Extremal (n,m)-Graphs w.r.t General Multiplicative Zagreb Indices

Area of Science:

• Chemical graph theory
• Mathematical chemistry
• Computational chemistry

Background:

• Topological indices quantify molecular properties.
• Multiplicative Zagreb indices are key graph invariants.
• Previous work by Xu et al. introduced graph transformations for extremal graphs.

Purpose of the Study:

• Extend Xu's graph transformations to general multiplicative Zagreb indices.
• Characterize extremal graphs for these indices.
• Apply transformations to acyclic, unicyclic, and bicyclic graphs.

Main Methods:

• Utilizing graph transformations that increase or decrease general multiplicative Zagreb indices.
• Systematically applying transformations to identify extremal graph structures.
• Analyzing the impact of transformations on molecular graph properties.

Main Results:

• Demonstrated that specific graph transformations alter the first and second general multiplicative Zagreb indices.
• Identified extremal acyclic, unicyclic, and bicyclic graphs based on these indices.
• Established a method for finding graphs with maximal and minimal general multiplicative Zagreb indices.

Conclusions:

• Graph transformations effectively modify general multiplicative Zagreb indices.
• Extremal acyclic, unicyclic, and bicyclic graphs were characterized.
• The study provides a framework for analyzing topological indices in specific molecular graph classes.