The polytope of degree sequences

Peled, Uri N.; Srinivasan, Murali K.

doi:10.1016/0024-3795(89)90470-9

Cited by 38 publications

(20 citation statements)

References 10 publications

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“…Peled and Srinivasan [29,Theorem 5.8] proved that a graph is degreemaximal if and only if it is threshold. But one of several equivalent definitions of threshold graphs is easily seen to be the definition of shifted graphs [8,Corollary 1A].…”

Section: Proposition 92 a Graph Is Degree-maximal If And Only If Itmentioning

confidence: 99%

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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Section: Proposition 92 a Graph Is Degree-maximal If And Only If Itmentioning

confidence: 99%

Shifted simplicial complexes are Laplacian integral

Duval

Reiner

2002

Trans. Amer. Math. Soc.

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“…Let

Ω_{n}

be the convex hull of all degree sequences of the simple graphs of order

n

. Then the extreme points of the polytope

Ω_{n}

are exactly the degree sequences of threshold graphs of order

n

(for another proof see ). Second, a nonnegative integer sequence is graphical if and only if it is majorized by the degree sequence of some threshold graph .…”

Section: Introductionmentioning

confidence: 99%

“…Then the extreme points of the polytope

Ω_{n}

are exactly the degree sequences of threshold graphs of order

n

(for another proof see ). Second, a nonnegative integer sequence is graphical if and only if it is majorized by the degree sequence of some threshold graph . Third, a graphical sequence has a unique labeled realization if and only if it is the degree sequence of a threshold graph [10, p. 72].…”

Section: Introductionmentioning

confidence: 99%

The minimum number of Hamilton cycles in a Hamiltonian threshold graph of a prescribed order

Qiao

Zhan

2019

Journal of Graph Theory

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We prove that the minimum number of Hamilton cycles in a Hamiltonian threshold graph of order n is 2 ⌊ ( n − 3 ) ∕ 2 ⌋ and this minimum number is attained uniquely by the graph with degree sequence n goodbreakinfix− 1 goodbreakinfix, n goodbreakinfix− 1 goodbreakinfix, n goodbreakinfix− 2 , … , ⌈ n ∕ 2 ⌉ goodbreakinfix, ⌈ n ∕ 2 ⌉ , … , 3,2 of n goodbreakinfix− 2 distinct degrees. This graph is also the unique graph of minimum size among all Hamiltonian threshold graphs of order n.

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“…Peled and Srinivasan [12] pursued this geometric perspective and investigated the relation to the polytope obtained by taking the convex hull of all the finitely many degree sequences in R n . The drawback of their construction was that not all integer points in this polytope corresponded to degree sequences.…”

Section: Undirected Graphsmentioning

confidence: 99%

“…These two fundamental results have led to a thorough understanding of degree sequences. A geometric perspective on degree sequences, pioneered by Peled and Srinivasan [12], turns out to be particularly fruitful. Here, degree sequences of graphs on n labelled nodes are identified with points in Z n and the collection of all such degree sequences can be studied by way of polytopes of degree sequences.…”

Section: Introductionmentioning

confidence: 99%

On degree sequences of undirected, directed, and bidirected graphs

Gellert

Sanyal

2017

European Journal of Combinatorics

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Abstract. Bidirected graphs generalize directed and undirected graphs in that edges are oriented locally at every node. The natural notion of the degree of a node that takes into account (local) orientations is that of net-degree. In this paper, we extend the following four topics from (un)directed graphs to bidirected graphs:-Erdős-Gallai-type results: characterization of net-degree sequences, -Havel-Hakimi-type results: complete sets of degree-preserving operations, -Extremal degree sequences: characterization of uniquely realizable sequences, and -Enumerative aspects: counting formulas for net-degree sequences. To underline the similarities and differences to their (un)directed counterparts, we briefly survey the undirected setting and we give a thorough account for digraphs with an emphasis on the discrete geometry of degree sequences. In particular, we determine the tight and uniquely realizable degree sequences for directed graphs.

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The polytope of degree sequences

Cited by 38 publications

References 10 publications

Shifted simplicial complexes are Laplacian integral

Shifted simplicial complexes are Laplacian integral

The minimum number of Hamilton cycles in a Hamiltonian threshold graph of a prescribed order

On degree sequences of undirected, directed, and bidirected graphs

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