, were responsible for the investigation and management of this outbreak, following Australian public health guidelines.Results: An outbreak of measles occurred in Australia after an infectious case arrived on a 12-hour flight from South Africa. Only one of four cases in the first generation exposed to the index case en route was sitting within the two rows recommended for contact tracing in Australian and other guidelines. The remaining four cases in subsequent generations, including two health care workers, were acquired in health care settings. Seven cases were young adults. Delays in diagnosis and notification hampered disease control and contact tracing efforts.Conclusion: Review of current contact tracing guidelines following in-flight exposure to an infectious measles case is required. Alternative strategies could include expanding routine contact tracing beyond the two rows on either side of the case’s row or expansion on a case-by-case basis depending on cabin layout and case and contact movements in flight. Releasing information about the incident by press release or providing generic information to everyone on the flight using e-mail or text messaging information obtained from the relevant airline, may also be worthy of consideration. Disease importation, inadequately vaccinated young adults and health care-related transmission remain challenges for measles control in an elimination era.

Abstract:
Complications related to cholecystectomy are well described. Most occur in the early postoperative period and are recognised either at the time of, or shortly after surgery. Clinical sequelae occurring years following cholecystectomy are rare and infrequently reported. In addition, most delayed complications are related to the continuing presence or new formation of gallstones. In this paper we present a unique case of an abscess of the common bile duct wall, presenting with painless obstructive jaundice more than 30 years following an open cholecystectomy, without the presence of gallstones. The clinical presentation, investigations, and treatment are discussed with a review of other relevant reported cases in the literature.

Abstract:
The principal permanent rank characteristic sequence is a binary sequence $r_0 r_1 \ldots r_n$ where $r_k = 1$ if there exists a principal square submatrix of size $k$ with nonzero permanent and $r_k = 0$ otherwise, and $r_0 = 1$ if there is a zero diagonal entry. A characterization is provided for all principal permanent rank sequences obtainable by the family of nonnegative matrices as well as the family of nonnegative symmetric matrices. Constructions for all realizable sequences are provided. Results for skew-symmetric matrices are also included.

Abstract:
We establish a conjecture of Graham and Lov\'asz that the (normalized) coefficients of the distance characteristic polynomial of a tree are unimodal; we also prove they are log-concave.