Graphs, mathematical objects

How the structural information in molecules is represented is crucial to the types of chemical questions that can be asked and answered. This is certainly true in MSA where different representations and their corresponding similarity measures can lead to dramatically different results (2). Four types of mathematical objects are typically used to represent molecules—sets, graphs, vectors, and functions. Sets are the most general objects and basically underlie the other three and are useful in their own right as will be seen below. Because of... 

Note that, being a labelled graph, the constitutional formula of an EM is also a mathematical object. Its algebraic counterpart, the fee-matrix, is suitable for mathematical manipulations by a computer, while the constitutional formula is more convenient for a chemist who prefers visual information. 

First we will focus attention on selected topics relating to the equivalence between benzenoid hydrocarbons, and special types of graphs and other mathematical objects that we can associate with benzenoids- In particular we will explore relations involving caterpillar trees [3] associated with catacondensed benzenoids and their line graphs [17] called, as already mentioned, Clar graphs [4]. Also relations involving "boards" (known technically as polyominos) of special properties such as those associated with "king" and "rook" pieces of chess... 

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 ... 

A graph is a mathematical object defined within the graph theory [Harary, 1969a Har-ary, 1969b Balaban and Harary, 1976 Rouvray and Balaban, 1979 Rouvray, 1990a Bonchev and Rouvray, 1991 Trinajstic, 1992]. 

The first step consists of associating a suitable mathematical object of fixed geometry with a map then, for the selected mathematical object a numerical representation is constructed in the form of a matrix once a matrix representing the map has been derived, local invariants and matrix invariants can be calculated in a similar way to local vertex invariants and graph invariants which encode information about a molecular graph. 

Common conventions of graph theory are followed here (see, e.g.. Refs. 8-10), but a rigid distinction is not always maintained between chemical and mathematical contexts. So the term graph is used interchangeably in both its strict sense as a mathematical object and as a shorthand term for its realization as an actual (usually carbon) molecule. Some equivalent pairs of terms are treated as being synonymous atom = vertex, bond = edge, valency = degree, and so on. 

Our next goal is to interpret the quotients of group actions as colimits. Assume, that X is some mathematical object (e.g., topological space, vector space, abstract simplicial complex, graph), and assume that a group G acts on X. Let us first informally contemplate what a quotient X/G should be. 

Definition A graph is a mathematical object de fined as a set of elements V, among which is defined a binary relation E. 

In our example, the object of evolution is a developmental program for a cell. This DP controls the growth of the cell into a graph of cells representing a mathematical function. Why would one want to effectively evolve a function A possible application is curve fitting. 

To overcome this weakness, we are developing a quantitative structure-activity strategy that is conceptually applicable to all chemicals. To be applicable, at least three criteria are necessary. First, we must be able to calculate the descriptors or Independent variables directly from the chemical structure and, presumably, at a reasonable cost. Second, the ability to calculate the variables should be possible for any chemical. Finally, and most importantly, the variables must be related to a parameter of Interest so that the variables can be used to predict or classify the activity or behavior of the chemical (j ) One important area of research is the development of new variables or descriptors that quantitatively describe the structure of a chemical. The development of these indices has progressed into the mathematical areas of graph theory and topology and a large number of potentially valuable molecular descriptors have been described (7-9). Our objective is not concerned with the development of new descriptors, but alternatively to explore the potential applications of a group of descriptors known as molecular connectivity indices (10). 

The Fourier transform describes precisely the mathematical relationship between an object and its diffraction pattern. In Figs. 2.7-2.10, the diffraction patterns are the Fourier transforms of the corresponding objects or arrays of objects. To put it another way, the Fourier transform is the lens-simulating operation that a computer performs to produce an image of molecules (or more precisely, of electron clouds) in the crystal. This view of p(x,y,z) as the Fourier transform of the structure factors implies that if we can measure three parameters— amplitude, frequency, and phase — of each reflection, then we can obtain the function p(x,y,z), graph the function, and "see" a fuzzy image of the molecules in the unit cell. 

Now let s look briefly at just enough of the mathematics of fiber diffraction to explain the origin of the X patterns. Whereas each reflection in the diffraction pattern of a crystal is described by a Fourier series of sine and cosine waves, each layer line in the diffraction pattern of a noncrystalline fiber is described by one or more Bessel functions, graphs that look like sine or cosine waves that damp out as they travel away from the origin (Fig. 9.3). Bessel functions appear when you apply the Fourier transform to helical objects. A Bessel function is of the form... 

The enumeration of Ld-sequences is equivalent to the enumeration of Gutman trees, which in the mathematical literature are called caterpillar trees (or caterpillars). These objects correspond to special trees in the graph-theoretical sense. 

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