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

Introduction

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, δ^v, 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 δ^v(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 K_p-(p-odd) set of indices is obtained, for q = 1 and p = sequential (1, 2, 3, …) a K_p-(p-seq) set is obtained, for q = p and p = odd a K_p-(pp-odd) set is obtained, and for q = p and p = seq a K_p-(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 f_δ takes care of the suppressed hydrogen atoms in a chemical graph or pseudograph, and is defined in the following way