On Realizability of a Set of Integers as Degrees of the Vertices of a Linear Graph. I

Hakimi, S. L.

doi:10.1137/0110037

Cited by 552 publications

(406 citation statements)

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“…Any degree sequence whose sum is even can be realized by a multigraph having loops [8]. In this paper we consider simple graphs (graphs without loops and multiple edges).…”

Section: Introductionmentioning

confidence: 99%

Transforming Graphs with the Same Graphic Sequence

Bereg

Ito

2017

Journal of Information Processing

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“…Any degree sequence whose sum is even can be realized by a multigraph having loops [8]. In this paper we consider simple graphs (graphs without loops and multiple edges).…”

Section: Introductionmentioning

confidence: 99%

Transforming Graphs with the Same Graphic Sequence

Bereg

Ito

2017

Journal of Information Processing

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“…Equivalently, graphicality can be tested by a recursive application of the Havel-Hakimi theorem [19,20]:…”

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confidence: 99%

All Scale-Free Networks Are Sparse

2011

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We study the realizability of scale free-networks with a given degree sequence, showing that the fraction of realizable sequences undergoes two first-order transitions at the values 0 and 2 of the power-law exponent. We substantiate this finding by analytical reasoning and by a numerical method, proposed here, based on extreme value arguments, which can be applied to any given degree distribution. Our results reveal a fundamental reason why large scale-free networks without constraints on minimum and maximum degree must be sparse.

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“…Thus our condition immediately yields an efficient linear-time algorithm for determining whether there exists a key set protocol to work for a given signature γ. Our condition looks in appearance to be similar to the condition for a given degree sequence to be "graphical," and the proof for our condition is complicated as well as those for a degree sequence [1,7,8,10].…”

Section: Introductionmentioning

confidence: 92%

Dealing Necessary and Sufficient Numbers of Cards for Sharing a One-Bit Secret Key (Extended Abstract)

Mizuki

Shizuya

Nishizeki

1999

Advances in Cryptology — EUROCRYPT ’99

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Abstract. Using a random deal of cards to players and a computationally unlimited eavesdropper, all players wish to share a one-bit secret key which is information-theoretically secure from the eavesdropper. This can be done by a protocol to make several pairs of players share one-bit secret keys so that all these pairs form a spanning tree over players. In this paper we obtain a necessary and sufficient condition on the number of cards for the existence of such a protocol. Our condition immediately yields an efficient linear-time algorithm to determine whether there exists a protocol to achieve such a secret key sharing.

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On Realizability of a Set of Integers as Degrees of the Vertices of a Linear Graph. I

Cited by 552 publications

References 1 publication

Transforming Graphs with the Same Graphic Sequence

Transforming Graphs with the Same Graphic Sequence

All Scale-Free Networks Are Sparse

Dealing Necessary and Sufficient Numbers of Cards for Sharing a One-Bit Secret Key (Extended Abstract)

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