Decoder-Transformers have achieved remarkable success and have laid the groundwork for the development of Large Language Models (LLMs). At the core of these models is the self-attention matrix, which allows different tokens to interact with each other. This process is remarkably similar to the message-passing mechanism used in Graph Neural Networks (GNNs), and as such decoder-Transformers suffer many of the optimization difficulties studied extensively in the GNN literature. In this paper, we present a unified graph perspective that bridges the theoretical understanding of decoder-Transformers and GNNs. We systematically examine how well-known phenomena in GNNs, such as over-smoothing and over-squashing, directly manifest as analogous issues like rank collapse and representational collapse in deep Transformer architectures. By interpreting Transformers' self-attention as a learned adjacency operator, we reveal shared underlying principles governing signal propagation and demonstrate how insights from one field can illuminate challenges and solutions in the other. We analyze the role of architectural components like residual connections, normalization, and causal masking in these issues. We aim to provide a framework for understanding how information flows through deep learning models that perform sequence mixing through an adjacency operator, and to highlight areas for cross-pollination of research, as well as to provide a comprehensive reference for researchers interested in the underpinnings of these architectures.