Spectral threshold dominance, Brouwer's conjecture and maximality of Laplacian energy

Abstract

The Laplacian energy of a graph is the sum of the distances of the eigenvalues of the Laplacian matrix of the graph to the graph's average degree. The maximum Laplacian energy over all graphs on $n$ nodes and $m$ edges is conjectured to be attained for threshold graphs. We prove the conjecture to hold for graphs with the property that for each $k$ there is a threshold graph on the same number of nodes and edges whose sum of the $k$ largest Laplacian eigenvalues exceeds that of the $k$ largest Laplacian eigenvalues of the graph. We call such graphs spectrally threshold dominated. These graphs include split graphs and cographs and spectral threshold dominance is preserved by disjoint unions and taking complements. We conjecture that all graphs are spectrally threshold dominated. This conjecture turns out to be equivalent to Brouwer's conjecture concerning a bound on the sum of the $k$ largest Laplacian eigenvalues.