Abstract
Each 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 descriptors of a physicochemical property in question. Some combinations of matrix elements are very good descriptors of physicochemical properties of octanes regardless how we enumerate their vertices.
Highlights
To check whether the topological indices derived in the vertex-degree vertex-distance weighted way are invariants or not, they were compared first for the case of one of the octanes with the quite nonsymmetric structure, the 2,3-dimethylhexane (23M6) – Structures (I), (II), and (III)
The ABwm(a,b,c) and ABL(a,b,c) index[14] using non-diagonal elements of the Universal matrix, which contain the mutual contributions of interior vertices only
The Cwm(a,b,c) and CL(a,b,c) index[14] using nondiagonal elements of the Universal matrix, which contain the mutual contributions of one interior and one terminal vertex
Summary
There is a general opinion that the molecular descriptors (topological indices) must be graph invariants.[1,2,3,4,5,6,7] on inclusion of vertex-degree vertex-distance weighted indices,[8,9,10,11,12,13,14] especially with variation of value of exponents on vertex degrees and vertex distances,[11,12,13,14] there appear situations, where some combinations of exponents give rise to invariance to molecular labeling, whereas other combinations do not. Regarding different labeling ways of graph vertices, particular matrix elements are not invariant to molecular labeling. Some of the matrix elements are the best topological indices of a physicochemical property in question.
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