scholarly journals How likely is an i.i.d. degree sequence to be graphical?

2005 ◽  
Vol 15 (1B) ◽  
pp. 652-670 ◽  
Author(s):  
Richard Arratia ◽  
Thomas M. Liggett
Keyword(s):  
Degree Sequence

The configuration model

2018 ◽  
Author(s):  
Mark Newman
Keyword(s):  
Degree Sequence ◽  
Graph Model ◽  
Network Models ◽  
Clustering Coefficient ◽  
Giant Component ◽  
Configuration Model ◽  
Bipartite Networks ◽  
Small Component ◽  
Random Graph Model ◽  
Average Size

A discussion of the most fundamental of network models, the configuration model, which is a random graph model of a network with a specified degree sequence. Following a definition of the model a number of basic properties are derived, including the probability of an edge, the expected number of multiedges, the excess degree distribution, the friendship paradox, and the clustering coefficient. This is followed by derivations of some more advanced properties including the condition for the existence of a giant component, the size of the giant component, the average size of a small component, and the expected diameter. Generating function methods for network models are also introduced and used to perform some more advanced calculations, such as the calculation of the distribution of the number of second neighbors of a node and the complete distribution of sizes of small components. The chapter ends with a brief discussion of extensions of the configuration model to directed networks, bipartite networks, networks with degree correlations, networks with high clustering, and networks with community structure, among other possibilities.

scholarly journals Random graphs with a given degree sequence

2011 ◽  
Vol 21 (4) ◽  
pp. 1400-1435 ◽  
Author(s):  
Sourav Chatterjee ◽  
Persi Diaconis ◽  
Allan Sly
Keyword(s):  
Random Graphs ◽  
Degree Sequence

scholarly journals New Results on Degree Sequences of Uniform Hypergraphs

10.37236/3414 ◽  
2013 ◽  
Vol 20 (4) ◽  
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.

scholarly journals A Turán Type Problem Concerning the Powers of the Degrees of a Graph

10.37236/1525 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Yair Caro ◽  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Degree Sequence ◽  
Upper And Lower Bounds ◽  
Exact Results ◽  
Type Problem ◽  
Maximum Value ◽  
Turán Number

For a graph $G$ whose degree sequence is $d_{1},\ldots ,d_{n}$, and for a positive integer $p$, let $e_{p}(G)=\sum_{i=1}^{n}d_{i}^{p}$. For a fixed graph $H$, let $t_{p}(n,H)$ denote the maximum value of $e_{p}(G)$ taken over all graphs with $n$ vertices that do not contain $H$ as a subgraph. Clearly, $t_{1}(n,H)$ is twice the Turán number of $H$. In this paper we consider the case $p>1$. For some graphs $H$ we obtain exact results, for some others we can obtain asymptotically tight upper and lower bounds, and many interesting cases remain open.

scholarly journals Switching in One-Factorisations of Complete Graphs

10.37236/3606 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Petteri Kaski ◽  
André de Souza Medeiros ◽  
Patric R.J. Östergård ◽  
Ian M. Wanless
Keyword(s):  
Degree Sequence ◽  
Clique Number ◽  
Latin Squares ◽  
Complete Graphs ◽  
Switching Operation ◽  
Isomorphism Classes ◽  
Switching Graph

We define two types of switchings between one-factorisations of complete graphs, called factor-switching and vertex-switching. For each switching operation and for each $n\le 12$, we build a switching graph that records the transformations between isomorphism classes of one-factorisations of $K_{n}$.  We establish various parameters of our switching graphs, including order, size, degree sequence, clique number and the radius of each component.As well as computing data for $n\le12$, we demonstrate several properties that hold for one-factorisations of $K_{n}$ for general $n$. We show that such factorisations have a parity which is not changed by factor-switching, and this leads to disconnected switching graphs. We also characterise the isolated vertices that arise from an absence of switchings. For factor-switching the isolated vertices are perfect one-factorisations, while for vertex-switching the isolated vertices are closely related to atomic Latin squares.

scholarly journals Minimal degree sequence for 2-bridge knots

2006 ◽  
Vol 190 ◽  
pp. 191-210 ◽  
Author(s):  
Prabhakar Madeti ◽  
Rama Mishra
Keyword(s):  
Degree Sequence ◽  
Minimal Degree
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