Vertex Degree Weighted Path Indices

Abstract

Vertex degree weighted path indices P N (a, b, ...), for example P 1 (a, b), P 2 (a, b, c), P 3 (a, b, c, d), and P 4 (a, b, c, d, e), are good topological indices for some of the physicochemical properties of octanes with |R| max up to 0.999. Mutually optimized combinations of them are even better, R (P 1 ..P 4 ) is in the worst tested case > 0.9. The best results are obtained in mutually optimized combinations of some of them with some of the elements of the Universal matrix, in the worst tested case R = 0.994. Some of the vertex degree weighted path one indices give rise to a »regular« sequence of octane isomers of increasing branching.

Usefulness of Combinations of Vertex-Degree Weighted Path Indices and Elements of a Universal Matrix

“…are the vertex degrees. 9 The vertex-degree vertex-distance weighted elements of the Universal matrix are defined as u ij (a ij , b ij , c ij ) = v i a ×v j b ×d ij c , where v i and v j are the vertex degrees and d ij c is the distance between the vertices i and j. 14 In the text, they are often presented in a shorthand form as PN and u ij , respectively.…”

“…The path indices 1,2 and the connectivity indices [5][6][7][8] were previously considered to be two different groups of indices. It has been, however, shown 9 that both groups of them are parts of the vertex-degree weighted path indices PN(a, b, ...). The mutually optimized combinations of vertex-degree weighted path indices gave rise in the case of 29 physicochemical properties of octanes to R (P1..P4) > 0.9.…”

“…The mutually optimized combinations of vertex-degree weighted path indices gave rise in the case of 29 physicochemical properties of octanes to R (P1..P4) > 0.9. 9 In the mutually optimized combinations with some of the elements of the Universal matrix the values were R = 0.994 or higher. Ivanciuc 10,11 described the Dval matrix and presented its characteristics.…”

Introduction of branching degrees of octane isomers

“…Branching degrees of octane isomers were not intended to represent a new index but to be only a quantitative illustration of the degree of branching of octane isomers as it is felt by a group of their physicochemical properties. Previous 12 results have shown that topological indices, which describe the »regular« sequence of octane isomers, are not good indices for physicochemical properties of octane isomers. This is reflected in Table 2 also for the branching degrees of octane isomers, where only the Pitzer's acentric factor ω exceeds the lower limit of usefulness of R = 0.99.…”

“…Later 8 were presented »regular«, in part intuitively derived sequences of octane isomers of increasing branching, one of them being Oct < 2M7 < 3M7 < 4M7 < 3Et6 < 25M6 < 24M6 < 23M6 < 34M6 < 3Et2M5 < 22M6 < 33M6 < 3Et3M5 < 234M5 < 224M5 < 223M5 < 233M5 < 2233M4, as well as the indices derived from a simplified version of the Universal matrix giving rise to such »regu-lar« sequences of octane isomers. Such a »regular« sequence is observed among some of the indices [9][10][11][12] V(a, b, c) ≡ V wm (a, b, c) and V L (a, b, c) as well as among some of the vertex degree weighted path one indices.…”