Edward is K's supervisor, and is the one who gave him a "Unit" in order to provide strength. It is unknown whether the two held anything other than a professional relationship, but K seemed to have been determined to fulfill Edward's orders at all costs, meaning that he might have been grateful towards him, at the very least.
In the Anime, Edward, humiliated by Sakuya after seeing K getting beaten by Tooru, started kicking K and berating him for losing. He planned to have K go berserk and destroy everything in the area. However K instead kills him by piercing his chest, and breaks the watch that Edward held, unlocking "Destruction Wings"
Edward appears to have a past with Lilith's family, who was originally a part of the "Unit" research until they suddenly abandoned it, leaving Edward to work on the project himself. He seems to hold a grudge against them for that, and swore to prove that they made a mistake in leaving him by creating a "god-killing army".
During the meeting in Volume 4, Edward appeared to have a mutual relationship towards Sakuya, offering a partnership between the two in order to achieve "Absolute Duo". However it later becomes apparent that he has a strong dislike towards not only Sakuya, but the entire Tsukumo family, which Sakuya mentions is because her grandfather's research surpassed his. Edward swore to prove that he was the better researcher by creating a "god-killing army"
Edward first appears in front of Miyabi as a kind old man who offered to give her power to help her with her problems. However it was just a ruse to use her as a test subject for a special "Unit" called "El Libel"