Mathematics graphs

One of the simplest ways of modeling a variety of physical structures, from molecules to rigid three-dimensional structures, is the mathematical graph. 

The merging of mathematical graph theory with chemical theory is the formalization of what most chemists do in a more or less intuitive mode. Chemists currently use graphical images to embody chemical information in compact form which can be transformed into algebraic sets. Chemical graph theory provides simple descriptive interpretations of complicated quantum mechanical calculations and is, thereby, in-itself-by-itself an important discipline of study. 

Thus, we have established that molecules can be represented by means of graphs. To proceed beyond this rather trivial statement, let us resort again to the mathematical graph theory and ask the following question is it possible to represent a graph without resort to a drawing, in other words, is it possible to show the relationship of the graph s vertices not by means of points and dashes but differently, by means of other mathematical objects ... 

The correspondence between mathematical graph theory and classical chemical structure theory is manifested in Table I. A widely used mathematical representation for graphs is the adjacency matrix (A). The rank of this matrix equals the number of the vertices (atoms), and its entries a,y are equal to either 0 or 1 ... 

The structural diagram of a molecule can be interpreted as a mathematical graph. Each atom therein is represented by a node in the graph, and accordingly, the bonds are represented... 

Graph Isomorphism is a method from mathematical graph theory that can be used for mapping a structure onto another to determine the identity between two structures. 

Isomorphism Algorithm is a mathematical graph theory method to determine the extent to whether two graphs can be mapped onto each other by permutation through the vertices of the graph. 

Visualization System This should ensure the adequate representation of the information retrieved from an information pool. This covers highly specialized representations like functional mathematical graphs, multidimensional graphs, and molecules as well as default representations in formatted texts or tables. In addition, some kind of three-dimensional (3D) information, like 3D views of departments, workflows, and operational procedures, can be helpful. 

One can immediately see the advantage of mathematical (graph-theoretical) techniques over quantum-chemical calculations in fields, which requires analysis of structure-property relation (such as environmental sciences). We quote the following paragraph from a paper by Randic et al. (1985). 

In general, molecular graphs in different forms are widely exploited in chemistry. This is due to the fact that the graphic presentations of molecules can serve as objects of mathematical graph theory. One of the methods based on the graph theory and currently widely used in the QSAR problems is the so-called method of topological indices (Tl). ... 

The Morgan algorithm was initially developed for the description of chemical structures that are searchable by computers [6], The molecule is treated as a mathematical graph. An ordered list of nodes that refers to the chemical symbols is created. This list contains entries that indicate the bond type and the list entry of atoms attached via these particular bonds. 

In this book, we describe, extend and apply methods of computer chemistry and chemoinformatics, suitable for molecular structure generation, structure elucidation, combinatorial chemistry, QSPRs, the generation of chemical patent libraries and so on. The tools come from discrete mathematics (graph theory, constructive combinatorics), stochastics (explorative data analysis, supervised and unsupervised learning), computer science (data structures, algorithms) and chemistry (combinatorial chemistry, molecular structure elucidation). 

The base of the mathematical representation of a flowsheet (Figure 2) is a mathematical graph (Figure 3). The vertexes of the graph are the feed (FI), the distillation columns (Cl, C2,...), the heat-exchangers (condensers Coni,. .. and reboilers Rebl,. ..), the mixers/splitters (MSI, MS2,. ..) and the products (PI, P2,. ..) the branches are the streams between the units. This graph can be represented in a matrix form (vertex-vertex matrix). In this matrix if a,/=l then there is a branch from vertex i to vertex j, if a,y=0, then there is no branch. 

The third type of CS representation is illnstrated by the mathematical graph depicted in Fig. 1.7c called a reflexive, labeled or simple, labeled graph. The term... 

An important type of matrix that plays a role in many procedures designed to determine graph/network properties is the adjacency matrix of mathematical graphs and netwoiks. The adjacency matrix corresponding to the CSN in Fig. 1.14b is given by... 

Data analysis of a water quality study Mathematics Graph interpretation Basic statistics Results assessment Calculation of mean value Collecting data from graph Comparison of mean Hardness of water Reliability of data... 

Chemical representation within computerized databases is a Byzantine and highly specialized area of research, and luckily for most scientific users, is buried invisibly within any particular database system. The most common representation of chemical connectivity relies upon the data structures and algorithms of mathematical graph theory (see Graph Theory in Chemistry). The graph must be transformed and stored in a searchable form that preserves atom identity, connectivity, and stereochemistry, yet one that permits the easy dissection of substructural subgraphs. 

---