Algebraic connectivity: local and global maximizer graphs

Algebraic connectivity is one way to quantify graph connectivity, which in turn gauges robustness as a network. In this paper, we consider the problem of maximising algebraic connectivity both local and globally over all simple, undirected, unweighted graphs with a given number of vertices and edges. We pursue this optimization by equivalently minimizing the largest eigenvalue of the Laplacian of the 'complement graph'. We establish that the union of complete subgraphs are largest eigenvalue "local" minimizer graphs. Further, under sufficient conditions satisfied by the edge/vertex counts we prove that this union of complete components graphs are, in fact, Laplacian largest eigenvalue "global" maximizers; these results generalize the ones in the literature that are for just two components. These sufficient conditions can be viewed as quantifying situations where the component sizes are either 'quite homogeneous' or some of them are relatively 'negligibly small', and thus generalize known results of homogeneity of components. We finally relate this optimization with the Discrete Fourier Transform (DFT) and circulant graphs/matrices.