Graph representation

Many of the descriptors which can be calculated from the 2D structure rely upon the molecular graph representation because of the need for rapid calculations. Kier and Hall have developed a large number of topological indices, each of which characterises the molecular structure as a single number [Hall and Kier 1991]. Every non-hydrogen atom ir the molecule is characterised by two delta values, the simple delta Si and the valence delta SJ ... 

Mass spectrum of neon (a) actual appearance, (b) bar graph representation. 

Sanmartf, E., Friedler, E, Puigjaner, L., 1998. Combinatorial technique for short term scheduling of multipurpose batch plants based on schedule-graph representation. Comput. Chem. Eng., 22(Suppl.) S847-S850... 

Fig. 2. The graph representation of various relations among proteins or genes. The pathway and the assembly are representations of biochemical knowledge about the network of molecular interactions. The genome represents the positional correlations of genes. The neighbor, cluster, and hierarchical tree are computationally obtained from a set of binary relations, namely, similarity relations of protein pairs or gene pairs.

Fig. 1.3. Three representations of the molecular ion signal in the field desorption mass spectrum (Chap. 8) of tetrapentacontane, C54H110 (a) profile spectrum, (b) bar graph representation, and (c) tabular listing.

Bar graph representations are much better suited for visualization of isotopic compositions than tables, and in fact they exactly show how such a distribution would appear in a mass spectrum (Fig. 3.1). This appearance gives rise to the term isotopic pattern. 

Fig. 3.1. Isotopic patterns of chlorine, bromine and xenon. The bar graph representations of the isotopic distributions have the same optical appearance as mass spectra.

Figure 2. MXC(R) and CXC(R) are general statements about the reaction R and are readily derived by graph manipulations of the system graph representation.

Three-Body Correlation Effects in Third-Order Reduced Density Matrices Table 1 Graph representation of elements of RDM s and related quantities... 

Table 2 Graph representation of elements of the correlation matrices...

A brief introduction to the types of molecular representations typically encountered in MSA is presented at the beginning of Subheading 2. followed in Subheading 2.1. by a discussion of similarity measures based on chemical-graph representations. Although graph-based representations are the most familiar to chemists, their use has been somewhat limited in similarity studies due to the difficulty of evaluating the appropriate similarity measures. This section is followed by a discussion of similarity measures based on finite vector representations, the most ubiquitous types of representations. In these cases, the vector components can be of four types ... 

Mezey, P. G. (1997) Chirality measures and graph representations. Computers Math. Applic. 34, 105-112. 

We here define our model and present a self-contained introduction to perturbation theory, deriving the Feynman graph representation of the cluster expansion. To deal with solutions of finite concentration we introduce the grand-canonical ensemble and resum the cluster expansion to construct the loop expansion. We Lhen show that without further insight the expansions can be applied only in the (9-region or for concentrated solutions since they diverge term by term in the excluded volume limit. 

Remark 1 Note that given the process units, inputs, and outputs such a graph representation includes all possible alterative interconnections among the units, the inputs, and the outputs. Note also that the one-way arcs directly from the inputs to the outputs correspond to by-pass streams of the process units. Arcs originating from a unit and returning to the same unit without going through any other node have not been incorporated for simplicity of the presentation. 

Figure 8.2 Graph representation of the example structure in Figure 8.1. Nodes (black dots) represent the atoms and edges (solid lines) represent the bonds. Note that standard graph representation disregards any extra information such atom type or bond order.

Koch et al. [111] have discussed the use of graph theoretical techniques in an attempt to find rules to relate beta sheet topology to amino acid sequence and for the comparison of beta sheet structures. They defined a graph representation for every protein in the PDB that contains beta sheets, notations and graphic representations for sheets which described the sequential and topological neighbourhoods of the strands, and constructed tools for substructure searches of this database. 

Figure 2. Signal graph representation for the system in Figure 1...

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