Some domination type invariants of lucas cubes

One of the reasons interconnection networks work is for the communication needs of multiple computers. Interconnection networks are mathematically represented as a graph $G=(V(G),E(G))$ with vertex set $V(G)$ and edge set $E(G)$. The $V(G)$ cluster represents the processors and the $E(G)$ cluster represents the communication networks. The hypercube with an $n$ dimensional that we can take as a basis for connection network models is the $Q_n$ diagram. While the vertices of the hypercube are represented by all binary sequences of length $n$, the edge set is obtained by matching vertices that differ by one bit. The Fibonacci cube with an $n$ dimensional $\Gamma_n$ is constructed by subtracting all vertices containing a consecutive one from the vertex set of $Q_n$, while the Lucas cube $ \Lambda_n$ is formed by subtracting the vertices that are one at the beginning and end of the $\Gamma_n$ diagram. In the literature, Lucas cubes and some domination type invariants have been studied and the domination numbers are known. In addition, there are also domination numbers that have not been studied so far. The signed domination number and the paired domination number and Roman type domination problem have not been studied in Lucas cubes before. In this thesis, unknown domination type invariants are discussed in five different Lucas cubes: Paired domination number, Signed domination number, Roman domination number, weak Roman domination number and double Roman domination number.By using integer linear programming problems, these dominance numbers are calculated as $n\leq 9$, $n\leq 10$ or $n\leq11$ in Lucas cubes, depending on the difficulty of the algorithm of the dominance type invariant and for uncomputable dimensions, the best lower and upper bounds were found up to $n\leq 13$.