Graph-Theoretical Indices based on Simple, General and Complete Graphs

Valence molecular connectivity indices are based on the concept of valence delta, δ v, that can be derived from general chemical graphs or chemical pseudographs. A general graph or pseudograph has multiple edges and loops and can be used to encode, through the valence delta, chemical entities. Two graph-theoretical concepts derived from chemical pseudographs are the intrinsic (I) and the electrotopological state (E) values, which are the used to define the valence delta of the pseudoconnectivity indices, ψI,S. Complete graphs encode, through a new valence delta, the core electrons of any atoms in a molecule. The connectivity indices, either valence connectivity or pseudoconnectivity, are the starting point to develop the dual connectivity indices. The dual indices show that not only can they assume negative values but also cover a wide range of numerical values. The central parameter of the molecular connectivity theory, the valence delta, defines a completely new set of connectivity indices, which can be distinguished by their configuration and advantageously used to model different properties and activities of compounds. molecular connectivity (MC) theory, by Kier and Hall (Kier & Hall, 1986, 1999). This theory has, since then, been enriched and discussed in many ways (Pogliani, 2000; Garcia-Domenech et al., 2008; Trinajstić, 1992; Devillers & Balaban, 1999, Todeschini & Consonni, 2000; Diudea, 2000; Estrada, 2001; Li & Gutman, 2006). A central parameter of the molecular connectivity algorithms is the valence delta number, δv, which has recently undergone a radical transformation in the hands of Pogliani (Garcia-Domenech et al., 2008). Today this graph-theoretical number is defined in a way that it is able to encode not only the sigma-, piand non-bonding nDOI: 10.4018/ijcce.2011010102 International Journal of Chemoinformatics and Chemical Engineering, 1(1), 12-28, January-March 2011 13 Copyright © 2011, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. electrons but also the core electrons and the contribute of the suppressed hydrogen atoms, and it does it by the aid of concepts like general graphs or pseudographs and complete graphs. This important number has been defined in the following way, d d d v n v q f ps pr = + + ( ) ( ) ( ) 1 (1) Parameter δv(ps) is the vertex degree number of an atom in a pseudograph. Parameter q equals 1 or p; parameter p⋅r equals the sum of all vertex degrees in a complete graph, and it equals twice the number of its connections. Notice that parameter q has fixed values and it is not used as an optimizing parameter, something like Randić’s variable index (Randić & Basak, 2001). The fact that q = 1 or p has the consequence that four possible sets of molecular connectivity indices can be obtained and they are: for q = 1 and p = odd (1, 3, 5, ..) a Kp-(podd) set of indices is obtained, for q = 1 and p = sequential (1, 2, 3, ...) a Kp-(p-seq) set is obtained, for q = p and p = odd a Kp-(pp-odd) set is obtained, and for q = p and p = seq a Kp-(ppseq) set of indices is obtained. The rationale of this distinction resides in the fact that in many cases MC indices derived with p odd-valued show a superior model quality. Perturbation parameter fδ takes care of the suppressed hydrogen atoms in a chemical graph or pseudograph, and is defined in the following way: fδ = [δ v m(ps)δ (ps)]/δm(ps) = 1 δ v(ps)/ δm(ps) = nH/δ v m(ps) (2) Here, δm(ps) is the maximal δ v(ps) value a heteroatom can have in a chemical HS-pseudograph when all bonded hydrogens are substituted by heteroatoms, and nH equals the number of hydrogen atoms normally bonded to any heteroatom (C inclusive). For completely substituted carbons (or heteroatoms): nH= 0, fδ = 0 [as: δ v m(ps) = δ v(ps)], while for alkanes δ and δv are now related as, δv(ps) = δ and δv = (1+ fδ n)δ (in this case p = 1). Only for quaternary carbons (with fδ = 0) δv = δ is obtained. Exponent n can have different values, and even negative values to allow for a strong perturbation value. The higher the value of n the lower the perturbation and for very high values (n ≥ 50) the perturbation is practically zero. The reader should notice that the hydrogen perturbation allows to differentiate among pand δv(ps)-similar atoms, which differ in the number of bonded hydrogen atoms as in the case of the HS chemical pseudographs plus complete graphs of LiF, BeHF, BH2F, and CH3F. For every n value a specific set of constant molecular connectivity index values is obtained, and this allows to check the amount of hydrogen perturbation, which is responsible for the best model. This parameter is not used as a variable parameter (even if it could) and this means that the number of molecular connectivity indices can be quite large as each set depends now on the values of q, p and n. These three values determine what could be defined a configuration of the molecular connectivity indices: Kp(p-odd/seq; pp-odd/seq)/fδ n. Chemical Graphs, Pseudographs and Complete Graphs Let us now spend two words about graphs, pseudographs (or general graphs) and complete graphs (Kier & Hall, 1986, 1999; Trinajstić, 1992; Devillers & Balaban, 1999, Todeschini & Consonni, 2000; Diudea, 2000; Estrada, 2001; Li & Gutman, 2006). Those interested in the history of chemical graph theory should consult Randić and Trinastić (1994). A graph G can informally be seen as a set of vertices together with a set of edges (connections) that connect these vertices. The degree of a vertex in a graph is the number of edges that occur with it. The degree of a vertex (= δ) known also as its valence, is the central concept of the molecular connectivity theory, which is the chemical counterpart of graph theory and it is based on the concept of chemical graph (also, molecular graph, structural graphs, constitutional graphs). 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