Topological indices are not necessarily invariant to graph labeling

Abstract

Dedicated to the memory of Prof. Dr. Jurij V. Bren~i~. AbstractEach element of the Universal matrix U (vertex-degree vertex-distance weighted matrix) represents the mutual contribution of two vertices weighted for the vertex degrees and the distance between them. Regarding different labeling ways of graph vertices, particular matrix elements are not invariant to molecular labeling. Regarding the structural features they are invariants since there are only particular vertex combinations representing particular structural features. Some of the matrix elements are the best topological indices of a physicochemical property in question. Some combinations of matrix elements are very good topological indices of physicochemical properties of octanes regardless how we enumerate their vertices.

Elements of an Universal Matrix as Topological Indices for Physicochemical Properties of Octanes

“…It has been demonstrated that although particular elements of the Universal matrix are not invariant to molecular labelling, they are invariant regarding the structural features of octanes, and the topological indices, which are not invariant to molecular labelling give rise to better correlations than the topological indices, which are invariant to molecular labeling. 18 For this reason, the elements of the Universal matrix and their mutually optimized combinations have been systematically studied and the results are presented here.…”

Vertex Degree Weighted Path Indices

“…Later 21 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 »regular« sequences of octane isomers. Such a »regular« sequence is observed among some of the indices 10,11,22 …”

Introduction of branching degrees of octane isomers

“…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.…”