Positive trees are crucial for partitioning a matrix-weighted graph into small connected components, from which they may be merged into larger clusters. If there exists a positive spanning tree, the matrix-weighted graph will be immediately connected; however, the reverse does not hold. Properties such as distances and product distances of positive spanning trees were studied in existing works; however, there is a lack of a corresponding study for general connected matrix-weighted graphs. This chapter firstly focuses on finding connected spanning subgraphs of a connected graph G with the least possible number of edges, which are termed expanded positive spanning trees. As an expanded positive spanning tree can only be sought via exhaustive searching, two greedy algorithms are proposed to find a heuristic solution, which is referred to as a quasi-positive spanning tree. The proposed algorithms hinge on the dependence of the graph spectrum with the addition or removal of edges and vertices in a matrix-weighted graph. Next, an algebraic version of the matrix-tree theorem is presented for connected matrix-weighted graphs. The theorem relates positive eigenvalues of the matrix-weighted Laplacian with the determinant of the grounded Laplacian. Specific formulas are then derived for path, cycle, and star matrix-weighted graphs. This chapter also provides several formulas on the minimum number of edges in an expanded positive spanning tree which only relies on the number of vertices and the dimension of the matrix weight. An algorithm for generating matrix-weighted graphs with a pre-selected number of clusters is also introduced.