Chemical Applications Of Topology And Graph Theory Pdf
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- Topological Indices, and Applications of Graph Theory
- Chemical applications of topology and group theory
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- Augmented Zagreb Index
Graph theory , branch of mathematics concerned with networks of points connected by lines.
Show all documents Unicyclic and bicyclic graphs with minimal augmented Zagreb index A description of the structure or shape of molecules is very helpful in predicting the ac- tivity and properties of molecules in complex experiments. Here, a relatively new topo- logical index is considered. Although it has been proved that for any connected graph with minimum value must be a tree, but there is still conundrum about its exact structure.
Topological Indices, and Applications of Graph Theory
Graph theory , branch of mathematics concerned with networks of points connected by lines. The subject of graph theory had its beginnings in recreational math problems see number game , but it has grown into a significant area of mathematical research, with applications in chemistry , operations research , social sciences , and computer science.
Euler argued that no such path exists. His proof involved only references to the physical arrangement of the bridges, but essentially he proved the first theorem in graph theory.
As used in graph theory, the term graph does not refer to data charts, such as line graphs or bar graphs. Instead, it refers to a set of vertices that is, points or nodes and of edges or lines that connect the vertices. When any two vertices are joined by more than one edge, the graph is called a multigraph. A graph without loops and with at most one edge between any two vertices is called a simple graph.
Unless stated otherwise, graph is assumed to refer to a simple graph. When each vertex is connected by an edge to every other vertex, the graph is called a complete graph. When appropriate, a direction may be assigned to each edge to produce what is known as a directed graph , or digraph.
An important number associated with each vertex is its degree, which is defined as the number of edges that enter or exit from it. Thus, a loop contributes 2 to the degree of its vertex. For instance, the vertices of the simple graph shown in the diagram all have a degree of 2, whereas the vertices of the complete graph shown are all of degree 3.
Knowing the number of vertices in a complete graph characterizes its essential nature. Another important concept in graph theory is the path , which is any route along the edges of a graph. A path may follow a single edge directly between two vertices, or it may follow multiple edges through multiple vertices. If there is a path linking any two vertices in a graph, that graph is said to be connected.
A path that begins and ends at the same vertex without traversing any edge more than once is called a circuit , or a closed path.
A circuit that follows each edge exactly once while visiting every vertex is known as an Eulerian circuit , and the graph is called an Eulerian graph. An Eulerian graph is connected and, in addition, all its vertices have even degree.
The puzzle involved finding a special type of path, later known as a Hamiltonian circuit , along the edges of a dodecahedron a Platonic solid consisting of 12 pentagonal faces that begins and ends at the same corner while passing through each corner exactly once. Hamiltonian graphs have been more challenging to characterize than Eulerian graphs, since the necessary and sufficient conditions for the existence of a Hamiltonian circuit in a connected graph are still unknown.
The histories of graph theory and topology are closely related, and the two areas share many common problems and techniques. The vertices and edges of a polyhedron form a graph on its surface, and this notion led to consideration of graphs on other surfaces such as a torus the surface of a solid doughnut and how they divide the surface into disklike faces.
Having considered a surface divided into polygons by an embedded graph, mathematicians began to study ways of constructing surfaces, and later more general spaces, by pasting polygons together. The connection between graph theory and topology led to a subfield called topological graph theory.
An important problem in this area concerns planar graphs. These are graphs that can be drawn as dot-and-line diagrams on a plane or, equivalently, on a sphere without any edges crossing except at the vertices where they meet. Complete graphs with four or fewer vertices are planar, but complete graphs with five vertices K 5 or more are not.
Nonplanar graphs cannot be drawn on a plane or on the surface of a sphere without edges intersecting each other between the vertices.
The use of diagrams of dots and lines to represent graphs actually grew out of 19th-century chemistry , where lettered vertices denoted individual atoms and connecting lines denoted chemical bonds with degree corresponding to valence , in which planarity had important chemical consequences.
The first use, in this context , of the word graph is attributed to the 19th-century Englishman James Sylvester , one of several mathematicians interested in counting special types of diagrams representing molecules. Another class of graphs is the collection of the complete bipartite graphs K m , n , which consist of the simple graphs that can be partitioned into two independent sets of m and n vertices such that there are no edges between vertices within each set and every vertex in one set is connected by an edge to every vertex in the other set.
In the Polish mathematician Kazimierz Kuratowski proved that any nonplanar graph must contain a certain type of copy of K 5 or K 3,3. While K 5 and K 3,3 cannot be embedded in a sphere, they can be embedded in a torus. The graph-embedding problem concerns the determination of surfaces in which a graph can be embedded and thereby generalizes the planarity problem.
It was not until the late s that the embedding problem for the complete graphs K n was solved for all n. Another problem of topological graph theory is the map-colouring problem. This problem is an outgrowth of the well-known four-colour map problem , which asks whether the countries on every map can be coloured by using just four colours in such a way that countries sharing an edge have different colours. Asked originally in the s by Francis Guthrie, then a student at University College London, this problem has a rich history filled with incorrect attempts at its solution.
In an equivalent graph-theoretic form, one may translate this problem to ask whether the vertices of a planar graph can always be coloured by using just four colours in such a way that vertices joined by an edge have different colours.
The result was finally proved in by using computerized checking of nearly 2, special configurations. Interestingly, the corresponding colouring problem concerning the number of colours required to colour maps on surfaces of higher genus was completely solved a few years earlier; for example, maps on a torus may require as many as seven colours. This work confirmed that a formula of the English mathematician Percy Heawood from correctly gives these colouring numbers for all surfaces except the one-sided surface known as the Klein bottle , for which the correct colouring number had been determined in Among the current interests in graph theory are problems concerning efficient algorithms for finding optimal paths depending on different criteria in graphs.
Two well-known examples are the Chinese postman problem the shortest path that visits each edge at least once , which was solved in the s, and the traveling salesman problem the shortest path that begins and ends at the same vertex and visits each edge exactly once , which continues to attract the attention of many researchers because of its applications in routing data, products, and people.
Work on such problems is related to the field of linear programming , which was founded in the midth century by the American mathematician George Dantzig. Graph theory Article Media Additional Info. While every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions.
Facebook Twitter. Give Feedback External Websites. Let us know if you have suggestions to improve this article requires login. External Websites. Stephan C. See Article History. Read More on This Topic. In the 18th century the Swiss mathematician Leonhard Euler was intrigued by the question of whether a route existed that would traverse each of the seven bridges exactly once. In demonstrating that the answer is no, he laid the foundation for graph theory. Get a Britannica Premium subscription and gain access to exclusive content.
Subscribe Now. A graph is a collection of vertices, or nodes, and edges between some or all of the vertices. When there exists a path that traverses each edge exactly once such that the path begins and ends at the same vertex, the path is known as an Eulerian circuit and the graph is known as an Eulerian graph. Eulerian refers to the Swiss mathematician Leonhard Euler, who invented graph theory in the 18th century.
A directed graph in which the path begins and ends on the same vertex a closed loop such that each vertex is visited exactly once is known as a Hamiltonian circuit. The 19th-century Irish mathematician William Rowan Hamilton began the systematic mathematical study of such graphs. K 5 is not a planar graph, because there does not exist any way to connect every vertex to every other vertex with edges in the plane such that no edges intersect.
With fewer than five vertices in a two-dimensional plane, a collection of paths between vertices can be drawn in the plane such that no paths intersect. With five or more vertices in a two-dimensional plane, a collection of nonintersecting paths between vertices cannot be drawn without the use of a third dimension. A bipartite map, such as K 3,2 , consists of two sets of points in the two-dimensional plane such that every vertex in one set the set of red vertices can be connected to every vertex in the other set the set of blue vertices without any of the paths intersecting.
The English recreational problemist Henry Dudeney claimed to have a solution to a problem that he posed in that required each of three houses to be connected to three separate utilities such that no utility service pipes intersected.
Dudeney's solution involved running a pipe through one of the houses, which would not be considered a valid solution in graph theory. In a two-dimensional plane, a collection of six vertices shown here as the vertices in the homes and utilities that can be split into two completely separate sets of three vertices that is, the vertices in the three homes and the vertices in the three utilities is designated a K 3,3 bipartite graph.
The two parts of such graphs cannot be interconnected within the two-dimensional plane without intersecting some paths. Learn More in these related Britannica articles:. The elements of V G , called vertices of G , may be represented by points. Number game , any of various puzzles and games that involve aspects of mathematics. Mathematical recreations comprise puzzles and games that vary from naive amusements to sophisticated problems, some of which have never been solved.
They may involve arithmetic, algebra, geometry, theory of numbers, graph theory, topology, matrices, group theory, combinatorics dealing…. Its applications extend to operations research, chemistry, statistical mechanics, theoretical physics, and socioeconomic problems.
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Chemical applications of topology and group theory
A topological index is actually designed by transforming a chemical structure into a number. Topological index is a graph invariant which characterizes the topology of the graph and remains invariant under graph automorphism. Eccentricity based topological indices are of great importance and play a vital role in chemical graph theory. In this article, we consider a graph non-zero component graph associated to a finite dimensional vector space over a finite filed in the context of the following eleven eccentricity based topological indices: total eccentricity index; average eccentricity index; eccentric connectivity index; eccentric distance sum index; adjacent distance sum index; connective eccentricity index; geometric arithmetic index; atom bond connectivity index; and three versions of Zagreb indices. Relationship of the investigated indices and their dependency with respect to the involved parameters are also visualized by evaluating them numerically and by plotting their results. Let a connected graph G having the sets V G and E G as the vertex set and the edge set, respectively. An emerging tool used, in the study of these phenomenal, is a topological index, which is invariant for chemical structures up to their symmetry automorphism.
E-mail: bako. Networks are increasingly recognized as important building blocks of various systems in nature and society. Water is known to possess an extended hydrogen bond network, in which the individual bonds are broken in the sub-picosecond range and still the network structure remains intact. Various characteristic properties e. We demonstrated that the topological properties of the hydrogen bond network found in liquid water systematically change with the temperature and that increasing temperature leads to a broader ring size distribution. We applied the studied topological indices to the network of water molecules with four hydrogen bonds, and showed that at low temperature K these molecules form a percolated or nearly-percolated network, while at ambient or high temperatures only small clusters of four-hydrogen bonded water molecules exist. Hydrogen bonded HB networks play an important role in determining the physical properties of many molecular liquids and solids, and play a crucial role in the structure and function of most biomolecules.
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Some topics in topological graph theory motivated by chemistry. Topological graph theory is a field of geometric topology. The mathematical objects of interest are embeddings of graphs in 3-space. The image is a so called spatial graphs. A spatial graph can be seen as a generalised knot.
Augmented Zagreb Index
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The 60 even permutations of the ligands in the five-coordinate complexes, ML 5 , form the alternating group A 5 , which is isomorphic with the icosahedral pure rotation group I. Using this idea, it is shown how a regular icosahedron can be used as a topological representation for isomerizations of the five-coordinate complexes, ML 5 , involving only even permutations if the five ligands L correspond either to the five nested octahedra with vertices located at the midpoints of the 30 edges of the icosahedron or to the five regular tetrahedra with vertices located at the midpoints of the 20 faces of the icosahedron. Applications to various fields of chemistry are briefly outlined. This is a preview of subscription content, access via your institution.
A topological graph index, also called a molecular descriptor, is a mathematical formula that can be applied to any graph which models some molecular structure. From this index, it is possible to analyse mathematical values and further investigate some physicochemical properties of a molecule. Therefore, it is an efficient method in avoiding expensive and time-consuming laboratory experiments. Molecular descriptors play a significant role in mathematical chemistry, especially in quantitative structure-property relationship QSPR , and quantitative structure-activity relationship QSAR investigations. An example of a molecular descriptor is a topological descriptor.
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