This chapter focuses on the connectedness of matrix-weighted graphs, one of their most fundamental properties. A matrix-weighted graph with n vertices is connected if and only if the rank of its matrix-weighted Laplacian is \(dn - d\) , where \(d \ge 2\) is the dimension of the edge weights. When this condition is not met, the graph can be decomposed into multiple clusters. However, the rank condition alone does not reveal the internal structure of these clusters. To address this limitation, several algorithms are proposed to identify the graph’s clustering behavior. One approach, based on an adaptation of the Warshall algorithm, uses algebraic computations involving the Laplacian matrix. Another method constructs connected components by merging positive trees, guided by algebraic conditions on the paths that connect them.