Strong equality of Roman and perfect Roman domination in trees

A Roman dominating function (RD-function) on a graph $G = (V, E)$ is a function $f: V \longrightarrow \{0, 1, 2\}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = 2$. An Roman dominating function $f$ in a graph $G$ is perfect Roman dominating function (PRD-function) if every vertex $u$ with $f(u) = 0$ is adjacent to exactly one vertex $v$ for which $f(v) = 2$. The (perfect) Roman domination number $\gamma_R(G)$ ($\gamma_{R}^{p}(G)$) is the minimum weight of an (perfect) Roman dominating function on $G$. We say that $\gamma_{R}^{p}(G)$ strongly equals $\gamma_R(G)$, denoted by $\gamma_{R}^{p}(G)\equiv \gamma_R(G)$, if every RD-function on $G$ of minimum weight is a PRD-function. In this paper we show that for a given graph $G$, it is NP-hard to decide whether $\gamma_{R}^{p}(G)= \gamma_R(G)$ and also we provide a constructive characterization of trees $T$ with $\gamma_{R}^{p}(T)\equiv \gamma_R(T)$.

An Enhanced Packet Transmission Framework for Handling Real Time Burst Traffic in Wireless Sensor Networks

Congestion control for real time traffic is an important network measure to be handled in case of repeated event triggers, continuous packet re-transmissions, node interference, node deaths and node failures in Wireless Sensor Networks (WSNs). Network modelling for transmission of packets from source node to sink using probabilistic M/Pareto and Poisson processes have been examined in the past. The existing methodologies are deficit in designing a queuing framework considering other network parameters such as energy consumption and delay for alleviating congestion and thereby efficiently routing packets to sink by reducing packet drops. To overcome this fall back, a Minimum Weight Estimation for Mitigating Congestion during Real Time Burst Traffic (MWCBT) framework is proposed. This gives a precautionary solution against heavy traffic occupancy among the interim and sink-neighbouring nodes in WSNs is proposed. Routing of packets using a congestion-free path is required to increase the node lifespan. An optimal M/Pareto stochastic traffic generator is used in combination with traffic factors such as energy and delay to predict amount of traffic across nodes. A simpler congestion prediction mechanism is performed to control the occurrence of heavy-tailed traffic distributions. A torrent weight value for incoming traffic is generated at each node periodically that directs routing of data packets to sink. The devised MWCBT framework supervises real-time traffic congestion and is found to be more optimal than the existing approaches for network traffic modelling. The proposed approach achieves greater packet delivery ratio and less node congestion compared to the existing network modelling techniques.

On the double Roman bondage numbers of graphs

For a graph [Formula: see text], a double Roman dominating function (DRDF) is a function [Formula: see text] having the property that if [Formula: see text] for some vertex [Formula: see text], then [Formula: see text] has at least two neighbors assigned [Formula: see text] under [Formula: see text] or one neighbor [Formula: see text] with [Formula: see text], and if [Formula: see text] then [Formula: see text] has at least one neighbor [Formula: see text] with [Formula: see text]. The weight of a DRDF [Formula: see text] is the sum [Formula: see text]. The minimum weight of a DRDF on a graph [Formula: see text] is the double Roman domination number of [Formula: see text] and is denoted by [Formula: see text]. The double Roman bondage number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality among all edge subsets [Formula: see text] such that [Formula: see text]. In this paper, we study the double Roman bondage number in graphs. We determine the double Roman bondage number in several families of graphs, and present several bounds for the double Roman bondage number. We also study the complexity issue of the double Roman bondage number and prove that the decision problem for the double Roman bondage number is NP-hard even when restricted to bipartite graphs.

Fundamental thresholds of realistic quantum error correction circuits from classical spin models

Mapping the decoding of quantum error correcting (QEC) codes to classical disordered statistical mechanics models allows one to determine critical error thresholds of QEC codes under phenomenological noise models. Here, we extend this mapping to admit realistic, multi-parameter noise models of faulty QEC circuits, derive the associated strongly correlated classical spin models, and illustrate this approach for a quantum repetition code with faulty stabilizer readout circuits. We use Monte-Carlo simulations to study the resulting phase diagram and benchmark our results against a minimum-weight perfect matching decoder. The presented method provides an avenue to assess fundamental thresholds of QEC circuits, independent of specific decoding strategies, and can thereby help guiding the development of near-term QEC hardware.

On the Double Roman Domination in Generalized Petersen Graphs P(5k,k)

A double Roman dominating function on a graph G=(V,E) is a function f:V→{0,1,2,3} satisfying the condition that every vertex u for which f(u)=0 is adjacent to at least one vertex assigned 3 or at least two vertices assigned 2, and every vertex u with f(u)=1 is adjacent to at least one vertex assigned 2 or 3. The weight of f equals w(f)=∑v∈Vf(v). The double Roman domination number γdR(G) of a graph G equals the minimum weight of a double Roman dominating function of G. We obtain closed expressions for the double Roman domination number of generalized Petersen graphs P(5k,k). It is proven that γdR(P(5k,k))=8k for k≡2,3mod5 and 8k≤γdR(P(5k,k))≤8k+2 for k≡0,1,4mod5. We also improve the upper bounds for generalized Petersen graphs P(20k,k).

Efficacy of Liraglutide 3.0 Mg Treatment on Weight-Loss in Patients with Weight Regain After Bariatric Surgery

Purpose Bariatric surgery, as Roux-en-Y gastric bypass (RYGB), laparoscopic gastric banding (LGB) and laparoscopic sleeve gastrectomy (LSG), is considered the gold standard treatment to achieve long-term weight-loss in severe obesity. In patients who fail to maintain the achieved weight, pharmacological treatment may be required. Here, we reported our real-life experience on efficacy of liraglutide therapy in 62 patients who regained weight after bariatric surgery.MethodsWe retrospectively evaluated 62 (60 F-2 M; mean age: 43.6±9.9 years) patients received liraglutide for weight-loss after bariatric surgery (17 RYGB, 22 LGB and 23 LSG). Body mass index (BMI) before and after surgery was respectively of 45.4±5.5 kg/m2 and 29.5±4.9 kg/m2. Patients were followed from 2016 until 2021. Liraglutide was administered after weight regain once-daily subcutaneously at starting dose of 0.6 mg and with weekly increases up to 3.0 mg. Treatments were administered when a weight regain of 10-15% occurred after reaching a minimum weight-loss from bariatric surgery or if weight-loss after bariatric surgery was unsatisfactory. ResultsAfter a mean of 70.7±43.7 months from any bariatric surgery, all patients started liraglutide therapy. At this time, mean BMI was 34.2±4.8 kg/m2 (mean increased BMI: 4.7±2.8 kg/m2). After a mean of 10.5±4.4 months from the beginning of liraglutide, 9 patients achieved normal weight (BMI 24.1±0.9 kg/m2), 28 were overweight (BMI 26.9±1.6 kg/m2). Twenty patients achieved grade I (BMI 32.1±1.5 kg/m2), 5 grade II (BMI 37.3±2.0 kg/m2) obesity, none had grade III obesity (mean BMI change: -5.1±2.5 kg/m2). The treatment was well tolerated, and no serious adverse events were recorded.ConclusionThese data confirm the efficacy and safety of liraglutide in patients who experienced weight regain after bariatric surgery. Considering the long-term follow-up, patients should be followed up regularly and the pharmacological treatment should be adapted to the weight fluctuations observed during the clinical history.

Synergistic effects of halloysite nanotubes with metal and phosphorus additives on the optimal design of eco-friendly sandwich panels with maximum flame resistance and minimum weight

The present work addresses the optimal design of sandwich panels made of flax fabric (FF)/vinyl ester (VE) composite face sheets and honeycomb VE core. The sandwich structures are first optimized in terms of flammability by obtaining the best combination of ammonium polyphosphate (APP), halloysite nanotube (HNT), and magnesium hydroxide (MH) as three flame retardants (FRs). Using the Taguchi method and horizontal burning test, it is shown that [6, 3, and 3%] and [1, 0.5, and 0%] are the optimal combinations of APP, HNT, and MH for the face sheets and core, respectively. Cone calorimeter test results indicate that the optimal FR combinations significantly decrease the mass lost rate (MLR), heat rate release (HRR), total smoke release (TSR), and maximum average release heat emission (MARHE). The FR sandwich structures are then geometrically optimized under compressive loads based on their weight. Different failure modes are considered as the design constraints of the optimization problem. Imperialist competitive algorithm (ICA), as a powerful meta-heuristic algorithm, is implemented to considerably reduce the computational cost of the optimization process. The results of this study show that proper combinations of FR additives can increase the flame retardancy while decreasing the weight of sandwich panels.

Estimating the Potential of Carbon Sequestration in Tree Species of Chintapalle Forest Range, Narsipatnam Division, Visakhapatnam, Andhra Pradesh, India

The potential of carbon sequestration of tree species in the Chintapalle forest range, of Narsipatnam Division, was estimated by using a non-destructive method. The sequestration of 6033 trees belonging to 22 species was investigated; the approximate height of tree species and the diameter at breast height (DBH) were measured for the estimation of CO2 sequestration. The maximum weight of carbon was observed in Pongamia pinnata (L.) Pierre species i.e (37987.06 kg) and the minimum weight of carbon was noted in Phyllanthus emblica L. species i.e is (61.8kg). The total carbon sequestrated by the entire tree species was (2370614.0 kg), The average carbon sequestered was (39865.81 kg). The highest sequestration was noted in the species P. pinnata (L.) Pierre i.e. (139271.95 kg) and the lowest (226.79 kg) was noted in the species P. emblica L. The maximum average DBH with maximum carbon sequestration potential was observed in Ficus benghalensis L. species, with higher total green (AGW) observed in all sites, whereas minimum average DBH with minimum carbon sequestration potential was noted in Bambusa vulgaris species. The regression analysis tests the relationship between two variables. The height of trees has no significant impact on the amount of CO2 sequestered F (32085087175.84, 12946607900) = 2.478262; P ? 0.05, which indicates that the tree height plays an insignificant role in CO2 sequestration (? = 2713.28 P ? 0.05). The dependent variable CO2 sequestered was also regressed on the predictor variable soil organic carbon (SOC) to test the relationship. SOC insignificantly predicted CO2 sequestrated F (5.83, 2.62) = 0.2236; P ? 0.25, indicating that the SOC has an insignificant role in CO2 sequestration (? = 102780.3 P ? 0.05). Insignificant relation was observed between the parameters SOC and height of tree species to the rate of carbon dioxide sequestered, and gave a regression equation of y = 10278x + 50863 with R2 = 0.100; y=2713.285803x-209800.8762 with R2 = 0.55