In this thesis, we study the topological structure in graph theory and various related results. Chapter one, contains fundamental concept of topology and basic definitions about near open sets and give an account of uncertainty rough sets theories also, we introduce the concepts of graph theory. Chapter two, deals with main concepts concerning topological structures using mixed degree systems in graph theory, which is M-space by using the mixed degree systems. In addition, the m-derived graphs, m-open graphs, m-closed graphs, m-interior operators, m-closure operators and M-subspace are defined and studied. In chapter three we study supra-approximation spaces using mixed degree systems and primary object in this chapter are two topological spaces, namely o-space and i-space. In chapter four we introduce two new approximation operators using mixed degree systems and comparing of them and we find the accuracy of the second new approximation operator is more thin the first new approximation operator. For reason we study in detail the properties of the second new operator. Finally, in chapter five we introduce new generalization of rough set theory using a finite number of graphs by using the second new approximation operators in the preiow chapter. Several characterizations and properties of these concepts are obtained.
On Topological Structures In Graph Theory
M-space
m-derived graphs
m-open graphs
m-closed graphs
m-interior operators
m-closure operators
M-subspace
o-space
i-space.
Publication Date
Sun Feb 02 2020
Journal Name
University Of Baghdad, College Of Education For Pure Sciences / Ibn Al-haitham, Department Of Mathematics
Some Types of Perfect Mappings
j-perfect mappings
j-ω-perfect mappings
-ω-perfect mappings
weakly -ω-perfect mappings
strongly-ω-perfect mappings
nm- j-ω-converges to a subset
nm- j-ω-directed toward a set
nm- j-ω-closed mappings
nm- j-ω-rigid set
nm- j-ω-continuous mappings
nm- j-ω-perfect mappings in bitopological.
The aims of this thesis are to study the topological space; we introduce a new kind of perfect mappings, namely j-perfect mappings and j-ω-perfect mappings. Furthermore, we devoted to study the relationship between j-perfect mappings and j-ω-perfect mappings. Finally, certain theorems and characterization concerning these concepts are studied. On the other hand, we studied weakly/ strongly forms of ω-perfect mappings, namely -ω-perfect mappings, weakly -ω-perfect mappings and strongly-ω-perfect mappings; also, we investigate their fundamental properties. We devoted to study the relationship between weakly -ω-perfect mappings and strongly -ω-perfect mappings. As well as, some new generalizations of some definitions wh
... Show MorePublication Date
Sat Sep 01 2018
Journal Name
Polyhedron
Novel dichloro (bis {2-[1-(4-methylphenyl)-1H-1, 2, 3-triazol-4-yl-κN3] pyridine-κN}) metal (II) coordination compounds of seven transition metals (Mn, Fe, Co, Ni, Cu, Zn and Cd)
The 1
3-dipolar cycloaddition “click” reaction between an azide and an alkyne to give a 1
2
3-triazole was reported by Huisgen in 1961 [1]. In 2002 the Copper(I)-catalyzed Azide-Alkyne Cycloaddition (CuAAC) to prepare a 1
2
3-triazole was reported [2]
[3]. The existence of relatively basic nitrogen atoms in the 1
2
3-triazole rings
and the possibility of introducing additional donor groups in the substituents (Fig. 1)
made the CuAAC “click” reaction an attractive method to prepare differently substituted 1
2
3-triazoles. These compounds have been used as ligands to coordinate to various metal ions that display a range of applications such as in electrochemical and photochemical studies
in supramolecular chemistry
magnetism
metal-ion sensing and catalysis [4]. The reasons for the success of the “click” reaction
is that it is easy to carry out and is widely applicable. It is not affected by a variety of functional groups
and can be carried out with a variety of Cu(I) catalysts and solvents
including aqueous conditions. The Cu(I) catalysts overcome the high activation energy barrier of the non-catalyzed Huisgen reaction by changing the mechanism of the reaction. A large variety of copper catalysts can be used for the CuAAC reaction
on condition that the maximum concentration of Cu(I) species is generated during the reaction. The pre-catalyst can be a Cu(II) salt (usually CuSO4) together with a reducing agent (often sodium ascorbate) or a Cu(I) compound in the presence of a base or amine ligand and a reducing agent to prevent oxidation to Cu(II). Some strong oxidising cupric salts or complexes such as Cu(OAc)2 also work. The solvent is very flexible from organic to aqueous
with the most commonly used combination water + an alcohol (t-BuOH
MeOH or EtOH). The key role of the solvent or solvent mixture is to solubilize the substrates and Cu(I) catalyst in order to ensure rapid reactions. Such aqueous conditions are very useful for biochemical conjugations
as well as for organic syntheses.
We recently reported on the synthesis of a series of differently substituted 1
2
3-triazole chromophores
the substituted 2-(1-phenyl-1H-1
2
3-triazol-4-yl)pyridine ligands [5]
see Fig. 2 middle
with substituents R = H (L1)
CH3 (L2)
OCH3 (L3)
COOH
F
Cl
CN
CF3
O(CH2)3CH3 and N(CH3)2. These versatile ligands were found to coordinate to various first row transition metals
such as manganese
cobalt and nickel [6]. Here we extend the series to include more first row transition metal(II) coordination compounds
iron
copper and zinc
as well as a second row transition metal(II) coordination compound
cadmium
containing the 2-(1-(4-methyl-phenyl)-1H-1
2
3-triazol-1-yl)pyridine chromophore (Fig. 2 right with R = CH3). This series of seven novel coordination compounds is the first series of pyridyl-triazole based transition metal coordination compounds where seven different transition metals are coordinated to the same 1
2
3-triazole chromophore
namely 2-(1-(4-methyl-phenyl)-1H-1
2
3-triazol-1-yl)pyridine.