Reducing Degeneracy of Degree-Based Topological Indices via Reciprocal Composite Descriptors: Extremal Theory and Exhaustive Enumeration

Abstract

Background:

Degree-based topological indices are important numerical descriptors of chemical graphs and are widely used in quantitative structure–property relationship and quantitative structure–activity relationship studies. However, many classical indices suffer from degeneracy, since non-isomorphic graphs may share the same descriptor value, thereby reducing their discriminative power.

Materials and Methods:

This study investigates reciprocal composite descriptors of the form

with particular emphasis on the reciprocal composite second Zagreb descriptor

where

The work combines extremal graph theory, edge-transfer transformations, and exhaustive computational enumeration of connected non-isomorphic graphs up to order . Sharp lower and upper bounds are established for connected, unicyclic, and -cyclic graph classes.

Results:

    The extremal analysis shows that minimum values are attained by paths and cycles, whereas maximum values are attained by graphs with high degree concentration. Exhaustive computational enumeration further shows that the reciprocal composite descriptor produces significantly more distinct values than the classical second Zagreb index. For graphs of order , the number of distinct values increases from for to for , and the degeneracy ratio is reduced by approximately .

Conclusion:
   The proposed reciprocal composite descriptors reduce degeneracy while preserving analytical tractability and clear extremal structure. These results indicate that reciprocal structural correction enhances the discriminative power of degree-based topological indices and provides a promising direction for developing more sensitive graph descriptors in chemical graph theory.