The Grone Merris Conjecture and a Quadratic Eigenvalue Problem 3

Given a graph G = (V, E), we define the transpose degree sequence d T j to be equal to the number of vertices of degree at least j. We define L G , the graph Laplacian, to be the matrix, whose rows and columns are indexed by the vertex set V , whose diagonal entry at v is the degree of v and whose value at a pair (v, w) is −1 if (v, w) ∈ E and 0 otherwise. Grone and Merris conjectured [GM] Conjecture. If λ 1 ,. .. , λ m are the eigenvalues of L G in (weakly) decreasing order, then for any 1 ≤ j ≤ m, we have j l=1 λ l ≤ j l=1 d T l. . The first inequality is well known and the last inequality is indeed always an equality. On the class of threshold graphs, all the inequalities are equalities. The second inequality was proved by Duval and Reiner [DR], and this paper grew out of an attempt to understand their proof. We will say that a graph is semi-bipartite if its vertex set is the union of a clique and an isolated set. We say a semi-bipartite graph is k regular, if every vertex in the isolated set has degree k. Duval and Reiner proved the second inequality by observing that it suffices to prove it for 1-regular semi-bipartite graph. They proved the second inequality by showing it was trivial for all but a few classes of 1 regular semi-bipartite graphs and then solving those cases one by one using a computer algebra system. However, if j is the number of vertices in the clique adjacent to some vertex in the isolated set, then it seems