The graph reconstruction conjecture: some new results and observations

A vertex-deleted subgraph (or simply a card) of graph G is an induced subgraph of G containing all but one of its vertices. The deck of G is the multiset of its cards. One of the best-known unanswered questions of graph theory asks whether G can be reconstructed in a unique way (up to isomorphism) from its deck. The likely positive answer to this question is known as the Reconstruction Conjecture. In the first part of the thesis two basic equivalence relations are considered on the set of vertices of the graph G to be reconstructed. The one is card equivalence, better known as removal equivalence, by which two vertices are equivalent if their removal results in isomorphic cards. The other equivalence is similarity, also called automorphism equivalence. Two vertices u and v are automorphism-equivalent (similar) if there exists an automorphism of G taking u to v. These relations are analyzed on various examples with special attention to vertices that are card-equivalent but not similar. Such vertices are called pseudo-similar, and they have been studied very extensively in the literature. The first result of the thesis is a structural characterization of card equivalence in terms of automorphism equivalence. A similar result was obtained by Godsil and Kocay in 1982 on the characterization of pseudosimilar vertices, which result is proved in the thesis as a corollary to the characterization theorem on card equivalence. In the second part of the thesis, the concept of relative degree-sequence is introduced for subgraphs of a graph G. By “relative” it is meant that each degree in the degree-sequence of the subgraph is coupled up with the original degree of the corresponding vertex in G. A new conjecture is formulated, which says that G is uniquely determined (up to isomorphism) by the multiset of the relative degree-sequences of its induced subgraphs. The new conjecture is then related to the Reconstruction Conjecture in a natural way. The third part of the thesis contains an original new result on graph reconstruction. Card-minimal graphs are investigated, the deck of which is a set. Thus, the deck of such graphs is free from duplicate cards. It is shown that every card-minimal graph G is reconstructible, provided that G does not have pseudo-similar couples of vertices. This condition is recognizable, that is, it can be checked by looking at the deck of G only. The results of this thesis have been partially published in [1]

Impact of environmental factors on COVID-19 pandemic in India

The present study has investigated the role of regional meteorology and air quality parameters in the outbreak pattern of COVID-19 pandemic in India. Using the remote sensing based dataset of 12 environmental variables we correlated infective case counts at a district level in India. Our investigation carried out on the circumstantial data from more than 300 major affected districts in India and found that air quality parameters are playing very crucial role in this outbreak. Among the air pollutants, O3 was better correlating with infection counts followed by AOD, CO, NO2, BC and SO2. We also observed that among the weather parameters air temperature, incoming shortwave radiation, wind speed are positively and significantly associate with outbreak pattern and precipitation and humidity are negatively correlated with confirmed cases; only cloud cover has no significant relation. We noted that coastal districts in the both coast of India and districts located in the plain and low-lying areas have experienced bitter situation during this pandemic. Our study suggests that improving air quality with proper strict regulations and complete lockdown during the peak of pandemic might reduce the misfortune in all over India

Facile synthesis, characterization and catalytic activity of nanoporous supports loaded with monometallic and bimetallic nanoparticles

When an aqueous platinum chloride or mixed rhodium and platinum chloride solutions are added to a mixture of lecithin, AOT and titanium isopropoxide (TIP) in isooctane, the surfactants partition the aqueous and oil phases into a nanoscale bicontinuous microstructure. Hydrolysis and condensation of the TIP generates a highly connected porous anatase support. Reduction of the metal salt/s in the confined aqueous nanodomains of the bicontinuous template results in monometallic or bimetallic alloy NPs. The NPs are well-distributed throughout the interior surfaces of the support. This route, where monometallic and bimetallic nanoparticles are synthesized together with the support material, can be exploited for generating diverse catalytically active materials for various applications

Oil emulsification using surface-tunable carbon black particles

Emulsification of oil from a subsurface spill and keeping it stable in the water is an important component of the natural remediation process. Motivated by the need to find alternate dispersants for emulsifying oil following a spill, we examine particle-stabilized oil-in-water emulsions. Emulsions that remain stable for months are prepared either by adding acid or salt to carboxyl-terminated carbon black (CB) suspension in water to make the particles partially hydrophobic, adding the oil to this suspension and mixing. When naphthalene, a model potentially toxic polycyclic aromatic hydrocarbon, is added to octane and an emulsion formed, it gets adsorbed significantly by the CB particles, and its transport into the continuous water is markedly reduced. In contrast to an undesirable seawater-in-crude oil emulsion produced using a commercially used dispersant, Corexit 9500A, we demonstrate the formation of a stable crude oil-in-seawater emulsion using the CB particles (with no added acid or salt), important for natural degradation. The large specific surface area of these surface functionalized CB particles, their adsorption capability and their ability to form stable emulsions are an important combination of attributes that potentially make these particles a viable alternative or supplement to conventional dispersants for emulsifying crude oil following a spill. © 2013 American Chemical Society

High capacity, stable silicon/carbon anodes for lithium-ion batteries prepared using emulsion-templated directed assembly

Silicon (Si) is a promising candidate for lithium ion battery anodes because of its high theoretical capacity. However, the large volume changes during lithiation/delithiation cycles result in pulverization of Si, leading to rapid fading of capacity. Here, we report a simple fabrication technique that is designed to overcome many of the limitations that deter more widespread adoption of Si based anodes. We confine Si nanoparticles in the oil phase of an oil-in-water emulsion stabilized by carbon black (CB). These CB nanoparticles are both oil- and water-wettable. The hydrophilic/hydrophobic balance for the CB nanoparticles also causes them to form a network in the continuous aqueous phase. Upon drying this emulsion on a current collector, the CB particles located at the surfaces of the emulsion droplets form mesoporous cages that loosely encapsulate the Si particles that were in the oil. The CB particles that were in the aqueous phase form a conducting network connected to the CB cages. The space within the cages allows for Si particle expansion without transmitting stresses to the surrounding carbon network. Half-cell experiments using this Si/CB anode architecture show a specific capacity of ∼1300 mAh/g Si + C and a Coulombic efficiency of 97.4% after 50 cycles. Emulsion-templating is a simple, inexpensive processing strategy that directs Si and conducts CB particles to desired spatial locations for superior performance of anodes in lithium ion batteries. © 2014 American Chemical Society

Shear free and blotless cryo-TEM imaging: A new method for probing early evolution of nanostructures

Cryogenic transmission electron microscopy (cryo-TEM) is a powerful method to image native state morphologies of nanoscale soft and hard objects suspended in solvents. Sample preparation is a critical step toward producing images at length and time scales of interest. We demonstrate a nearly shear-free sample thinning method which simultaneously allows imaging of evolving nanostructures at subsecond time scales. This device breaks the trade-off between high shear and short time scales typical in current cryo-TEM sample preparation methods. We demonstrate the low-shear feature of the new method by imaging wormlike micelles, showing an interconnected network, in contrast to the traditional sample preparation method which shows aligned micelles at similar time points. The time resolution of this method is demonstrated by imaging morphologies of calcium carbonate (formed through the reaction of calcium chloride with sodium carbonate) at subsecond time scales, capturing its evolution from an amorphous to a crystalline state. The impact of hyperbranched polyglycerol additives on the amorphous to crystalline transition in calcium carbonate at short times is examined. Early images at low shear provide unique fundamental insights into mechanisms of nanostructure evolution, thus offering a new paradigm for research in materials sciences, soft matter, and biological sciences. © 2012 American Chemical Society

New insights into the transformation of calcium sulfate hemihydrate to gypsum using time-resolved cryogenic transmission electron microscopy

We use time-resolved cryogenic transmission electron microscopy (TR-cryo-TEM) on a supersaturated solution of calcium sulfate hemihydrate to examine the early stages of particle formation during the hydration of the hemihydrate. As hydration proceeds, we observe nanoscale amorphous clusters that evolve to amorphous particles and then reorganize to crystalline gypsum within tens of seconds. Our results indicate that a multistep particle formation model, where an amorphous phase forms first, followed by the transformation into a crystalline product, is applicable even at time scales of the order of tens of seconds for this system. The addition of a small amount of citric acid significantly delays the reorganization to gypsum crystals. We hypothesize that available calcium ions form complexes with the acid by binding to the carboxylic groups. Their incorporation into a growing particle produces disorder and extends the time over which the amorphous phase exists. We see evidence of patches of trapped amorphous phase within the growing gypsum crystals at time scales of the order of 24 h. This is confirmed by complementary X-ray diffraction experiments. Direct imaging of nanoscale samples by TR-cryo-TEM is a powerful technique for a fundamental understanding of crystallization and many other evolving systems. © 2012 American Chemical Society