Speaker
Description
Within Spectral Graph Theory, Brouwer’s Conjecture (BC) is a fundamental problem concerning Laplacian eigenvalues and graph invariants. It proposes a relationship between the sum of the largest Laplacian eigenvalues of a graph and its number of edges, with direct implications for the study of Laplacian energy. More precisely, for a graph ( G = (V, E) ) with ( n = |V| ) vertices and ( m = |E| ) edges, the conjecture states that, for ( k = 1, \ldots, n ), the sum ( S(k) ) of the ( k ) largest eigenvalues of the Laplacian matrix of ( G ) satisfies
S(k) ≤ m + k(k − 1)/2.
In this talk, we discuss the role of Brouwer’s Conjecture in the area of Spectral Graph Theory and describe recent progress on the problem, highlighting results and contributions obtained in this framework.