A1) Kagome would never fall in love with Sesshomaru, because he hates humans, 2) He can be mature unlike Inuyasha, but he's so very ruthless. 3) He tricked Inuyasha (his own half younger brother) to demand the location of their father's (Great Dog Demon) grave, to find their father's incredible and powerful sword: Tessaiga for himself. Sesshomaru used this demon called "The Un-Mother" to take the form of Inuyasha's deceased human mother, Izayoi to fool Inuyasha to demand the path to their father's tomb. 4) In ep. 7; Sesshomaru at first, tried to kill Kagome with his toxic claws, because only she pull out the Tessaiga from the stone, since he and Inuyasha can't & another reason why because she's human not a demon, and the Great Dog Demon used the Tetsusaiga to protect humans (like Izayoi - his human lover/wife & Inuyasha's mom). 5) Kagome knows that Inuyasha deserves the Tessagia not Sesshomaru. 6) Kagome dislikes Sesshomaru for his cruel deeds towards Inuyasha, herself and other humans too. 7) Plus, she and Sesshomaru are enemies since they first met.