Quantifying Algebraic Connectivity: Sombor Index and Polynomial in Some Graphs of Commutative Ring Z p.

This work explores the unit graph of the commutative ring Z p for prime p and explores the significance of the Sombor index and the Sombor polynomial in comprehending its structural details. We systematically investigate the unit graph and the identity graph of Z p , exposing subtle patterns and symmetries within its vertices and edges by utilizing the extensive linkages between algebraic structures and graph theory. Using an in-depth examination, we prove the importance of the Sombor index and the Sombor polynomial as indispensable instruments for describing the algebraic and combinatorial characteristics inherent in the graph. Our results clarify the basic characteristics of the unit graph and the identity graph of Z p and highlight its function in expressing the ring's underlying algebraic structure. This work provides an avenue for further investigations into the interplay between algebraic structures and graph-theoretic concepts and also makes contributions to the subject of algebraic graph theory. We clear the path for further research and developments in this fascinating field of study by explaining the complexities regarding the unit and the identity graphs of Z p and emphasizing the significance of the Sombor index and the Sombor polynomial. [ABSTRACT FROM AUTHOR]