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On sum of powers of Laplacian eigenvalues and Laplacian Estrada index of graphs
 MATCH Commun. Math. Comput. Chem
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A Common Recursion For Laplacians of Matroids and Shifted Simplicial Complexes
 DOCUMENTA MATH.
, 2005
"... A recursion due to Kook expresses the Laplacian eigenvalues of a matroid M in terms of the eigenvalues of its deletion M − e and contraction M/e by a fixed element e, and an error term. We show that this error term is given simply by the Laplacian eigenvalues of the pair (M −e,M/e). We further show ..."
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A recursion due to Kook expresses the Laplacian eigenvalues of a matroid M in terms of the eigenvalues of its deletion M − e and contraction M/e by a fixed element e, and an error term. We show that this error term is given simply by the Laplacian eigenvalues of the pair (M −e,M/e). We further show that by suitably generalizing deletion and contraction to arbitrary simplicial complexes, the Laplacian eigenvalues of shifted simplicial complexes satisfy this exact same recursion. We show that the class of simplicial complexes satisfying this recursion is closed under a wide variety of natural operations, and that several specializations of this recursion reduce to basic recursions for natural invariants. We also find a simple formula for the Laplacian eigenvalues of an arbitrary pair of shifted complexes in terms of a kind of generalized degree sequence.
On the sum of Laplacian eigenvalues of graphs
"... Let k be a natural number and let G be a graph with at least k vertices. A.E. Brouwer conjectured that the sum of the k largest Laplacian eigenvalues of G is at most e(G) + () k+1 2 where e(G) is the number of edges of G. We prove this conjecture for k = 2. We also show that if G is a tree, then the ..."
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Let k be a natural number and let G be a graph with at least k vertices. A.E. Brouwer conjectured that the sum of the k largest Laplacian eigenvalues of G is at most e(G) + () k+1 2 where e(G) is the number of edges of G. We prove this conjecture for k = 2. We also show that if G is a tree, then the sum of the k largest Laplacian eigenvalues of G is at most e(G) + 2k − 1.
Near Threshold Graphs
"... A conjecture of Grone and Merris states that for any graph G, its Laplacian spectrum, Λ(G), is majorized by its conjugate degree sequence, D ∗ (G). That conjecture prompts an investigation of the relationship between Λ(G) and D ∗ (G), and Merris has characterized the graphs G for which the multisets ..."
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A conjecture of Grone and Merris states that for any graph G, its Laplacian spectrum, Λ(G), is majorized by its conjugate degree sequence, D ∗ (G). That conjecture prompts an investigation of the relationship between Λ(G) and D ∗ (G), and Merris has characterized the graphs G for which the multisets Λ(G) and D ∗ (G) are equal. In this paper, we provide a constructive characterization of the graphs G for which Λ(G) and D ∗ (G) share all but two elements. 1