If I be serious:
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Even though I'm into Zed x Syndra, it has some sense. They're power hungry, have similar goals and similar personality. Those may be attractive for each other. In other side, Syndra is beyond power hungry, she wanna becomes power itself. So, I don't think a person like this would be interested in love. She is probably asexual, like Zed. They still have possibility to be ally though.
Veigar and Lulu are known as close friends in lore. But I'm also not sure on this relationship either. Lulu is ok, have possibility, crazy, but ok. On the other hand, Veigar is against weakness. And he probably sees love as weakness.
Graves and TF are beyond close friends IMO.
Graves: Going to chase after escaping from the jail immediately, having so much chance to kill TF but still not pulling the gun (even while he says I'm going to kill this motherfucker when he finds him in second), for a retarded emotional dude, saving TF in the back from a bilgewater dog (According to TF, he goes too emotional and doesn't think about the situations, so he probably had no plan to survive with TF and kill him later), saving TF without hope to survive, saving TF again, looking dead in the eye and feel like it is like the old days.
TF: Trying to save Graves with work friends while he doesn't need to do, saving Graves from drowning while he is afraid of diving a lot. He could get really rich and he didn't have to face with his biggest nightmare. He literally sacrificed his live for Graves. The person who was going to be saved was also Graves who tried to kill him for a lot times (It is the biggest proof IMO). Don't forget, TF is a miser guy. He probably sacrificed and will probably continue to sacrifices a lot people for his benefits.
They're the closest bromance ever or hardcore gay couple.
Gone too deep for Graves x TF because they have the most information.
Cait and Vi are the the closest female friends ever. It is that much. No information.
But don't say me it is not cute and sad at the same time:
http://i.imgur.com/Lpl0Znm.jpg
Also this
http://pre14.deviantart.net/56a7/th/pre/f/2014/039/a/c/_lol__one_for_the_road___by_chisuukei-d75mzsa.png