\begin{origine}
I hereby certify that I am the author of this dissertation. To the best of my knowledge, there are not any parts of this research infringing anyone's copyright and intellectual property. The dissertation does not contain any materials previously written or published by other people except appropriate references for the sake of completeness.



I declare that the graphs $ \mathcal{F}^K_{u,n} $ and $ \widehat{\mathcal{F}}^K_{u,n} $ are the new mathematical objects generalized from $ \mathcal{F}_{u,n} $ and $ \widehat{\mathcal{F}}_{u,n} $. Therefore, the results for $ \mathcal{F}^K_{u,n} $ and $ \widehat{\mathcal{F}}^K_{u,n} $ are new. We also investigated some new properties of $ \mathcal{F}_{u,n} $ and $ \widehat{\mathcal{F}}_{u,n} $ including $ \mathcal{F}^K_{u,n} $ and $ \widehat{\mathcal{F}}^K_{u,n} $ and then extended them to the graphs $ \mathcal{G}_{u,n} $ and $ \widehat{\mathcal{G}}_{u,n} $.



This is a true copy of my dissertation including any final corrections approved by the dissertation examining committee and the Graduate School. The dissertation has not been accepted for a degree or diploma at any educational institution or university.
\end{origine}