Studies on some graph operators and related topics

Department of Mathematics, Cochin University of Science and Technology

‘Studies on some topics in product graphs’

For routing problems in interconnection networks it is important to find the shortest containers between any two vertices, since the w-wide diameter gives the maximum communication delay when there are up to w−1 faulty nodes in a network modeled by a graph. The concept of ‘wide diameter’ was introduced by Hsu [41] to unify the concepts of diameter and The concept of ‘domination’ has attracted interest due to its wide applications in many real world situations [38]. A connected dominating set serves as a virtual backbone of a network and it is a set of vertices that helps in routing. In this thesis, we make an earnest attempt to study some of these notions in graph products. This include, the diameter variability, the diameter vulnerability, the component factors and the domination criticality.connectivity

Intraobserver and interobserver variability of ascending aorta diameter measurements as assessed by ECG-gated MDCT : automatic versus manual measurements : P15

Purpose: Revolutionary endovascular treatments are on the verge of being available for management of ascending aortic diseases. Morphometric measurements of the ascending aorta have already been done with ECG-gated MDCT to help such therapeutic development. However the reliability of these measurements remains unknown. The objective of this work was to compare the intraobserver and interobserver variability of CAD (computer aided diagnosis) versus manual measurements in the ascending aorta. Methods and materials: Twenty-six consecutive patients referred for ECG-gated CT thoracic angiography (64-row CT scanner) were evaluated. Measurements of the maximum and minimum ascending aorta diameters at mid-distance between the brachiocephalic artery and the aortic valve were obtained automatically with a commercially available CAD and manually by two observers separately. Both observers repeated the measurements during a different session at least one month after the first measurements. Intraclass coefficients as well the Bland and Altman method were used for comparison between measurements. Two-paired t-test was used to determine the significance of intraobserver and interobserver differences (alpha = 0.05). Results: There is a significant difference between CAD and manual measurements in the maximum diameter (p = 0.004) for the first observer, whereas the difference was significant for minimum diameter between the second observer and the CAD (p <0.001). Interobserver variability showed a weak agreement when measurements were done manually. Intraobserver variability was lower with the CAD compared to the manual measurements (limits of variability: from -0.7 to 0.9 mm for the former and from -1.2 to 1.3 mm for the latter). Conclusion: In order to improve reproductibility of measurements whenever needed, pre- and post-therapeutic management of the ascending aorta may benefit from follow-up done by a unique observer with the help of CAD.

Intraobserver and interobserver variability of ascending aorta diameter as assessed with ECG-gated MDCT: automatic versus manual measurements

Purpose: Recently morphometric measurements of the ascending aorta have been done with ECG-gated MDCT to help the development of future endovascular therapies (TCT) [1]. However, the variability of these measurements remains unknown. It will be interesting to know the impact of CAD (computer aided diagnosis) with automated segmentation of the vessel and automatic measurements of diameter on the management of ascending aorta aneurysms. Methods and Materials: Thirty patients referred for ECG-gated CT thoracic angiography (64-row CT scanner) were evaluated. Measurements of the maximum and minimum ascending aorta diameters were obtained automatically with a commercially available CAD and semi-manually by two observers separately. The CAD algorithms segment the iv-enhanced lumen of the ascending aorta into perpendicular planes along the centreline. The CAD then determines the largest and the smallest diameters. Both observers repeated the automatic measurements and the semimanual measurements during a different session at least one month after the first measurements. The Bland and Altman method was used to study the inter/intraobserver variability. A Wilcoxon signed-rank test was also used to analyse differences between observers. Results: Interobserver variability for semi-manual measurements between the first and second observers was between 1.2 to 1.0 mm for maximal and minimal diameter, respectively. Intraobserver variability of each observer ranged from 0.8 to 1.2 mm, the lowest variability being produced by the more experienced observer. CAD variability could be as low as 0.3 mm, showing that it can perform better than human observers. However, when used in nonoptimal conditions (streak artefacts from contrast in the superior vena cava or weak lumen enhancement), CAD has a variability that can be as high as 0.9 mm, reaching variability of semi-manual measurements. Furthermore, there were significant differences between both observers for maximal and minimal diameter measurements (p<0.001). There was also a significant difference between the first observer and CAD for maximal diameter measurements with the former underestimating the diameter compared to the latter (p<0.001). As for minimal diameters, they were higher when measured by the second observer than when measured by CAD (p<0.001). Neither the difference of mean minimal diameter between the first observer and CAD nor the difference of mean maximal diameter between the second observer and CAD was significant (p=0.20 and 0.06, respectively). Conclusion: CAD algorithms can lessen the variability of diameter measurements in the follow-up of ascending aorta aneurysms. Nevertheless, in non-optimal conditions, it may be necessary to correct manually the measurements. Improvements of the algorithms will help to avoid such a situation.

Variability of ascending aorta diameter measurements as assessed with electrocardiography-gated multidetector computerized tomography and computer assisted diagnosis software.

Recently, morphometric measurements of the ascending aorta have been done with ECG-gated multidector computerized tomography (MDCT) to help the development of future novel transcatheter therapies (TCT); nevertheless, the variability of such measurements remains unknown. Thirty patients referred for ECG-gated CT thoracic angiography were evaluated. Continuous reformations of the ascending aorta, perpendicular to the centerline, were obtained automatically with a commercially available computer aided diagnosis (CAD). Then measurements of the maximal diameter were done with the CAD and manually by two observers (separately). Measurements were repeated one month later. The Bland-Altman method, Spearman coefficients, and a Wilcoxon signed-rank test were used to evaluate the variability, the correlation, and the differences between observers. The interobserver variability for maximal diameter between the two observers was up to 1.2 mm with limits of agreement [-1.5, +0.9] mm; whereas the intraobserver limits were [-1.2, +1.0] mm for the first observer and [-0.8, +0.8] mm for the second observer. The intraobserver CAD variability was 0.8 mm. The correlation was good between observers and the CAD (0.980-0.986); however, significant differences do exist (P<0.001). The maximum variability observed was 1.2 mm and should be considered in reports of measurements of the ascending aorta. The CAD is as reproducible as an experienced reader.

Genetic variability in S(1) families from different maize populations.

Genetic variability in S(1) families from different maize populations. The objectives of the present work were directed towards the study of genetic: variablilty In seven maize populations with a broad genetic base, as a guide for population improvement. The field evaluation was conducted in completely randomized blocks, at one location (Anhembi, Sao Paulo state) with different groups, of S(1) families Obtained from seven populations (GO-D: dent type, GO-F: flint type, GO-L: long car, GO-G: thick Car; and composites G3, G4 and GO-S). Estimates were obtained for genetic variance (progeny mean basis), phenotypic variance of families means, and coefficient of heritability (broad sense) for progeny means. Estimates of heritability were high for Car weight (0,89 to 0.94), car length (0.77 to 0.88) and car diameter (0.77 to 0.92); and lower for plant height (0.58 to 0.80) and Car height (0.54 to 0.84), thus showing the high Potential of the populations for recurrent selection based oil S, families. Ear yield in the base populations used as controls varied front 11,200 kg ha(-1) (GO-D) to 12,800 kg ha(-1) (G3). The means of S(1) families varied from 6,070 kg ha(-1) (GO-F) to 7,380 kg ha(-1) (G4); the Inbreeding depression in S(1) Families varied front 37.5% (G4) to 48.0% (G3) relative to the non-inbred population.

Seasonal variability on the structure of sublittoral macrozoobenthic association in the Patos Lagoon estuary, southern Brazil

The aim of this study is to analyze and relate the spatial-temporal variability of macrozoobenthic assemblages to bottom characteristics and salinity fluctuations, in an estuarine shallow water region of Patos Lagoon. Monthly samples, between September 2002 and August 2003, were taken on six sampling stations (distant 90 m). Three biological samples with a 10 cm diameter corer, one sample for sediment analysis, fortnightly bottom topography measurements, and daily data of temperature and salinity were taken from each station. Two biotic and environmental conditions were identified: the first corresponding to spring and summer months, with low macrozoobenthos densities, low values of salinity, small variations in bottom topographic level and weak hydrodynamic activity. A second situation occurred in the months of fall and winter, which showed increased salinity, hydrodynamics and macrobenthos organisms. These results which contrast with previous studies carried out in the area, were attributed to failure in macrozoobenthos recruitments during summer period, especially of the bivalve Erodona mactroides Bosc, 1802 and the tanaid Kalliapseuses schubartii Mañe-Garzón, 1949. This results showed that recruitments of dominant species were influenced by salinity and hydrodynamic conditions.

Interobserver variability of ultrasound parameters in portal hypertension

The aim of this study was to assess interobserver agreement of ultrasound parameters for portal hypertension in hepatosplenic mansonic schistosomiasis. Spleen size, diameter of the portal, splenic and superior mesenteric veins and presence of thrombosis and cavernous transformation were determined by three radiologists in blinded and independent fashion in 30 patients. Interobserver agreement was measured by the kappa index and intraclass correlation coefficient. Interobserver agreement was considered substantial (κ = 0.714-0.795) for portal vein thrombosis and perfect (κ = 1) for cavernous transformation. Interobserver agreement measured by the intraclass correlation coefficient was excellent for longitudinal diameter of the spleen (r = 0.828-0.869) and splenic index (r = 0.816-0.905) and varied from fair to almost perfect for diameter of the portal (r = 0.622-0.675), splenic (r = 0.573-0.913) and superior mesenteric (r = 0.525-0.607) veins. According to the results, ultrasound is a highly reproducible method for the main morphological parameters of portal hypertension in schistosomiasis patients.

Spatial variability of aggregate stability in latosols under sugarcane

The spatial variability of soils under a same management system is differentiated, as expressed in the properties. The spatial variability of aggregate stability of a eutrophic Red Latosol (ERL) and a dystrophic Red Latosol (DRL) under sugarcane was characterized. Samples were collected in a regular 10 m grid, in the layers 0.0-0.2 and 0.2-0.4 m, with 100 points per area, and the following properties were determined: geometric mean diameter (GMD) of aggregates, mean weight diameter (MWD) of aggregates, percent of aggregates in the > 2.0 mm class and organic matter (OM) content. The eutrophic Red Latosol (ERL) had a higher aggregate stability thn the dystrophic Red Latosol (DRL), which may be attributed to the higher clay and OM content and the gibbsitic mineralogy of this soil class. The differentiated evolution of the studied Oxisols explains the wider range and lower variation coefficient and variability, for all properties studied in the eutrophic Red Latosol.

Spatial variability of the chemical, physical and biological properties in lowland cultivated with irrigated rice

In the areas where irrigated rice is grown in the south of Brazil, few studies have been carried out to investigate the spatial variability structure of soil properties and to establish new forms of soil management as well as determine soil corrective and fertilizer applications. In this sense, this study had the objective of evaluating the spatial variability of chemical, physical and biological soil properties in a lowland area under irrigated rice cultivation in the conventional till system. For this purpose, a 10 x 10 m grid of 100 points was established, in an experimental field of the Embrapa Clima Temperado, in the County of Capão do Leão, State of Rio Grande do Sul. The spatial variability structure was evaluated by geostatistical tools and the number of subsamples required to represent each soil property in future studies was calculated using classical statistics. Results showed that the spatial variability structure of sand, silt, SMP index, cation exchange capacity (pH 7.0), Al3+ and total N properties could be detected by geostatistical analysis. A pure nugget effect was observed for the nutrients K, S and B, as well as macroporosity, mean weighted diameter of aggregates, and soil water storage. The cross validation procedure, based on linear regression and the determination coefficient, was more efficient to evaluate the quality of the adjusted mathematical model than the degree of spatial dependence. It was also concluded that the combination of classical with geostatistics can in many cases simplify the soil sampling process without losing information quality.

Spatial variability of physical attributes of an alfisol under different hillslope curvatures

The influence of relief forms has been studied by several authors and explains the variability in the soil attributes of a landscape. Soil physical attributes depend on relief forms, and their assessment is important in mechanized agricultural systems, such as of sugarcane. This study aimed to characterize the spatial variability in the physical soil attributes and their relationship to the hillslope curvatures in an Alfisol developed from sandstone and growing sugarcane. Grids of 100 x 100 m were delimited in a convex and a concave area. The grids had a regular spacing of 10 x 10 m, and the crossing points of this spacing determined a total of 121 georeferenced sampling points. Samples were collected to determine the physical attributes related to soil aggregates, porosity, bulk density, resistance to penetration and moisture within the 0-0.2 and 0.2-0.4 m depth. Statistical analyses, geostatistics and Student's t-tests were performed with the means of the areas. All attributes, except aggregates > 2 mm in the 0-0.2 m depth and macroporosity at both depths, showed significant differences between the hillslope curvatures. The convex area showed the highest values of the mean weighted diameter, mean geometric diameter, aggregates > 2 mm, 1-2 mm aggregates, total porosity and moisture and lower values of bulk density and resistance to penetration in both depth compared to the concave area. The number of soil attributes with greater spatial variability was higher in the concave area.

Genetic parameters and variability in physic nut accessions during early developmental stages

The objective of this work was to estimate the genetic parameters and variability among accessions (half-sib families) of physic nut (Jatropha curcas) during the early stages of development. For this study, 110 accessions in the first year of development of the physic nut germplasm bank, maintained at Embrapa Cerrados, DF, Brazil, were evaluated in situ. The experiment was established in a randomized complete block design, with two replicates and five plants per plot arranged in rows at 4x2 m spacing. Grain yield, total number of branches per plant, plant height, stem diameter, canopy projection on the row, canopy projection between rows, canopy volume, number of days until first flowering and height of the first inflorescence were evaluated. Estimates of vegetative genetic parameters showed the existence of genetic variability in the physic nut germplasm bank. Physic nut accessions of the germplasm bank were grouped into five similarity groups based on character divergence. Although preliminary, the obtained results are promising for showing potential for Jatropha improvement with selective efficiency.

Morphological and functional variability in the root system of Quercus ilex L. subject to confinement: consequences for afforestation

We examined root morphological and functional differences caused by restrictions imposed to vertical growth in the root system of holm oak (Quercus ilex L.) seedlings to assess the consequences of using nursery containers in the development of a confined root system for this species. Thus, root morphological, topological and functional parameters, including hydraulic conductance per leaf unit surface area (K $_{\rm RL})$, were investigated in one-year seedlings cultivated in three PVC tubes differing in length (20, 60 and 100 cm). Longer tubes showed greater projected root area, root volume, total and fine root lengths, specific root length (SRL) and K$_{\rm RL}$ values than did shorter tubes. On the other hand, the length of coarse roots (diameter > 4.5 mm) and the average root diameter were greater in shorter tubes. The strong positive correlation found between K$_{\rm RL}$ and SRL (r=+0.69; P<0.001) indicated that root thickness was inversely related to water flow through the root system. We concluded that root systems developed in longer tubes are more efficient for plant water uptake and, therefore, changes in root pattern produced in standard forest containers (i.e. about 20 cm length) may in fact prevent a proper establishment of the holm oak in the field, particularly in xeric environments.

Spatial variability of yield and other parameters associated with pear trees

In the last few years, precision agriculture has become commonly used with many crops, particularly cereals, and there is also interest in precision horticulture. Pear is a seasonal fruit and well appreciated by Brazilian people, although it is mostly imported. Brazilian farmers are nowadays trying to increase pear production. Thus, this research aimed at mapping the yield of pear trees in order to study the spatial variability of yield as well as its comparison with spatial variability of soil and plant attributes. The experimental field had 146 pear trees, variety 'Pêra d'água', distributed on a 1.24 ha. Four harvests were performed according to the fruit ripening and from each tree; only the ripe fruits were harvested. In each harvest, all the fruits were weighed and the total yield was obtained based on the sum of each harvest. The soil attributes analyzed were P, K, Ca, Mg, pH in CaCl2, C, Cu, Zn, Fe, Mn and base saturation, and the plant attributes were fruit length, diameter and yield. Yield had low correlation with soil and plant attributes. An index of spatial variability was suggested in this study and helped in classifying levels of spatial dependence of the various soil and plant attributes: very low (fruit length); low (P, fruit diameter), medium (Mg, pH, Cu, Zn, Fe), high (Ca, K, base saturation and yield), and very high (Mn and C).