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The minimal-ABC trees with B1-branches.

Darko Dimitrov1, Zhibin Du2, Carlos M da Fonseca3

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The atom-bond connectivity (ABC) index characterizes molecular topology. This study shows minimal-ABC trees do not contain 4 or 3 B1-branches, advancing tree characterization.

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Area of Science:

  • Chemical graph theory
  • Mathematical chemistry
  • Molecular structure analysis

Background:

  • The atom-bond connectivity (ABC) index is a key molecular descriptor linking chemistry and graph theory.
  • It is widely applied in predicting bioactivity, chemical thermodynamics, and analyzing various chemical structures.
  • Characterizing trees with minimal ABC index remains an open problem with significant research interest.

Purpose of the Study:

  • To advance the characterization of trees with minimal atom-bond connectivity (ABC) index.
  • To investigate the structural properties of trees that minimize the ABC index.
  • To contribute to solving the open problem of minimal ABC index trees.

Main Methods:

  • Graph theory principles applied to molecular structures.
  • Analysis of tree structures and their topological indices.
  • Investigating the impact of specific graph features (B1-branches) on the ABC index.

Main Results:

  • Demonstrated that minimal-ABC trees cannot possess 4 B1-branches.
  • Proved that minimal-ABC trees cannot possess 3 B1-branches.
  • Initiated the analysis of minimal-ABC trees with 2 B1-branches.

Conclusions:

  • Significant progress made towards the full characterization of minimal-ABC trees.
  • Established constraints on the number of B1-branches in trees with the minimum ABC index.
  • Opens avenues for further research into the structural properties of extremal trees in chemical graph theory.