Topological graphs

Another approach applies graph theory. The analogy between a structure diagram and a topological graph is the basis for the development of graph theoretical algorithms to process chemical structure information [33-35]. 

Construct the reduced pipe network corresponding to the topological graph of the worst-case state. 

In the transformed graph, the only nodes are pipes and supersources and an edge leads from one node to another iff there is a path connecting the corresponding nodes in the topological graph, and no other pipe or source lies on this path. 

To this end we use a formula in topological graph theory which fixes an upper bound on the Euler characteristic and thus a lower bound on the genus of a graph [22] ... 

J. L. Gross and T. W. Tucker, Topological Graph Theory, Dover, New York, 2001. 

For a brief introduction into the field of Topological Graph Theory, see the survey by Dan Archdeacon on the webpage http //www.emba.uvm.edu/ archdeac/papers/papers.html... 

S. H. Bertz, Chem.Appl.Topology Graph Theory, 28 (1983) 206. 

In developing madiematical modeling in chemistry and physics various branches of mathematics have been widely employed, frequently each with a well defined domain of applicability This sometimes has not been sufficiently emphasized or recognized, particularly when less familiar or less frequently used mathematical disciplines are involved, such as combinatorics, topology, graph theory, set theory or category theory The lack of appreciation of the specifics of each such discipline... 

The only information that structural formulas provide is how a given atom is bonded (if at all) to others, or, in the words of Butlerov, how the chemical interaction is distributed . Moreover, when chemists began using the graph theory, they simplified the representation of a molecule as compared to the classical one. In distinction to a structural formula a topological graph, as a rule, has no... 

As we shall show later, many properties of molecules are defined by the shape of their topological graphs regardless of the geometry of molecules or the nature of chemical bonding in them. 

Let us try to write a topological graph of a molecule using a matrix. To do this, we shall first of all enumerate the vertices of the graph (i.e. the atoms). Then we shall compose a certain matrix A with elements where i and / are the numbers of the vertices (atoms). 

With Eq. (21) in mind, compare the Hiickel equation (9) with Eq. (16) defining the ectrum of topological graph of molecule ... 

Theorem one — on the number of zeros. Suppose a certain alternant hydrocarbon possesses p marked and q unmarked atoms. Then the number of zeros (A o) in th spectrum of the topological graph of this molecule obeys the inequality... 

Theorem two — on deletion of two vertices. If in the topological graph of an alternant hydrocarbon there is... 

Theorem five—almost obvious. If the topological graph G of a molecule can be represented in the form of two sulj-... 

Theorem six — on one important fragment. If the topological graph of a molecule has the fragment... 

As an example, let us consider the allyl radical tGHj = = CH — CHj) whose topological graph has the form... 

The geometrical symmetry group of a topological graph of a diphenyl molecule includes four elements the identity transformation E and three symmetry axes of second order C and... 

The application of these equations can be illustrated for the case of the dianion BjoH . According to our notation, it should be written as BioHJj+g.j. This ion is represented by the following topological graph... 

From the Lipscomb theory one can get the topological graphs for any borane of assigned composition and thus predict new structures. Certain limitations should be imposed, though, on the quantities appearing in the balance equations (31)-(34), for example,... 

Each symmetry operation of the group (the symmetry group Cg for a graph) is associated with a certain permutation of vertices in the topological graph of the molecule (Table 2). 

Beachhead. However, basically this discrepance does not disparage the explorations of the English mathematician. Cayley was the first to represent a molecule as a topological graph and to start developing the methods of isomer enumeration on the basis of the graph theory. Moreover, Cayley forwarded the notion of enumerative polynomial for rooted trees, i.e. the polynomial whose coefficients Ai specify the number of rooted trees with i vertices ... 

Table 6 shows the relationship between the symmetry group operations of topological graph of a benzene molecule and permutations of their vertices the cyclic type of each permutation is also indicated. Using the data of this table, we can obtain the following expression similar to (37) ... 

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