Characterizing Rings via Domination in Compressed Zero-Divisor Graphs and Applications to Network Optimization

Compressed zero-divisor graphs provide a symmetry-preserving representation of the algebraic structure of finite commutative rings by merging vertices with identical annihilator behavior. This compression highlights structural regularities and reduces graph complexity while retaining essential algebraic information. Although domination parameters have been widely studied in classical graph theory, their role in characterizing rings through compressed zero-divisor graphs remains largely unexplored. This work investigates how domination numbers and related variants encode algebraic properties of rings and how these parameters can be translated into efficient models for real-world network optimization.

Methods:
For several families of finite commutative rings, we construct compressed zero-divisor graphs using equivalence classes defined by identical annihilator sets. We compute classical domination, Roman domination, and double Roman domination numbers and analyze how these values correlate with ring-theoretic features such as annihilator chains, idempotent decompositions, and local ring behavior. Exact results are obtained through structural lemmas and combinatorial reductions, supported by computational verification using integer linear programming formulations.

Results:
The study shows that domination parameters in compressed zero-divisor graphs serve as effective invariants for distinguishing ring structures. Rings with symmetries induced by annihilator equivalence display predictable domination patterns, while irregular annihilator distributions yield unique domination signatures. Furthermore, compressed graphs significantly reduce computational complexity without loss of domination-relevant information.

Conclusion:
The results demonstrate that domination in compressed zero-divisor graphs provides a novel framework for characterizing finite rings and for modeling optimal resource deployment in symmetric network environments. These findings bridge algebraic graph theory with practical optimization, offering new perspectives for analyzing highly structured networks.