Emotions play a critical role in our everyday lives by altering how we perceive, process and respond to our environment. Affective computing aims to instill in computers the ability to detect and act on the emotions of human actors. A core aspect of any affective computing system is the classification of a user's emotion. In this study we present a novel methodology for classifying emotion in a conversation. At the backbone of our proposed methodology is a pre-trained Language Model (LM), which is supplemented by a Graph Convolutional Network (GCN) that propagates information over the predicate-argument structure identified in an utterance. We apply our proposed methodology on the IEMOCAP and Friends data sets, achieving state-of-the-art performance on the former and a higher accuracy on certain emotional labels on the latter. Furthermore, we examine the role context plays in our methodology by altering how much of the preceding conversation the model has access to when making a classification.