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The Laplacian energy of a graph is the sum of the distances of theeigenvalues 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 isconjectured to be attained for threshold graphs. We prove the conjecture tohold for graphs with the property that for each $k$ there is a threshold graphon the same number of nodes and edges whose sum of the $k$ largest Laplacianeigenvalues exceeds that of the $k$ largest Laplacian eigenvalues of the graph.We call such graphs spectrally threshold dominated. These graphs include splitgraphs and cographs and spectral threshold dominance is preserved by disjointunions and taking complements. We conjecture that all graphs are spectrallythreshold dominated. This conjecture turns out to be equivalent to Brouwer'sconjecture concerning a bound on the sum of the $k$ largest Laplacianeigenvalues.

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