Brute-force search and Warshall algorithms for matrix-weighted graphs

Although research on the control of networked systems has grown considerably, graph-theoretic and algorithmic studies on matrix-weighted graphs remain limited. To bridge this gap in the literature, this work introduces two algorithms-the brute-force search and the Warshall algorithm-for determining connectedness and clustering in undirected matrix-weighted graphs. The proposed algorithms, which are derived from a sufficient condition for connectedness, emphasize a key distinction between matrix-weighted and scalar-weighted graphs. While the existence of a path between two vertices guarantees connectedness in scalar-weighted graphs, connectedness in matrix-weighted graphs is a collective contribution of all paths joining the two vertices. Proofs of correctness and numerical examples are provided to illustrate and demonstrate the effectiveness of the algorithms.