Degree maximal graphs are Laplacian integral

Merris, Russell

doi:10.1016/0024-3795(94)90361-1

Cited by 122 publications

(70 citation statements)

References 3 publications

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“…This generalizes the case k = 2 considered by Merris [26], who refers to shifted graphs as degree-maximal graphs.…”

Section: Conjecture 12 For Any K-family K the Spectrum S Defined supporting

confidence: 68%

“…Let s(K) or just s denote the -equivalence class of the spectrum of As discussed in the introduction, in the case k = 2, the integrality of s is a result of Kelmans (see [20,Corollary 4.1]). Kelmans' result was independently rediscovered by Merris [26], and elegantly reformulated as in the above theorem. We devote the remainder of this section to its proof.…”

Section: The Main Theoremmentioning

confidence: 94%

“…Merris [26] defined a graph to be degree-maximal (or maximal ) if its degree sequence is majorized by no other graphic partition.…”

Section: Shifted Versus Degree-maximal Complexesmentioning

confidence: 99%

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Shifted simplicial complexes are Laplacian integral

Duval

Reiner

2002

Trans. Amer. Math. Soc.

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Abstract. We show that the combinatorial Laplace operators associated to the boundary maps in a shifted simplicial complex have all integer spectra. We give a simple combinatorial interpretation for the spectra in terms of vertex degree sequences, generalizing a theorem of Merris for graphs.We also conjecture a majorization inequality for the spectra of these Laplace operators in an arbitrary simplicial complex, with equality achieved if and only if the complex is shifted. This generalizes a conjecture of Grone and Merris for graphs.

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“…This generalizes the case k = 2 considered by Merris [26], who refers to shifted graphs as degree-maximal graphs.…”

Section: Conjecture 12 For Any K-family K the Spectrum S Defined supporting

confidence: 68%

Section: The Main Theoremmentioning

confidence: 94%

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Shifted simplicial complexes are Laplacian integral

Duval

Reiner

2002

Trans. Amer. Math. Soc.

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“…. , v n } and edge set E. Let The readers can refer to [9,12] for a few papers that study the Laplacian integrability of certain graphs.…”

Section: Laplacian Spectrum Of Weakly Quasi-threshold Graphsmentioning

confidence: 99%

“…This conjecture has been studied by several researchers and a few partial results are known (for example, see [1,12]). We now prove that this conjecture holds true for all cographs.…”

Section: Grone-merris Conjecture Holds For Cographsmentioning

confidence: 99%

Laplacian Spectrum of Weakly Quasi-threshold Graphs

Bapat

Lal

Pati

2008

Graphs and Combinatorics

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Abstract. In this paper we study the class of weakly quasi-threshold graphs that are obtained from a vertex by recursively applying the operations (i) adding a new isolated vertex, (ii) adding a new vertex and making it adjacent to all old vertices, (iii) disjoint union of two old graphs, and (iv) adding a new vertex and making it adjacent to all neighbours of an old vertex. This class contains the class of quasi-threshold graphs. We show that weakly quasi-threshold graphs are precisely the comparability graphs of a forest consisting of rooted trees with each vertex of a tree being replaced by an independent set. We also supply a quadratic time algorithm in the the size of the vertex set for recognizing such a graph. We completely determine the Laplacian spectrum of weakly quasi-threshold graphs. It turns out that weakly quasi-threshold graphs are Laplacian integral. As a corollary we obtain a closed formula for the number of spanning trees in such graphs. A conjecture of Grone and Merris asserts that the spectrum of the Laplacian of any graph is majorized by the conjugate of the degree sequence of the graph. We show that the conjecture holds for cographs.

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