Degree Square Research Articles

Optimizing distance constraints frequency assignment with relaxation

In a typical wireless telecommunication network, a large number of communication links is established with a limited number of available frequencies. The problem that addresses assigning available frequencies to transmitters such that interference is avoided as far as possible is called the frequency assignment problem. The problem is usually modeled as a graph coloring (labeling) problem. We study in this paper the (s, t)-relaxed L(2, 1)-labeling of a graph which considers the situation where transceivers that are very close receive frequencies that differ by at least two while transceivers that are close receive frequencies that differ by at least one. In addition, the model allows at most s (resp. t) anomalies at distance one (resp. two). The objective of the model is to minimize the span of frequencies in a corresponding network. We show that it is NP-complete to decide whether the minimal span of a (1, 0)-relaxed L(2, 1)-labeling of a graph is at most k. We also prove that the minimal span of this labeling for two classes of graphs is bounded above with the the square of the largest degree in the graph of interest. These results confirm Conjecture 6 and partially confirm Conjecture 3 stated in Lin [J. Comb. Optim. 31 (2016) 1–22].

Complexity of some graphs generated by square

Complexity plays a vital and significant role when designing communication networks (graphs). The more quality and perfect the network, the greater the number of trees spanning this network, which leads to greater possibilities of connection between two vertices, and this ensures good rigidity and resistance. In this work, we present nine network designs created by a square of different average degree 4, 6 and 8, then we deduce a simpler and evident formula expressing the number of spanning trees of these networks using some basic properties of orthogonal polynomials, block matrix analysis technique, and recurrence relations. In addition, we compute the entropy of each network and determine the best by comparing these designs using network entropy. Finally, we compare the entropy of spanning trees on our networks with other triangle and Apollonian networks and observe the entropy of our networks, which is the highest among the triangle and Apollonian networks studied.

Computing Forgotten Topological Index of Extremal Cactus Chains

Abstract The F-index is the whole of 3D squares of vertex degrees in a chart G. It as of late turned into the subject of a few investigates because of its extraordinary capability of uses. The point of this paper is to register the F-record of triangular prickly plant chain, square desert flora chains, 6-sided cactus restraints and polyomino restraints. In addition, we decided the extremal chains in the desert plant chains and polyomino fastens as for the F-index.

MASTER Optical Observation of LIGO/VIRGO S200302c Event

Abstract The results of LIGO/Virgo S200302c (O3) error-box optical inspection by MASTER are presented. We observed 4242 square degrees square degrees inside 3σ error box during 1 month. We present MASTER OT J141401.39–483305.7, MASTER OT J215856.95–392909.6, MASTER OT J061642.05+435617.9, MASTER OT J052817.95+672801.4—optical transients found by the MASTER auto-detection system during this inspection as the result of a survey of large error-fields, indicating the efficiency of MASTER observational strategy.

Do Centres of Endemism provide a spatial context for predicting and preserving plant phylogeographic patterns in the Cape Floristic Region, South Africa?

AimThe evolutionary forces that gave rise to the exceptional plant species richness of the Cape Floristic Region (CFR) have also likely played a role at the intraspecific level (i.e. plant populations)—and thereby generating shared phylogeographic patterns among taxa. Here we test whether plant populations in the CFR exhibit phylogeographic breaks across the boundaries between Centres of Endemism (CoEs). The boundaries between CoEs (derived from the distribution ranges of endemic taxa and currently mapped at a coarse, Quarter Degree Square scale) represent a spatial proxy for the evolutionary diversifying drivers acting on plant taxa in the CFR.LocationThe CFR, located along the southern Cape of South Africa.MethodsPublished phylogeographic literature were compiled and spatial patterns of genetic divergence re-analysed to assess the frequency at which CFR plant taxa exhibit phylogeographic breaks either (1) across or (2) within CoE boundaries. Population pairs from each study were compared across and within CoEs and scored as either exhibiting a phylogeographic break or not.ResultsPhylogeographic breaks in Cape plants were found to occur across the boundaries of CoEs more often than not. Significantly more population pairs exhibited phylogeographic breaks across CoE boundaries (506 of the 540, χ2 = 886, p < 0.001) and fewer breaks within CoEs (94 of 619, χ2 = 300, p < 0.001) than would be expected if there was equal probability of a genetic break occurring across CoE boundaries.Main conclusionsThe evolutionary forces that have produced and maintained the exceptional plant diversity in the CFR appear to have operated at the population level, producing similar patterns of phylogeographic structuring of plant lineages regardless of life history or taxonomy. This tendency for Cape plants to exhibit shared patterns of spatially structured genetic diversity that match the distribution of endemic taxa may assist CFR phylogeographers to streamline sampling efforts and test novel hypotheses pertaining to the distribution of genetic diversity among CFR plant taxa. Additionally, the resolution at which CoEs are mapped should be refined, which may provide a valuable tool for future conservation planning and the development of precautionary guidelines for the translocation of genetic material during species reintroductions and commercial cultivation of Cape endemic crops. Thus, to answer the question ‘Do Centres of Endemism provide a spatial context for predicting and preserving plant phylogeographic patterns in the Cape Floristic Region, South Africa?’—yes, CoEs do appear to be an important tool for Cape phylogeographers. However, the data is limited and more plant phylogeography work is needed in the CFR.

P‐70: Elimination of Double Images and Stray Light of a Light Field Virtual Reality Near‐to‐Eye Display

A light field near‐to‐eye virtual reality display is proposed. The field of view is 86 degrees square and the PPD is 13.3. To eliminate the double images and stray light, a collimated backlight and a black is adopted. The ANSI contrast is enhanced

Extremal multiplicative Zagreb indices among trees with given distance k-domination number

The first multiplicative Zagreb index [Formula: see text] of a graph [Formula: see text] is the product of the square of every vertex degree, while the second multiplicative Zagreb index [Formula: see text] is the product of the products of degrees of pairs of adjacent vertices. In this paper, we give sharp lower bound for [Formula: see text] and upper bound for [Formula: see text] of trees with given distance [Formula: see text]-domination number, and characterize those trees attaining the bounds.

GRANDMA observations of advanced LIGO’s and advanced Virgo’s third observational campaign

ABSTRACT GRANDMA (Global Rapid Advanced Network Devoted to the Multi-messenger Addicts) is a network of 25 telescopes of different sizes, including both photometric and spectroscopic facilities. The network aims to coordinate follow-up observations of gravitational-wave (GW) candidate alerts, especially those with large localization uncertainties, to reduce the delay between the initial detection and the optical confirmation. In this paper, we detail GRANDMA’s observational performance during Advanced LIGO/Advanced Virgo Observing Run 3 (O3), focusing on the second part of O3; this includes summary statistics pertaining to coverage and possible astrophysical origin of the candidates. To do so, we quantify our observation efficiency in terms of delay between GW candidate trigger time, observations, and the total coverage. Using an optimized and robust coordination system, GRANDMA followed-up about 90 per cent of the GW candidate alerts, that is 49 out of 56 candidates. This led to coverage of over 9000 deg2 during O3. The delay between the GW candidate trigger and the first observation was below 1.5 h for 50 per cent of the alerts. We did not detect any electromagnetic counterparts to the GW candidates during O3, likely due to the very large localization areas (on average thousands of degrees squares) and relatively large distance of the candidates (above 200 Mpc for 60 per cent of binary neutron star, BNS candidates). We derive constraints on potential kilonova properties for two potential BNS coalescences (GW190425 and S200213t), assuming that the events’ locations were imaged.

Métodos geoestatísticos e determinísticos na espacialização da altura dominante em povoamento de eucalipto

The dominant height of eucalyptus can present strong spatial autocorrelation, being indispensable to investigate the interpolation methods in its spatialization. In this context, this study aimed to evaluate deterministic and geostatistical methods in estimating the dominant height in eucalyptus stands. In 14 management units of the genus (377 hectares), 233 sample units were allocated, obtaining the variables dominant height and diameter at 1.30 m of the soil. The spatial interpolators evaluated were geostatistics (kriging and ordinary cokriging) and deterministic (inverse of the distance square and global polynomial of the 10th degree). Furthermore, indicator kriging was applied to spatialize the probability of locations with above-average dominant height values. The results showed that the dominant eucalyptus height presents with a strong spatial dependence, being possible its spatialization by both ordinary kriging and cokriging. In the presence of the spatial component, deterministic methods are not recommended, especially the global polynomial models. Indicator kriging enables the mapping of the probability of a given class of productivity to occur, subsidizing forest rotation planning.

Degree square sum equienergetic and hyperenergetic graphs

Degree square sum equienergetic and hyperenergetic graphs

A bound on the spectral radius of graphs in terms of their Zagreb indices

The first and the second Zagreb index of a graph, usually defined as the sum of the squares of degrees over all vertices and the sum of the products of degrees of edge endvertices over all edges, respectively, are tightly related to the numbers of walks of length two and three in the graph. We provide here a lower bound on the spectral radius of adjacency matrix A of graph in terms of its Zagreb indices, based on the properties of the least square approximation of the vector A2j with the vectors Aj and j, where j is the all-one vector. The bound is sharp for all graphs with two main eigenvalues, surpassing the range of sharpness of other bounds among connected graphs.

A note on larger orbit size

Abstract In this note, we present an improvement on the large orbit result of Halasi and Podoski, and then answer an open question raised in [X. Chen, J. P. Cossey, M. Lewis and H. P. Tong-Viet, Blocks of small defect in alternating groups and squares of Brauer character degrees, J. Group Theory 20 2017, 6, 1155–1173].

Some conjectures on Brauer character degrees

Let G be a finite group and let p be a prime. Willems conjectured that the sum of the squares of all Brauer character degrees of a finite group G is bounded below by the p′-part of the order of G. In this note, we propose a ‘projective’ version of this conjecture and prove it for p-solvable groups. Using this, we manage to reduce the proof of Willems conjecture to a question concerning quasi-simple groups. We also introduce a projective version of a conjecture due to Martínez-Pérez and Willems on projective indecomposable characters which, if true, implies the veracity of the projective version of the Willems conjecture above.

On certain topological indices of signed graphs

The first Zagreb index of a graph $G$ is the sum of squares of the vertex degrees in a graph and the second Zagreb index of $G$ is the sum of products of degrees of adjacent vertices in $G$. The imbalance of an edge in $G$ is the numerical difference of degrees of its end vertices and the irregularity of $G$ is the sum of imbalances of all its edges. In this paper, we extend the concepts of these topological indices for signed graphs and discuss the corresponding results on signed graphs.

On certain topological indices of signed graphs

The first Zagreb index of a graph G is the sum of squares of the vertex degrees in a graph and the second Zagreb index of G is the sum of products of degrees of adjacent vertices in G. The imbalance of an edge in G is the numerical difference of degrees of its end vertices and the irregularity of G is the sum of imbalances of all its edges. In this paper, we extend the concepts of these topological indices for signed graphs and discuss the corresponding results on signed graphs.

P5651Association of left atrial fibrosis extent with left ventricular geometric remodeling and diastolic dysfunction in patients with paroxysmal atrial fibrillation

Abstract Introduction Whether left atrial fibrosis (LAf) in patients with atrial fibrillation (AF) is a consequence of left ventricular (LV) diastolic dysfunction or primary atrial pathology continues to be a debatable issue. Electroanatomical mapping (EAM) allows to image and to define LAf as a substrate of AF. Purpose To study the relationship of LAf extent with LV diastolic function and geometric remodeling in patients (pts) with paroxysmal AF. Methods 56 pts with paroxysmal AF (mean age 57.1±8.4 years, 31 males), undergone catheter ablation, were enrolled in the study, including 30 pts with arterial hypertension (AH), 15 – with coronary artery disease (CAD) and AH. Comprehensive transthoracic echocardiography was carried out in all pts to assess chamber volumes, systolic and LV diastolic functions and geometry patterns according to Recommendations of ASE and EACVI. Before ablation, EAM was performed in sinus rhythm. The bipolar low voltage areas of LAf were identified with the cut-off &lt;0.5 mV. For the LAf quantification following indicators were calculated: total square of LAf (Sf, cm2) and LAf degree, estimated as an analog of the UTAH staging system, by selection of UTAH I: &lt;5% fibrosis; II: 5–19%; III: 20–35% and IV: &gt;35%. Results All patients had preserved systolic LV function. To assess the influence of LV geometry on LAf extent all pts were distributed in accordance to LV geometry patterns (p): normal geometry (pI) – 27 pts, concentric remodeling (pII) – 13, eccentric hypertrophy (pIII) – 10, concentric hypertrophy (pIV) – 6. Pts with pIII were older than pI pts: 60.8±6.4 vs 53.9±10.4 (p=0.048). All pts with pIII and pIV had AH. From 11 pts without AH, 10 had pI of LV geometry. PIII was revealed more often in CAD pts compared to those without CAD: 29.2 vs 10.5% (p=0.04). PIII pts had bigger LA volume compared to pI pts (74.3±22.5 vs 58.8±19.4 ml, p=0.019) and pII pts (61.9±14.9, p=0.05), but LA volume of pIII pts didn't differ from pIV pts (71.9±14.5, p=0.78). PIII pts had more extent Sf than pI pts (28.32±8.9 vs 13.4±6.5, p=0.05), while Sf of pII (17.3±8.7, p=0.495) and pIV pts (16.4±9.5, p=0.699) didn't differ significantly from Sf of pI pts. As for the degree of LAf, UTAH I was absent in pts with pIII and UTAH IV was revealed in 40% of these pts, while in pts with pI UTAH I was in 26% and UTAH IV - in 14.8% (p=0.049). However, Sf and UTAH degree did not depend on age, CAD and heart failure presence. As for diastolic dysfunction, in pIII and pIV pts e∼septal and e∼lateral were lower compared to pI pts: 6.3±1.9, 5.5±2.4 vs 8.5±2.2 (p&lt;0.01) and 8.2±2.7, 8.0±3.8 vs 11.3±2.9 (p&lt;0.01), respectively, while E/e∼ in pIII pts didn't differ from pI pts (8.0±1.6 vs 7.2±1.6, p=0.17), but in pIV was more than in pI pts (10.4±2.8, p=0.003). Conclusion LAf extent in paroxysmal AF is associated more with such LV geometry pattern as eccentric hypertrophy, than with diastolic disorders, which accompany both eccentric and concentric hypertrophy.

Sharp bounds on the reduced second Zagreb index of graphs with given number of cut vertices

The first Zagreb index M1 of a graph is defined as the sum of the square of every vertex degree, and the second Zagreb index M2 of a graph is defined as the sum of the product of vertex degrees of each pair of adjacent vertices. In Furtula et al. (2014) Furtula, Gutman and Ediz first proposed the reduced second Zagreb index as RM2(G)=∑uv∈E(G)(d(u)−1)(d(v)−1)=ΔM(G)+m(G),where ΔM(G) is the difference between M2 and M1 of graph G and m(G) is the size of G. In this paper, some graph transformations are introduced. We study the monotonicity of RM2(G) with respect to these graph transformations. Then based on these properties, we present sharp upper and lower bounds on RM2(G) of connected n-vertex cyclic graphs with given number of cut vertices. All the corresponding extremal graphs are identified.

Runge–Kutta–Gegenbauer explicit methods for advection-diffusion problems

In this paper, Runge–Kutta–Gegenbauer ( RKG) stability polynomials of arbitrarily high order of accuracy are introduced in closed form. The stability domain of RKG polynomials extends in the real direction with the square of polynomial degree, and in the imaginary direction as an increasing function of Gegenbauer parameter. Consequently, the polynomials are naturally suited to the construction of high order stabilized Runge–Kutta ( SRK) explicit methods for systems of PDEs of mixed hyperbolic-parabolic type.We present SRK methods composed of L ordered forward Euler stages, with complex-valued stepsizes derived from the roots of RKG stability polynomials of degree L. Internal stability is maintained at large stage number through an ordering algorithm which limits internal amplification factors to 10L2. Test results for mildly stiff nonlinear advection-diffusion-reaction problems with moderate (≲1) mesh Péclet numbers are provided at second, fourth, and sixth orders, with nonlinear reaction terms treated by complex splitting techniques above second order.

Distribution and habitats of the alien invader freshwater snail &lt;i&gt;Physa acuta&lt;/i&gt; in South Africa

This article focuses on the geographical distribution and habitats of the invader freshwater snail species Physa acuta as reflected by samples taken from 758 collection sites on record in the database of the National Freshwater Snail Collection (NFSC) at the Potchefstroom Campus of the North-West University. This species is currently the second most widespread alien invader freshwater snail species in South Africa. The 121 different loci (1/16- degree squares) from which the samples were collected, reflect a wide but discontinuous distribution mainly clustered around the major ports and urban centres of South Africa. Details of each habitat as described by collectors during surveys were statistically analysed, as well as altitude and mean annual air temperatures and rainfall for each locality. This species was reported from all types of water-bodies represented in the database, but the largest number of samples was recovered from dams and rivers. Chi-square and effect size values were calculated and an integrated decision tree constructed from the data which indicated that temperature, altitude and types of water-bodies were the important factors that significantly influenced the distribution of P. acuta in South Africa. Its slow progress in invading the relatively undisturbed water-bodies in the Kruger National Park as compared to the recently introduced invader freshwater snail species, Tarebia granifera, is briefly discussed.

Computing Zagreb Indices of the Subdivision-Related Generalized Operations of Graphs

Mathematical modeling or numerical coding of the molecular structures play a significant role in the studies of the quantitative structure-activity relationships (QSAR) and quantitative structures property relationships (QSPR). In 1972, the entire energy of π-electron of a molecular graph is computed by the addition of square of degrees (valencies) of its vertices (nodes). Later on, this computational result was called by the first Zagreb index and became well studied topological index in the field of molecular graph theory. In this paper, for k ∈ N (set of counting numbers), we define four subdivision-related operations of graphs in their generalized form named by S k , R k , Q k and T k . Moreover, using these operations and the concept of the cartesian product of graphs, we construct the generalized F k -sum graphs Γ 1+Fk Γ 2 , where F k ∈ {S k , R k , Q k , T k } and Γ i are any connected graphs for i ∈ {1, 2}. Finally, the first and second Zagreb indices are computed for the generalized F k -sum graphs in terms of their factor graphs. In fact, the obtained results are a general extension of the results Eliasi et al. and Deng et al. who studied these operations for exactly k = 1 and computed the Zagreb indices for only F 1 -sum graphs respectively.