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

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

Lionello Pogliani
DOI: 10.4018/ijcce.2011010102
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Valence molecular connectivity indices are based on the concept of valence delta, d 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, ?I,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.
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The many new types of molecular connectivity indices (Pogliani, 2000, 2002, 2004, 2005, 2006, 2007, 2009, 2010; Garcia-Domenech et al., 2008), which will be discussed and used for model purposes in this paper are defined by a family of formally similar algorithms. The very first graph-theoretical algorithm for these indices was devised by Randić in 1975 (Randić, 1975), and was, soon later, developed into a completely new chemical theory known as the 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-, pi- and non-bonding n-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,


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-(p-odd) 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-(pp-seq) 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

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