With Social Media platforms establishing themselves as the de facto destinations for their customers views and opinions, brands around the World are investing heavily on invigorating their customer connects by utilizing such platforms to their fullest. In this paper, we develop a novel technique for mining conversations in Twitter by weaving together all conversations around an event into one unified graph (Conversation Graph, henceforth). The structure of the Conversation Graph emerges as a variant of the BOWTIE structure (dubbed ASKEWBOWTIE henceforth) as a result of the complex communication patterns amongst these players. Finally, we investigate the structural properties of the ASKEWBOWTIE structure to understand the configuration of the components and their temporal evolution.