Shifted simplicial complexes are Laplacian integral

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On the sum of Laplacian eigenvalues of graphs (2010, background): “� 1(G) 6 |V (G)| and the equality obviously occurs for k = n − 1, n. Moreover, the conjecture has been proved whenever k = 2 [ 3, Theorem 7.1 ] or G is a tree [9].”
Simplicial matrix-tree theorems (2008, methodology): “As in [13], we begin by calculating the algebraic finely weighted eigenvalues of a near-cone in terms of the eigenvalues of its link and deletion with respect to its apex (Proposition 7.”
Spectra of Graphs (2011, background): “Duval & Reiner [159] generalized the Grone-Merris conjecture.”
The Laplacian eigenvalues of graphs: a survey (2011, background): “In 2002, Duval and Reiner [20] investigated the combinatorial Laplace operators associated to the boundary maps in a shifted simplicial complex.”
Spectra of combinatorial Laplace operators on simplicial complexes (2011, methodology): “Therefore, the combinatorial Laplace operator analysed by Duval, Reiner [11], Friedmann [14] and others [29],[9], is the combinatorial Laplace operator Li for the identity matricesas weight matricesWi (−1 ≤ i ≤ dimK), i.”