On the Symmetry and Domination Integrity of Some Bidegreed Graphs

Graphs are one of the dynamic tools used to solve network-related problems and real-time application models. The stability of the network plays a crucial role in ensuring uninterrupted data flow. A network becomes vulnerable when a node or a link becomes non-functional. To maintain a stable network connection, it is essential for the nodes to be able to interact with each other. The vulnerability of a network can be defined as the level of resistance it exhibits following the failure of communication links. Graphs serve as vital tools for depicting molecular structures, where atoms are shown as vertices and bonds as edges. The domination number quantifies the least number of atoms (vertices) required to dominate the entire molecular framework. Domination integrity reflects the impact of removing specific atoms on the overall molecular structure. This concept is valuable for forecasting fragmentation and decomposition pathways. In contrast to the domination number, domination integrity evaluates the extent to which the molecule remains intact following the removal of reactive or controlling atoms. It aids in assessing stability, particularly in the contexts of drug design, polymer analysis, or catalytic systems. This work focuses on the vulnerability parameter, specifically examining the domination integrity of a specific group of bidegreed hexagonal chemical network systems such as pyrene PY(p), prolate rectangle Rp,q, honeycomb HC(p), and hexabenzocoronene HBC(p). This work also extends to the calculation of the domination integrity value for Cyclic Silicate CCp and Chain Silicate CSp chemical structure networks.