Algebraic Connectivity: Local and Global Maximizer Graphs

Abstract

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 maximizing algebraic connectivity both locally and globally overall 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">local</i> 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 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">global</i> 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. While a conjecture about global optimality of complete bipartite graphs' from the literature continues to remain open, assuming appropriate constraints we prove the conjecture and also formulate/prove a variant of this claim. We finally relate this central optimization problem in this paper with the Discrete Fourier Transform (DFT) and circulant graphs/matrices.