Many Thanks for your Posting...
Your Grandmother, Violette Frangie, on the Klimos Tree is Violette Frangie, Born About 1932, Daughter of Fouad Frangie and Bahige Regina of Greek Orthodox Religion, registered and identified in File #477 in the Beirut Province, Beirut 1 Caza, City of Rmeil.
It is unlikely that there is a relation between Violette and our tree in the current generation, but this is not to say that there is no relation many generations back. I can see that your Great Grandfather Fouad's bloodline is El-Badoui, then Youssef, then El-Badoui again, then Sleiman, then Aghnatious, then Moussi, then Antonious, then Youssef, then Girgis - If we used an average age for them all, the names would take you back a very long time... None of those names correlate with names on our tree, however, I have no names for any siblings of my Great Grandfather and above, so it may be possible that a link could be found somewhere along the bloodline which would be amazing if we can find one...
Do you have any more information about the Frangie portions of your family tree at all that may help with identifying a link? If you send me your email, I'm happy to give you access to our tree and see if there's any identifying names that may help identify a link.