Graph invariant theory

Figure 7 Entropy vs. temperature from graph invariant theory (thick solid line). The horiz,ontal line is the Pauling entropy for fully disordered ice. Entropy calculated using the occupational probabilities obtained from our simulations, is plotted using the expressions of Nagle( A and Howe and Whitworth(TJ...

With the help of the graph invariant theory described in section, we were able to identify the most stable H-bond configurations surround ing the defect pair. 

Topological information indices are graph invariants, based on information theory and calculated as - information content of specified equivalence relationships on the molecular graph. 

However, the assumptions needed to facilitate calculation and numerical approximation are themselves associated with an inherent error, although in most cases the direction, but not the magnitude, of the error is known. Moreover, within a series of related compounds the error term is usually considered to be approximately constant. All kinds of error are absent only for the most simple theoretical descriptors such as count descriptors or for descriptors directly derived from exact mathematical theories such as graph invariants. 

Balaban, A.T. (1987a). Numerical Modelling of Chemical Structures Local Graph Invariants and Topological Indices. In Graph Theory and Topology in Chemistry (King, R.B. and Rouvray, D.H., eds.), Elsevier, Amsterdam (The Netherlands), pp. 159-176. 

ETA indices [Roy and Ghosh, 2003] are both local vertex and graph invariants, defined in the framework of the Valence Electron Mobile environment (VEM environment) theory [Pal, Sengupta et al, 1988 Pal, Sengupta et al, 1989], according to which a vertex in the H-depleted molecular graph is considered to be consisted of a core and a valence electronic environment. 

Ordering implies a comparison, and instead of actual structures, one normally compares sequences of numbers characterizing a molecular graph of a chemical structure. Frequently the required sequences are derived from an enumeration of selected graph invariants. If the selected invariants lead to integers, then the ordering theory of Muirhead (1903) is most suited for these special cases ... 

According to Graph Theory, a molecule may be represented by its skeletal molecular graph which, from a mathematical point of view, is the union of a set of points, symbolising the atoms other than hydrogens, and a set of lines, symbolising bonds. Its properties can be then expressed in terms of graph-theoretical invariants, Njj, which have been defined as "the number of distinct ways in which skeleton i... 

Graph theory is largely applied to the characterization of chemical structures, as well as to structure-property and structure-activity correlations, by means of so-called topological indices. These are numerical quantities based on various invariants or characteristics of molecular graphs. 

In this review we discuss the scaling properties of topological invariants and the relations between them. First, we recall the basic concepts of the chemical graph theory. Then, we introduce reduced topological invariants and discuss the problem of size extensivity. Following a brief overview of the known approximate relations between topological invariants, we move to considerations of their scaling properties. Finally, we discuss some practical aspects of the present formalism. 

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