Edge Roman domination on graphs

An edge Roman dominating function of a graph $G$ is a function $f\colon E(G) \rightarrow \{0,1,2\}$ satisfying the condition that every edge $e$ with $f(e)=0$ is adjacent to some edge $e'$ with $f(e')=2$. The edge Roman domination number of $G$, denoted by $\gamma'_R(G)$, is the minimum weight $w(f) = \sum_{e\in E(G)} f(e)$ of an edge Roman dominating function $f$ of $G$. This paper disproves a conjecture of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad stating that if $G$ is a graph of maximum degree $\Delta$ on $n$ vertices, then $\gamma_R'(G) \le \lceil \frac{\Delta}{\Delta+1} n \rceil$. While the counterexamples having the edge Roman domination numbers $\frac{2\Delta-2}{2\Delta-1} n$, we prove that $\frac{2\Delta-2}{2\Delta-1} n + \frac{2}{2\Delta-1}$ is an upper bound for connected graphs. Furthermore, we provide an upper bound for the edge Roman domination number of $k$-degenerate graphs, which generalizes results of Akbari, Ehsani, Ghajar, Jalaly Khalilabadi and Sadeghian Sadeghabad. We also prove a sharp upper bound for subcubic graphs. In addition, we prove that the edge Roman domination numbers of planar graphs on $n$ vertices is at most $\frac{6}{7}n$, which confirms a conjecture of Akbari and Qajar. We also show an upper bound for graphs of girth at least five that is 2-cell embeddable in surfaces of small genus. Finally, we prove an upper bound for graphs that do not contain $K_{2,3}$ as a subdivision, which generalizes a result of Akbari and Qajar on outerplanar graphs

Predicting Stock Volatility Using After-Hours Information

We use realized volatilities based on after hours high frequency returns to predict next day volatility. We extend GARCH and long-memory forecasting models to include additional information: the whole night, the preopen, the postclose realized variance, and the overnight squared return. For four NASDAQ stocks (MSFT, AMGN, CSCO, and YHOO) we find that the inclusion of the preopen variance can improve the out-of-sample forecastability of the next day conditional day volatility. Additionally, we find that the postclose variance and the overnight squared return do not provide any predictive power for the next day conditional volatility. Our findings support the results of prior studies that traders trade for non-information reasons in the postclose period and trade for information reasons in the preopen period.

Locoregional therapy in luminal-like and HER2-enriched patients with de novo stage IV breast cancer

BACKGROUND: Locoregional therapy is rarely the standard of care for De Novo stage IV breast cancer but usually used for palliation of symptoms. This retrospective study aimed to determine whether surgery or radiation would contribute to survival benefit for this group of patients by examining the survival outcome through the disease molecular subtypes. MATERIALS AND METHODS: We reviewed 246 patients with de novo stage IV (M1) breast cancer treated at our hospital between 1990 and 2009. Multivariable Cox Analysis was used to evaluate the survival association with subtypes and clinicopathologic factors. RESULTS: Patients with luminal-like subtype are mostly premonopausal (66.9%, P = 0.0002), with abnormal CA 15–3 level at initial diagnosis (58.7%, P = 0.01), a higher rate of bony metastases (78.5%, P = 0.02), and a lower rate of liver metastases (22.3%, P < 0.0001). Patients with HER2-enriched and triple negative showed higher rate of nuclear grade III, up to 35% and 40%, respectively (P = 0.01). There is no difference in treatment options patient received: systemic chemotherapy up to 82.2 ~ 95% (p = 0.0705), locoregional treatment up to 40.0 ~ 51.2% (P-0.2571). The median overall survival was 23.1 months: luminal-like subtype 39.6 months, HER2-enriched subtype 17.9 months, and triple negative subtype 13.3 months, respectively (P < 0.0001). In multivariate analysis, poor prognostic factors included HER2-enriched (HR 2.2, P < 0.0001) and triple negative subtype (HR 4.3, P < 0.0001), liver metastasis (HR 1.9, P < 0.0001), lung metastasis (HR 1.4, P = 0.0153), and bone metastasis (HR 1.8, P = 0.0007). Subgroup analysis revealed that local treatments (surgery or radiotherapy) to primary/regional tumors achieved better survival in patients with luminal-like (3-year survival 66.4% vs. 34.4%, p = 0.0001) and HER2-enriched (3-year survival 41.6% vs. 8.8%, p = 0.0012) subtypes, but not in triple negative subtype (P = 0.9575). CONCLUSIONS: For better survival outcome, De Novo Stage IV breast cancer patients with luminal-like or HER2-enriched subtype should be offered local treatments when surgery and/or radiotherapy presents an option for proper control of the primary and regional tumors