Maximizing the smallest eigenvalue of grounded Laplacian matrices via edge addition

Abstract

The smallest eigenvalue of the grounded Laplacian matrix holds pivotal significance across various practical scenarios, particularly in characterizing the convergence rate of leader–follower networks in multi-agent systems, with a larger smallest eigenvalue indicating a faster convergence rate. This paper focuses on maximizing the smallest eigenvalue of the grounded Laplacian matrix via adding edges for both undirected and directed networks. For undirected networks, under intuitive conditions, we prove that adding one edge between two vertices that correspond to the smallest and largest eigenvector components for the smallest eigenvalue will maximize the smallest eigenvalue of the grounded Laplacian matrix. In addition, the discussion is extended to the case of multiple edge addition, where a suboptimal algorithm is proposed to maximize the eigenvalue with a low time complexity. For directed networks, when fixing a vertex $i$ and adding an edge pointing to it, choosing the vertex, if there is any, that has the smallest eigenvector component than that of $i$ leads to the maximal increase of the smallest eigenvalue. We apply the derived results to the distributed neighbor selection for directed networks.