This chapter investigates the algebraic properties of matrix-weighted graphs, emphasizing how matrix-valued edge weights generalize corresponding results in scalar-weighted graphs. A large portion of this chapter is devoted to the study of the matrix-weighted Laplacian. In undirected graphs, the associated properties extend many well-established results from the scalar-weighted setting. For directed graphs, the discussion focuses on specific graph topologies such as acyclic leader–follower, generalized balance, and directed cycles. The joint influence of graph topologies and values of the matrix weights on the kernel of the matrix-weighted Laplacian are established and illustrated via numerical examples.