In this context, imperiously means arrogantly.
This text is a description of the life and behavior of the protagonist, Buck, at the beginning of The Call of the Wild by Jack London. We are told that the dog lives comfortably with his master's family and perceives himself as being their favorite pet. In the quote, Buck's inflated ego ("king") comes across as arrogant because he "utterly ignores" the other dogs, which indicates that he thinks he is too important to be concerned by them.
