The Alkanes with Maximum Wiener Polarity Index

Abstract

The Wiener polarity index (usually denoted by Wp ) of an alkane is the number of unordered pairs of carbon atoms which are separated by three carbon-carbon bonds. This topological index Wp is useful for predicting the boiling points of alkanes. Deng [MATCH Commun. Math. Comput. Chem. 66 (2011) 305] proved that the maximum Wp value among all alkanes, with n carbon atoms, is 3n-15 . The main purpose of present paper is to find all those alkanes with n carbon atoms, which attain the maximum value of Wp .

Viscosity Prediction of Lubricants by a General Feed-Forward Neural Network

“…(2) 5 topological descriptors, namely, longest chain, longest aliphatic chain, Wiener path number, 61 Wiener polarity number, 61,62 and fragment complexity; 63 (3) 4 geometric descriptors, namely, Petitjean number, 64 eccentricity connectivity index 65 (hereby as eccentricity index), McGowan characteristic volume 66 (hereby as McGowan volume), and VABC; 67 and (4) 2 hydrophobic descriptors, namely, Crippen log P 68 and Crippen MR. 69 Temperature, which we understand to be inversely related to viscosity, is included as the 15th feature. We explain our selection of descriptors: (1) molecular weight: a power-law relationship is exhibited between polymer viscosity and its entanglement molecular weight; 70 as polymer entanglement is difficult to measure or calculate, the molecular weight is used instead, (2) rotatable bonds and rotatable bond fraction: the ease of rotation of the backbone bonds in polymers (i.e., chain flexibility) affects the degree of entanglement of polymer chains, 71 and hence the viscosity of the polymers, (3) the longest chain and longest aliphatic chain: the polymer viscosity is directly proportional to its chain length, 72 (4) Wiener path number, Wiener polarity number, and fragment complexity: these topological descriptors have a strong correlation with viscosity, 73,74 (5) Petitjean number and eccentricity index: distance-based descriptors often exhibit a high degree of predictability of physicochemical properties such as viscosity, 75 (6) McGowan volume and VABC: most physicochemical properties such as viscosity are size-related; 66 the McGowan volume and VABC are variants to calculate the van der Waals volume, (7) Crippen log P and Crippen MR: the solute hydrophobicity in two immiscible or partially miscible solvents as described by the Crippen indices relates to the interaction strength between the solute and the solvents 68 and therefore viscosity.…”

“…A total of 15 descriptors (abbreviated as D1–D15) are selected as the first pool of features to train the neural network to predict the dynamic viscosity. They include 14 molecular descriptors: (1) 3 constitutional descriptors, namely, molecular weight, number of rotatable bonds, and rotatable bond fraction; (2) 5 topological descriptors, namely, longest chain, longest aliphatic chain, Wiener path number, Wiener polarity number, , and fragment complexity; (3) 4 geometric descriptors, namely, Petitjean number, eccentricity connectivity index (hereby as eccentricity index), McGowan characteristic volume (hereby as McGowan volume), and VABC; and (4) 2 hydrophobic descriptors, namely, Crippen log P and Crippen MR . Temperature, which we understand to be inversely related to viscosity, is included as the 15th feature.…”

A Note on the Minimum Wiener Polarity Index of Trees with a Given Number of Vertices and Segments or Branching Vertices

“…e index W p for a graph G is defined as the number of unordered pairs of those vertices of G that are at distance 3. In the previous decade, W p has attracted much attention from researchers; for example, see the surveys [6,7], papers [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25], and related references therein.…”

Towards the Solution of an Extremal Problem Concerning the Wiener Polarity Index of Alkanes

“…Detail about the chemical applications of W p can be found in the papers [7,16,19,23,27,[29][30][31][32]. Considering the importance of this topological index, many researchers have devoted their attention towards it and studied its mathematical properties, for example see the papers [4,6,15,21,22,24,25,33,34] and references listed therein.…”

“…Deng et al in [5] determined the maximum and minimum W p value of the chemical trees on n vertices with n ≥ 4. For, further references on extremal results on W p we refer the reader to the references [1,2,6,10,17].…”