Algebraic connectivity: local and global maximizer graphs

TL;DR

This paper investigates how to maximize algebraic connectivity in graphs by analyzing local and global optimal structures, establishing conditions for optimality, and relating the problem to spectral properties and circulant graphs.

Contribution

It characterizes local and global maximizers of algebraic connectivity in graphs, generalizing previous results to multiple components and linking the problem to spectral graph theory and DFT.

Findings

Union of complete subgraphs are local maximizers.

Under certain conditions, these unions are also global maximizers.

Results extend to multiple components and relate to circulant 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 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.