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Maximum Topological Distances Based Indices As Molecular Descriptors for QSPR

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Maximum Topological Distances Based Indices As Molecular Descriptors for QSPR Int. J. Mol. Sci. 2001, 2, 121-132 International Journal of Molecular Sciences ISSN 1422-0067 www.mdpi.org/ijms/ Maximum Topological Distances Based Indices as Molecular Descriptors for QSPR. 4. Modeling the Enthalpy of Formation of Hydrocarbons from Elements Andrés Mercader 1, Eduardo A. Castro 1,* and Andrey A. Toropov2 1 CEQUINOR, Departamento de Química, Facultad de Ciencias Exactas, Universidad Nacional de la Plata, C.C. 962, La Plata 1900, Argentina. e-mail: [email protected] 2 Andrey A. Toropov, Vostok Innovation Company, S. Azimstreet 4, 100047 Tashkent, Uzbekistan. e-mail: [email protected] * Author to whom correspondence should be addressed. e-mail: [email protected] Received: 9 April 2001 Accepted: 15 June 2001/ Published: 28 June 2001 Abstract: The enthalpy of formation of a set of 60 hydroarbons is calculated on the basis of topological descriptors defined from the distance and detour matrices within the realm of the QSAR/QSPR theory. Linear and non-linear polynomials fittings are made and results show the need to resort to higher-order regression equations in order to get better concordances between theoretical results and experimental available data. Besides, topological indices computed from maximum order distances seems to yield rather satisfactory predictions of heats of formation for hydrocarbons. Keywords: QSPR theory, Detour matrix, Enthalpy of formation, Hydrocarbons, Topologi- cal indices 1. Introduction Graphs have found considerable employment in several chemistry fields, particularly in modeling molecular structure [1-10]. The applications of graphs to the study of structure-property relationships implies the representation of molecules by selected molecular descriptors, often referred to as topo- logical indices [11,12]. These topological indices, which often have a direct structural interpretation, are defined in terms of selected structural parts and hopefully should help one in building molecular models for structure-property relationships. Among hundreds of possible descriptors a few have arisen again and again as the most useful for characterization of molecules [13-16] Int. J. Mol. Sci. 2001, 2 122 The graph theoretical characterization of molecular structure is realized by means of various matri- ces, polynomials, spectra, spectral moments, sequences counting distances, paths and walks. The mo- lecular matrices represent an important source of structural descriptors. Usually, a small number of matrices is used to characterize the molecular topology, namely the adjacency, the distance and some- times, the Laplacian matrix. Novel matrices were developed in recent years, encoding in various ways the topological information [17]. However, distance matrices remain being the most relevant ones within the realm of the Quantitative Structure Activity (Property) Relationships (QSAR/QSPR) theory. Any matrix, the elements of which satisfy the axioms of distance, can be referred to as a distance matrix D ≡ {Dij} [18]. The axioms of distance require that a) Distance is a positive quantity, Dij ≥ 0, assigned to a pair of elements (points in an n- dimensional vector space). b) Dii = 0 ∀i = 1, 2, ......N; N = number of elements c) Distance does not depend on the direction of measurement, i.e. Dij = Dji d) Distance satisfies the triangular inequality, i.e. Dij ≤ Dik + Dkj Two distance matrices are particularly important, both of them based on the topological distance for vertices within a graph: the distance matrix and the detour matrix. The detour matrix, together with the distance matrix, was introduced into the mathematical literature in 1969 by Frank Harary [19]. Both matrices were also briefly discussed in 1990 by Buckley and Harary [20]. The detour matrix was introduced into the chemical literature in 1994 under the name "the maximum path matrix of a molecu- lar graph" [21,22]. The graph-theoretical detour matrix have found some interest in chemistry [23] and a valuable variation pertaining to the definition of the diagonal elements was proposed by Rücker and Rücker [24]. They found this matrix in combination with the Wiener index W is more useful than Ho- soya's Z index for regression of the boiling points of a large sample of compounds containing all acyclic and cyclic alkanes with known boiling points from methane up to polycyclic octanes. In three previous papers [25-27] we have analyzed the relative merits of these distances when they are used to define molecular descriptors in order to calculate physical chemistry properties within the realm of QSAR/QSPR theory. The aim of this paper is to continue with this sort of discussion resorting to the calculation of heats of formation for a representative set of 60 hydrocarbons. The paper is organized as follows: next section deals with some basic mathematical definitions and the computational procedure; then we present the results and discuss them; and finally we state the conclusion of the present study as well as some possible extensions. 2. Basic Definitions The adjacency matrix: The adjacency matrix A = A(G) of a graph G with N vertices is the square N x N symmetric matrix whose entry in the i th row and j th column is defined as [19] Int. J. Mol. Sci. 2001, 2 123 1 if i ≠ j and (i,j) ∈ E(G) Aij = (1) 0 if i = j or (i,j) ∉ E(G) where E(G) represents the set of edges of G. The sum of entries over row i or column j in A(G) is the degree of vertex i, degi. An example of molecular graph and the adjacency matrix is given below for the 1-ethyl-2- methylcyclopropane. The vertices and edges are labeled from 1 to 6 and from a to f, respectively. e 5 4 d 1 a c f 6 b 2 3 0 1 1 1 0 0 1 0 1 0 0 1 1 1 0 0 0 0 A = 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 The distance matrices The distance matrix D(G) can be defined for G with elements Dij, the distance, or the number of least steps from vertex i to vertex j. Similarly, the detour matrix ∆(G) of a labeled connected graph G is a real symmetric N x N matrix whose (i,j) entry is the length of the longest path from vertex i to ver- tex j (i.e., the maximum number of edges in G between vertices i and j [23]). For example, for the previous graph corresponding to 1-ethyl-2-methyl cyclopropane molecule, ma- trices A and ∆ are: 0 1 1 1 2 2 0 2 2 1 2 3 1 0 1 2 3 1 2 0 2 3 4 1 1 1 0 2 3 2 2 2 0 3 4 3 A = 1 2 2 0 1 3 ∆ = 1 3 3 0 1 4 2 3 3 1 0 4 2 4 4 1 0 5 2 1 2 3 4 0 3 1 3 4 5 0 Int. J. Mol. Sci. 2001, 2 124 Topological molecular descriptors We present the basic definitions related to the topological descriptors chosen for the present study. Wiener index [28] W = 0.5 ∑ Dij (2) i,j -1 Harary index [29] H = 0.5 ∑ Dij (3) i,j MTI index [30, 31] MTI = ∑ ei (4) i where ei are elements of the row matrix v[A + D] = [e1, e2, e3, ...., eN]. v is the so-called valence ma- trix. -1/2 Balaban index [32, 33] J = [q/(µ + 1)] ∑ (di dj) (5) i,j where di = ∑ Dij, q is the number of edges and µ the number of cycles in the graph. The j summations in formulas (2), (3) and (5) are over all edges i-j in the hydrogen-depleted graph. 0 -1/2 Zero order connectivity index [34,35] χ = ∑ (δi) (6) i where δi = ∑ Aij is the degree of the i-th vertex. j 1 -1/2 Randic's connectivity index [32] χ = ∑ (δiδj) (7) i, j h -1/2 Generalized connectivity index [35] χ = ∑ (δvo δv1 ....δvh) (8) paths where the summation is taken over all possible paths of lengths 0, 1, ...., h. 2 The Zagreb group indices [36,37] M1 = ∑ δi (9) i Int. J. Mol. Sci. 2001, 2 125 M2 = ∑ δiδj (10) i,j The employment of these topological descriptors has proven to be extremely useful in QSAR/QSPR studies giving simple correlations between the selected properties and the molecular structure [39-40]. Multiple regression analysis is often employed in such studies in the hope that it might point to struc- tural factors that influence a particular property. Naturally, regression analysis results do no establish any sort of causal relationships between structural components and molecular properties. However, it may be helpful in model building and be useful in the design of molecules with prescribed desirable properties [41]. An interesting alternative to the previous definitions based upon distance matrix D is replace it by the detour matrix ∆, defining the associated topological indices W*, H*, etc. on the basis of this last matrix in a similar fashion as done in Eqs. (2-5). 3. Results and Discussion 0 1 2 We have employed two sets of topological descriptors; a) {Nc, χ, χ, χ, M1, M2, W, H, H, J, 0 1 2 MTI} and b) {Nc, χ, χ, χ, M1, M2, W, H, J, MTI, W*, H*, J*, MTI*}. Nc stands for the number of C atoms. This particular choice was made since we want to know the relative merits of topological in- dices defined on the basis of the two distance matrices (i.e. D and ∆). A way to assess it is via this choice, although there are other options. The molecular set comprises 60 hydrocarbons containing from 1 to 18 carbon atoms and they are presented in Table 1 together with their corresponding topological parameters.
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