Constraints on Brouwer's Laplacian spectrum conjecture

Abstract

Brouwer's Conjecture states that, for any graph G, the sum of the k largest (combinatorial) Laplacian eigenvalues of G is at most |E(G)|+(k+12), 1≤k≤n. We present several interrelated results establishing Brouwer's conjecture ▪ for a wide range of graphs G and parameters k. In particular, we show that (1) ▪ is true for low-arboricity graphs, and in particular for planar G when k≥11; (2) ▪ is true whenever the variance of the degree sequence is not very high, generalizing previous results for G regular or random; (3) ▪ is true if G belongs to a hereditarily spectrally-bounded class and k is sufficiently large as a function of k, in particular k≥32n for bipartite graphs; (4) ▪ holds unless G has edge-edit distance <k2n=O(n3/2) from a split graph; (5) no G violates the conjectured upper bound by more than O(n5/4), and bipartite G by no more than O(n); and (6) ▪ holds for all k outside an interval of length O(n3/4). Furthermore, we show that a natural generalization of Brouwer's conjecture surprisingly is quite false: asymptotically almost surely, a uniform random signed complete graph violates the conjectured bound by Ω(n).