This work explores the properties of the edge variant of the graph Laplacian in the context of the edge agreement problem. We show that the edge Laplacian, and its corresponding agreement protocol, provides a useful perspective on the well-known node agreement, or the consensus algorithm. Specifically, the dynamics induced by the edge Laplacian facilitates a better understanding of the role of certain subgraphs, e. g., cycles and spanning trees, in the original agreement problem. Using the edge Laplacian, we proceed to examine graph-theoretic characterizations of the H-2 and H-infinity performance for the agreement protocol. These results are subsequently applied in the contexts of optimal sensor placement for consensus-based applications. Finally, the edge Laplacian is employed to provide new insights into the nonlinear extension of linear agreement to agents with passive dynamics.