Enumerating degree sequences in digraphs and a cycle–cocycle reversing system

We investigate the Laplacian eigenvalues of sparse random graphs Gnp. We show that in the case that the expected degree d = (n-1)p is bounded, the spectral gap of the normalized Laplacian $\LL(\gnp)$ is o(1). Nonetheless, w.h.p. G = Gnp has a large subgraph core(G) such that the spectral gap of $\LL(\core(G))$ is as large as 1-O (d−1/2). We derive similar results regarding the spectrum of the combinatorial Laplacian L(Gnp). The present paper complements the work of Chung, Lu and Vu [8] on the Laplacian spectra of random graphs with given expected degree sequences. Applied to Gnp, their results imply that in the ‘dense’ case d ≥ ln2n the spectral gap of $\LL(\gnp)$ is 1-O (d−1/2) w.h.p.

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Degree sequences of random graphs

Discrete Mathematics ◽

10.1016/0012-365x(81)90253-3 ◽

1981 ◽

Vol 33 (1) ◽

pp. 1-19 ◽

Cited By ~ 57

Author(s):

Béla Bollobás

Keyword(s):

Random Graphs ◽

Degree Sequences

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Parallel Algorithms for Generating Random Networks with Given Degree Sequences

International Journal of Parallel Programming ◽

10.1007/s10766-015-0389-y ◽

2015 ◽

Vol 45 (1) ◽

pp. 109-127 ◽

Cited By ~ 6

Author(s):

Maksudul Alam ◽

Maleq Khan

Keyword(s):

Parallel Algorithms ◽

Random Networks ◽

Degree Sequences

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New Results on Degree Sequences of Uniform Hypergraphs

The Electronic Journal of Combinatorics ◽

10.37236/3414 ◽

2013 ◽

Vol 20 (4) ◽

Cited By ~ 9

Author(s):

Sarah Behrens ◽

Catherine Erbes ◽

Michael Ferrara ◽

Stephen G. Hartke ◽

Benjamin Reiniger ◽

...

Keyword(s):

Efficient Algorithm ◽

Necessary Conditions ◽

Degree Sequence ◽

Sufficient Conditions ◽

Degree Sequences ◽

Uniform Hypergraph ◽

Graphic Sequence ◽

Uniform Hypergraphs ◽

Open Question

A sequence of nonnegative integers is $k$-graphic if it is the degree sequence of a $k$-uniform hypergraph. The only known characterization of $k$-graphic sequences is due to Dewdney in 1975. As this characterization does not yield an efficient algorithm, it is a fundamental open question to determine a more practical characterization. While several necessary conditions appear in the literature, there are few conditions that imply a sequence is $k$-graphic. In light of this, we present sharp sufficient conditions for $k$-graphicality based on a sequence's length and degree sum.Kocay and Li gave a family of edge exchanges (an extension of 2-switches) that could be used to transform one realization of a 3-graphic sequence into any other realization. We extend their result to $k$-graphic sequences for all $k \geq 3$. Finally we give several applications of edge exchanges in hypergraphs, including generalizing a result of Busch et al. on packing graphic sequences.

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