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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 $PY\left(p\right)$, prolate rectangle $R_{p,q}$, honeycomb $HC\left(p\right)$, and hexabenzocoronene $HBC\left(p\right)$. This work also extends to the calculation of the domination integrity value for Cyclic Silicate $C{C}_p$ and Chain Silicate $C{S}_p$ chemical structure networks. Full article

It is a well-known fact that a graph of diameter d has at least $d+1$ eigenvalues. A graph is d-extremal (resp. $d_{\alpha }$-extremal) if it has diameter d and exactly $d+1$ distinct eigenvalues (resp. $\alpha$-eigenvalues), and a graph is split if its vertex set can be partitioned into a clique and a stable set. Such graphs have a diameter of at most three. If all vertex degrees in a split graph are either $\tilde{d}$ or $\widehat{d}$, then we say it is $(\tilde{d},\widehat{d})$-bidegreed. In this paper, we present a complete classification of the connected bidegreed $3_{\alpha }$-extremal split graphs using the association of split graphs with combinatorial designs. This result is a natural generalization of Theorem 4.6 proved by Goldberg et al. and Proposition 3.8 proved by Song et al., respectively. Full article