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

Transition metal complexes of some N4-phenylsemicarbazones: crystal structures and spectral studies

The unusual coordination modes of semicarbazones when bound to metals, the wide applications and structural diversity of metal complexes of semicarbazones provoked us to synthesize and characterize the tridentate ONO and NNO-donor semicarbazones and their transition metal complexes. This work is focused on the studies on complexes of three N4-phenylsemicarbazones synthesized by changing the carbonyl compounds. This work is concerned with the studies of two new semicarbazones, 2- formylpyridine-N4-phenylsemicarbazone (HL1) and 3-ethoxysalicylaldehyde- N4-phenylsemicarbazone (H2L2) and a reported semicarbazone 2-benzoylpyridine-N4-phenylsemicarbazone (HL3) [29]. The compositions of these semicarbazones were determined by the CHN analyses and IR, UV and NMR spectral studies were used for the characterization of these compounds. The molecular structure of 3-ethoxysalicylaldehyde-N4-phenylsemicarbazone (H2L2) was obtained by single crystal X-ray diffraction studies. Also, we have synthesized Cu(II), Cd(II), Zn(II) and Ni(II) complexes of these three semicarbazones. The complexes were characterized by various spectroscopic techniques, magnetic and conductivity studies. We could isolate single crystals of some complexes of all metals suitable for X-ray diffraction studies. This thesis is divided into six chapters.

Transition metal complexes derived from ketone based N(4)-substituted thiosemicarbazones: Crystal structures and spectral studies

Thiosemicarbazones have recently attracted considerable attention due to their ability to form tridentate chelates with transition metal ions through either two nitrogen and sulfur atoms, N–N–S or oxygen, nitrogen and sulfur atoms, O–N–S. Considerable interest in thiosemicarbazones and their transition metal complexes has also grown in the areas of biology and chemistry due to biological activities such as antitumoral, fungicidal, bactericidal, antiviral and nonlinear optical properties. They have been used for metal analyses, for device applications related to telecommunications, optical computing, storage and information processing.The versatile applications of metal complexes of thiosemicarbazones in various fields prompted us to synthesize the tridentate NNS-donor thiosemicarbazones and their metal complexes. As a part of our studies on transition metal complexes with these ligands, the researcher undertook the current work with the following objectives. 1. To synthesize and physico-chemically characterize the following thiosemicarbazone ligands: a. Di-2-pyridyl ketone-N(4)-methyl thiosemicarbazone (HDpyMeTsc) b. Di-2-pyridyl ketone-N(4)-ethyl thiosemicarbazone (HDpyETsc) 2. To synthesize oxovanadium(IV), manganese(II), nickel(II), copper(II), zinc(II) and cadmium(II) complexes using the synthesized thiosemicarbazones as principal ligands and some anionic coligands. 3. To study the coordination modes of the ligands in metal complexes by using different physicochemical methods like partial elemental analysis, thermogravimetry and by different spectroscopic techniques. 4. To establish the structure of compounds by single crystal XRD studies

Ligation properties of N4-phenylsemicarbazones towards transition metals: Synthesis and spectral studies

Semicarbazones and their transition metal complexes have been receiving considerable attention because of their biological relevance and applications in the field of analysis and in the field of organic NLO materials. Their structural diversity also attracted inorganic chemists. A good deal of work has been reported on the synthesis and structural investigation of semicarbazones and their complexes. This is due partially to their capability of acting as multidentate, NO, NNO, ONO and ONNO donors with the formation of either mono or bi or polynuclear complexes. Their chemistry and pharmacological applications have been extensively investigated. Appreciable biological applications as well as diverse stereochemistry of their metal complexes prompted us to synthesize two new tridentate ONO donor N4-phenyl semicarbazones derived from 2-hydroxy-4-methoxyacetophenone and 2-hydroxy-4-methoxybenzophenone and their transition metal complexes. These ketones were selected since they can provide a further binding site from phenolic–OH and can thus increase the denticity. Introduction of heterocyclic bases like 1,10-phenanthroline, 2,2′-bipyridine, 4,4′-dimethyl- 2,2′-bipyridine and 4-picoline and some pseudohalides like azide and thiocyanate ion can result in mixed ligand metal chelates with different geometries in coordination compounds In the present study, oxovanadium(IV), manganese(II), cobalt (II/III), nickel(II), copper(II) and zinc(II) complexes of 2-hydroxy-4- methoxyacetophenone-N4-phenylsemicarbazone (H2ASC) and 2-hydroxy-4- methoxybenzophenone-N4-phenylsemicarbazone (H2BSC) were synthesized and characterized.

Crystal Structures and Spectral Aspects of Mixed Ligand Transition Metal Chelates Derived from Aldehyde based ONS Donor Thiosemicarbazones

Thiosemicarbazones have emerged as an important class of ligands over a period of time, for a variety of reasons, such as variable donor properties, structural diversity and biological applications. Interesting as the coordination chemistry may be, the driving force for the study of these ligands has undoubtedly been their biological properties and the majority of the 3000 or so publications on thiosemicarbazones since 2000 have alluded to this feature. Thiosemicarbazones with potential donor atoms in their structural skeleton fascinate coordination chemists with their versatile chelating behavior. The thiosemicarbazones of aromatic aldehydes and ketones form stable chelates with transition metal cations by utilizing both their sulfur and azomethine nitrogen as donor atoms. They have been shown to possess a diverse range of biological activities including anticancer, antitumor, antibacterial, antiviral, antimalarial and antifungal properties owing to their ability to diffuse through the semipermeable membrane of the cell lines. The enhanced effect may be attributed to the increased lipophilicity of the metal complexes compared to the ligand alone.

‘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

Studies On Some New Transition Metai. Complexes Of The Schiff Bases Derived From Ouinoxaline - 2 - Carboxaldehyde

This thesis deals with the synthesis, characterisation and catalytic activity studies of some new transition metal complexes of the Schiff bases, derived from quinoxaline—2—carboxaldehyde. The model complexes derived from specially designed and synthesised Schiff bases help us to understand the chemistry of biological systems. Schiff bases derived from heterocyclic aldehydes like quinoxaline-2-carboxaldehyde provide great structural diversity during complexation. The Schiff bases synthesised in the present study ' are quinoxaline—2—carboxa.lidene-2-aminophenol (QAP). quinoxaline—2carboxaldehyde semicarbazone (QSC), quinoxaline-2—carboxalidene—o— phenylenediamine (QOD) and quinoxaline-2-carboxalidene-2-furfurylamine (QFA). The elucidation of the structure of these complexes is done using conductance, magnetic susceptibility measurements. infrared, UV—Vis and EPR spectral studies.

Mono- and binuclear transition metal complexes of n(4)-substituted thiosel\licarbazones derived frol\i salicylaldehyde and 2-hydroxyacetophenone synthesis, spectral and structural studies

The work embodied in the thesis is divided into eight chapters. Chapter I gives a brief introduction about metal complexes of thiosemicarbazones, including their structural and bonding properties. Chapter 2 deals with the synthesis and single crystal X-ray diffraction studies of various thiosemicarbazones used up for the present investigations and various characterization techniques. Chapter 3 deals with synthesis, spectral and structural studies of Cu(U) complexes with ONS donor thiosemicarbazones. Chapter 4 deals with synthesis and spectral studies of Ni(II) complexes \vith 2-hydroxyacetophenone N(4)-cyclohexyl thiosemicarbazone as the ligand. Chapter 5 includes synthesis and spectral studies of Mn(II) complexes. Chapter 6 deals with synthesis, spectral and structural studies of Zn(II) complexes. Chapter 7 includes synthesis and spectral studies of oxovanadium(IV) complexes. Chapter 8 deals with synthesis, spectral and single crystal X-ray diffraction studies of dioxomolybdenum(VI) complexes.

“Catalysis by Nanocrystalline Ceria Modified with Transition Metals”

The objective of the present work is to improve the textural and structural properties of cerium oxide by the incorporation of transition metals as well as sulphate ions. We have incorporated tungsten, molybdenum and chromium oxide into pure as well as sulphated cerium oxide and the catalytic systems thus prepared were characterised using various techniques. lndustrially important reactions such as acetalization and deacetalization, oxidative dehydrogenation of ethylbenzene, MTBE synthesis and Beckmann rearrangement of cinnamaldoxime and salicylaldoxime have been selected for the measurement of the catalytic activity of the systems. The work is presented in eight chapters

"Catalysis by pillared montmorillonites exchanged with transition metals"

The present work is oriented to obtain a comparative evaluation of the physicochemical properties and catalytic activities of iron, aluminium and iron aluminium mixed pillared montmorillonites and their transition metal exchanged analogues. Reactions of industrial importance like Friedel Crafts alkylations, catalytic wet peroxide oxidation of phenol and MTBE synthesis have been selected for the present study. The thesis is structured into seven chapters. First chapter deals with a brief introduction and literature survey on pillared clays. Second chapter explains the materials and methods employed in the work. Results and discussions on the characterisation techniques are described in the third chapter. The subsequent three chapters describe the catalytic activities of pillared clays in the industrially important reactions. Last chapter comprises the summary of the investigations and the conclusions drawn from the earlier chapters

‘‘catalysis by nano sulfated titania modified by transition metals"

in the present study, we have prepared and evaluated the physical and chemical properties and catalytic activities of transition metal loaded sulfated titania via the sol-gel route. Sol-gel method is widely used for preparing porous materials having controlled properties and leads to the formation of oxide particles in nano range, which are spherical or interconnected to each other. Characterization using various physico-chemical techniques and a detailed study of acidic properties are also carried out. Some reactions of industrial importance such as Friedel-Crafts reaction, fen-butylation of phenol,Beckmann rearrangement of cyclohexanone oxime, nitration of phenol and photochemical degradation of methylene blue have been selected for catalytic activity study in the present venture. The work is organized into eight chapters

Computing median and antimedian sets in median graphs

The median (antimedian) set of a profile π = (u1, . . . , uk) of vertices of a graphG is the set of vertices x that minimize (maximize) the remoteness i d(x,ui ). Two algorithms for median graphs G of complexity O(nidim(G)) are designed, where n is the order and idim(G) the isometric dimension of G. The first algorithm computes median sets of profiles and will be in practice often faster than the other algorithm which in addition computes antimedian sets and remoteness functions and works in all partial cubes

On the remoteness function in median graphs

A profile on a graph G is any nonempty multiset whose elements are vertices from G. The corresponding remoteness function associates to each vertex x 2 V.G/ the sum of distances from x to the vertices in the profile. Starting from some nice and useful properties of the remoteness function in hypercubes, the remoteness function is studied in arbitrary median graphs with respect to their isometric embeddings in hypercubes. In particular, a relation between the vertices in a median graph G whose remoteness function is maximum (antimedian set of G) with the antimedian set of the host hypercube is found. While for odd profiles the antimedian set is an independent set that lies in the strict boundary of a median graph, there exist median graphs in which special even profiles yield a constant remoteness function. We characterize such median graphs in two ways: as the graphs whose periphery transversal number is 2, and as the graphs with the geodetic number equal to 2. Finally, we present an algorithm that, given a graph G on n vertices and m edges, decides in O.mlog n/ time whether G is a median graph with geodetic number 2

Strongly distance-balanced graphs and graph products

A graph G is strongly distance-balanced if for every edge uv of G and every i 0 the number of vertices x with d.x; u/ D d.x; v/ 1 D i equals the number of vertices y with d.y; v/ D d.y; u/ 1 D i. It is proved that the strong product of graphs is strongly distance-balanced if and only if both factors are strongly distance-balanced. It is also proved that connected components of the direct product of two bipartite graphs are strongly distancebalanced if and only if both factors are strongly distance-balanced. Additionally, a new characterization of distance-balanced graphs and an algorithm of time complexity O.mn/ for their recognition, wheremis the number of edges and n the number of vertices of the graph in question, are given

Simultaneous Embeddings Of Graphs As Median And Antimedian Subgraphs

The distance DG(v) of a vertex v in an undirected graph G is the sum of the distances between v and all other vertices of G. The set of vertices in G with maximum (minimum) distance is the antimedian (median) set of a graph G. It is proved that for arbitrary graphs G and J and a positive integer r 2, there exists a connected graph H such that G is the antimedian and J the median subgraphs of H, respectively, and that dH(G, J) = r. When both G and J are connected, G and J can in addition be made convex subgraphs of H.