  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.
