On the spectral moments of trees with a given bipartition

For two given positive integers $p$ and $q$ with $p\leqslant q$, we denote \mathscr{T}_n^{p, q}={T: T is a tree of order $n$ with a $(p, q)$-bipartition}. For a graph $G$ with $n$ vertices, let $A(G)$ be its adjacency matrix with eigenvalues $\lambda_1(G), \lambda_2(G), ..., \lambda_n(G)$ in non-increasing order. The number $S_k(G):=\sum_{i=1}^{n}\lambda_i^k(G)\,(k=0, 1, ..., n-1)$ is called the $k$th spectral moment of $G$. Let $S(G)=(S_0(G), S_1(G),..., S_{n-1}(G))$ be the sequence of spectral moments of $G$. For two graphs $G_1$ and $G_2$, one has $G_1\prec_s G_2$ if for some $k\in {1,2,...,n-1}$, $S_i(G_1)=S_i(G_2) (i=0,1,...,k-1)$ and $S_k(G_1)<S_k(G_2)$ holds. In this paper, the last four trees, in the $S$-order, among $\mathscr{T}_n^{p, q} (4\leqslant p\leqslant q)$ are characterized.Comment: 11 pages, 7 figure

TransModel: An R Package for Linear Transformation Model with Censored Data

Linear transformation models, including the proportional hazards model and proportional odds model, under right censoring were discussed by Chen, Jin, and Ying (2002). The asymptotic variance of the estimator they proposed has a closed form and can be obtained easily by plug-in rules, which improves the computational efficiency. We develop an R package TransModel based on Chen's approach. The detailed usage of the package is discussed, and the function is applied to the Veterans' Administration lung cancer data

spBayesSurv: Fitting Bayesian Spatial Survival Models Using R

Spatial survival analysis has received a great deal of attention over the last 20 years due to the important role that geographical information can play in predicting survival. This paper provides an introduction to a set of programs for implementing some Bayesian spatial survival models in R using the package spBayesSurv. The function survregbayes includes the three most commonly-used semiparametric models: proportional hazards, proportional odds, and accelerated failure time. All manner of censored survival times are simultaneously accommodated including uncensored, interval censored, current-status, left and right censored, and mixtures of these. Left-truncated data are also accommodated. Time-dependent covariates are allowed under the piecewise constant assumption. Both georeferenced and areally observed spatial locations are handled via frailties. Model fit is assessed with conditional Cox-Snell residual plots, and model choice is carried out via the log pseudo marginal likelihood, the deviance information criterion and the WatanabeAkaike information criterion. The accelerated failure time frailty model with a covariatedependent baseline is included in the function frailtyGAFT. In addition, the package also provides two marginal survival models: proportional hazards and linear dependent Dirichlet process mixtures, where the spatial dependence is modeled via spatial copulas. Note that the package can also handle non-spatial data using non-spatial versions of the aforementioned models

General Trinajstić Index

In memory of the outstanding theoretical chemist Nenad Trinajstić, Furtula introduced a new distance-based molecular structure descriptor "Trinajstić index" in chemical graph theory. In this paper, we propose the general Trinajstić index, and give the calculation formula of the general Trinajstić index for double-star graphs, double brooms, Kragujevac trees, firefly graphs and wheel graphs. As an application, we calculate the general Trinajstić index for some hydrocarbons