GALLAI AND ANTI-GALLAI GRAPHS OF A GRAPH

Abstract. The paper deals with graph operators-the Gallai graphs and the anti-Gallai graphs. We prove the existence of a finite family of forbidden subgraphs for the Gallai graphs and the anti-Gallai graphs to be H-free for any finite graph H. The case of complement reducible graphs-cographs is discussed in detail. Some relations between the chromatic number, the radius and the diameter of a graph and its Gallai and anti-Gallai graphs are also obtained.

Antimedian graphs

Antimedian graphs are introduced as the graphs in which for every triple of vertices there exists a unique vertex x that maximizes the sum of the distances from x to the vertices of the triple. The Cartesian product of graphs is antimedian if and only if its factors are antimedian. It is proved that multiplying a non-antimedian vertex in an antimedian graph yields a larger antimedian graph. Thin even belts are introduced and proved to be antimedian. A characterization of antimedian trees is given that leads to a linear recognition algorithm.

Energies of some non-regular graphs

The energy of a graph G is the sum of the absolute values of its eigenvalues. In this paper, we study the energies of some classes of non-regular graphs. Also the spectrum of some non-regular graphs and their complements are discussed.

Studies on the dynamics and rheology of dilute suspensions of Brownian particles in simple shear flow under the action of an external force

We present a novel approach to computing the orientation moments and rheological properties of a dilute suspension of spheroids in a simple shear flow at arbitrary Peclct number based on a generalised Langevin equation method. This method differs from the diffusion equation method which is commonly used to model similar systems in that the actual equations of motion for the orientations of the individual particles are used in the computations, instead of a solution of the diffusion equation of the system. It also differs from the method of 'Brownian dynamics simulations' in that the equations used for the simulations are deterministic differential equations even in the presence of noise, and not stochastic differential equations as in Brownian dynamics simulations. One advantage of the present approach over the Fokker-Planck equation formalism is that it employs a common strategy that can be applied across a wide range of shear and diffusion parameters. Also, since deterministic differential equations are easier to simulate than stochastic differential equations, the Langevin equation method presented in this work is more efficient and less computationally intensive than Brownian dynamics simulations.We derive the Langevin equations governing the orientations of the particles in the suspension and evolve a procedure for obtaining the equation of motion for any orientation moment. A computational technique is described for simulating the orientation moments dynamically from a set of time-averaged Langevin equations, which can be used to obtain the moments when the governing equations are harder to solve analytically. The results obtained using this method are in good agreement with those available in the literature.The above computational method is also used to investigate the effect of rotational Brownian motion on the rheology of the suspension under the action of an external force field. The force field is assumed to be either constant or periodic. In the case of con- I stant external fields earlier results in the literature are reproduced, while for the case of periodic forcing certain parametric regimes corresponding to weak Brownian diffusion are identified where the rheological parameters evolve chaotically and settle onto a low dimensional attractor. The response of the system to variations in the magnitude and orientation of the force field and strength of diffusion is also analyzed through numerical experiments. It is also demonstrated that the aperiodic behaviour exhibited by the system could not have been picked up by the diffusion equation approach as presently used in the literature.The main contributions of this work include the preparation of the basic framework for applying the Langevin method to standard flow problems, quantification of rotary Brownian effects by using the new method, the paired-moment scheme for computing the moments and its use in solving an otherwise intractable problem especially in the limit of small Brownian motion where the problem becomes singular, and a demonstration of how systems governed by a Fokker-Planck equation can be explored for possible chaotic behaviour.

Study on the determination of molecular distance in organic dye mixtures using dual beam thermal lens technique

A sensitive method based on the principle of photothermal phenomena to study the energy transfer processes in organic dye mixtures is presented. A dual beam thermal lens method can be very effectively used as an alternate technique to determine the molecular distance between donor and acceptor in fluorescein–rhodamine B mixture using optical parametric oscillator.

A simple linear magnetic field sweep generator for magnetic resonance experiments

A simple and inexpensive linear magnetic field sweep generating system suitable for magnetic resonance experiments is described. The circuit, utilising a modified IC bootstrap configuration, generates field sweep over a wide range of sweep durations with excellent sweep linearity.

A Beacon based Localization System for Autonomous Mobile Robots and its Applications in Traffic and Transport Control

ACCURATE sensing of vehicle position and attitude is still a very challenging problem in many mobile robot applications. The mobile robot vehicle applications must have some means of estimating where they are and in which direction they are heading. Many existing indoor positioning systems are limited in workspace and robustness because they require clear lines-of-sight or do not provide absolute, driftfree measurements.The research work presented in this dissertation provides a new approach to position and attitude sensing system designed specifically to meet the challenges of operation in a realistic, cluttered indoor environment, such as that of an office building, hospital, industrial or warehouse. This is accomplished by an innovative assembly of infrared LED source that restricts the spreading of the light intensity distribution confined to a sheet of light and is encoded with localization and traffic information. This Digital Infrared Sheet of Light Beacon (DISLiB) developed for mobile robot is a high resolution absolute localization system which is simple, fast, accurate and robust, without much of computational burden or significant processing. Most of the available beacon's performance in corridors and narrow passages are not satisfactory, whereas the performance of DISLiB is very encouraging in such situations. This research overcomes most of the inherent limitations of existing systems.The work further examines the odometric localization errors caused by over count readings of an optical encoder based odometric system in a mobile robot due to wheel-slippage and terrain irregularities. A simple and efficient method is investigated and realized using an FPGA for reducing the errors. The detection and correction is based on redundant encoder measurements. The method suggested relies on the fact that the wheel slippage or terrain irregularities cause more count readings from the encoder than what corresponds to the actual distance travelled by the vehicle.The application of encoded Digital Infrared Sheet of Light Beacon (DISLiB) system can be extended to intelligent control of the public transportation system. The system is capable of receiving traffic status input through a GSM (Global System Mobile) modem. The vehicles have infrared receivers and processors capable of decoding the information, and generating the audio and video messages to assist the driver. The thesis further examines the usefulness of the technique to assist the movement of differently-able (blind) persons in indoor or outdoor premises of his residence.The work addressed in this thesis suggests a new way forward in the development of autonomous robotics and guidance systems. However, this work can be easily extended to many other challenging domains, as well.

Studies on some generalizations of line graph and the power domination problem in graphs

Department of Mathematics, Cochin University of Science and Technology

Studies on Simple and Encapsulated Transition Metal Complexes of Schiff Bases Derived from Diaminonaphthalene

Schiff base complexes of transition metal ions have played a significant role in coordination chemistry.In the present study we have synthesized some new Mn(II),Co(II) and Cu(II) complexes of Schiff bases derived from 1,8-diaminonaphthalene.Even though we could not isolate theses Schiff bases (as they readily cyclise to form the perimidine compounds),we were able to characterize unequivacally the complexes synthesized from these compounds as complexes of Schiff Bases. We Synthesized three perimidine derivatives ,2-(quinoxalin-2-yl)-2,3-dihydro-1H-perimidine,2-(2,3-dihydro-1H-perimidin-2-yl)-6-methoxyphenol and 4-(2,3-dihyro-1H-perimidin-2-yl)-2-methoxyphenol by the condensation of 1,8-diaminonaphthalene with quinoxaline-2-carboxaldehyde,2- hydroxy-3-methoxybenzaldehyde or 4-hydroxy-3-methoxybenzaldehyde respectively.Theses compounds were used as precursor ligands for the preparation of Schiff base complexes.The complexes were characterized by using elemental analysis ,conductance and magnetic susceptibility measuremets ,infrared and UV-Visible spectroscopy ,thermogravimetric analysis and EPR spectroscopy .We also encapsulated the complexes in zeolite Y matrix and these encapsulated complexes were also characterized. We have also tried theses complexes as catalysts in the oxidation of cyclohexanol and decomposition of hydrogen peroxide.

Full complex Fresnel holograms displayed on liquid cristal displays

We propose a method to display full complex Fresnel holograms by adding the information displayed on two analogue ferroelectric liquid crystal spatial light modulators. One of them works in real-only configuration and the other in imaginary-only mode. The Fresnel holograms are computed by backpropagating an object at a selected distance with the Fresnel transform. Then, displaying the real and imaginary parts on each panel, the object is reconstructed at that distance from the modulators by simple propagation of light. We present simulation results taking into account the specifications of the modulators as well as optical results. We have also studied the quality of reconstructions using only real, imaginary, amplitude or phase information. Although the real and imaginary reconstructions look acceptable for certain distances, full complex reconstruction is always better and is required when arbitrary distances are used.

Studies on the Spectrum and the Energy of Graphs

Department of Mathematics, Cochin University of Science and Technology

Simple and Encapsulated Transition Metal Complexes of Schiff Bases Derived from Quinoxaline-2-Carboxaldehyde and Diamines: Biological and Catalytic Activity Studies

This thesis is mainly concerned with the synthesis and characterisation of new simple and zeolite encapsulated transition metal (manganese(II),nickel(II),and copper(II)complexes of quinoxaline based double Schiff base ligands.Theses ligands are N,N'-bis(quinoxaline-2-carboxalidene)hydrazine,N,N'-bis(quinoxaline-2-carboxalidene)-1,2-diaminoethane,N,N'-bis(quinoxaline-2-carboxalidene)-1,3-diamonopropane,N,N'-bis(quinoxaline-2-carboxalidene)-1,4-diaminobutane,N,N'-bis(quinoxaline-2-carboxalidene)-1,2-diaminocyclohexane and N,N'-bis(quinoxaline-2-carboxalidene)-1,2-diaminobenzene.The Schiff base ligands have been characterised by spectral and single crystal XRD studies.Theses ligands provide great structural diversity during complexation.Mn(II) and Ni(II) form octahedral with these Schiff bases,whereas Cu(II) forms both octahedral and tetrahedral complexes.Studies on the biological and Catalytic activity of the copper(ll) complexes are also presented in this thesis.

On infinite graphs and related matrices

This thesis Entitled On Infinite graphs and related matrices.ln the last two decades (iraph theory has captured wide attraction as a Mathematical model for any system involving a binary relation. The theory is intimately related to many other branches of Mathematics including Matrix Theory Group theory. Probability. Topology and Combinatorics . and has applications in many other disciplines..Any sort of study on infinite graphs naturally involves an attempt to extend the well known results on the much familiar finite graphs. A graph is completely determined by either its adjacencies or its incidences. A matrix can convey this information completely. This makes a proper labelling of the vertices. edges and any other elements considered, an inevitable process. Many types of labelling of finite graphs as Cordial labelling, Egyptian labelling, Arithmetic labeling and Magical labelling are available in the literature. The number of matrices associated with a finite graph are too many For a study ofthis type to be exhaustive. A large number of theorems have been established by various authors for finite matrices. The extension of these results to infinite matrices associated with infinite graphs is neither obvious nor always possible due to convergence problems. In this thesis our attempt is to obtain theorems of a similar nature on infinite graphs and infinite matrices. We consider the three most commonly used matrices or operators, namely, the adjacency matrix