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On the structure of extremal graphs of high girth

Felix Lazebnik, Ping Wang
1997 Journal of Graph Theory  
Let n ≥ 3 be a positive integer, and let G be a simple graph of order v containing no cycles of length smaller than n + 1 and having the greatest possible number of edges (an extremal graph).  ...  On the other hand we prove that the answer to the question is negative for v = 2n + 2 ≥ 26.  ...  Hartnell for his suggestions on the proof of Theorem 3, and to referees, whose comments helped them to improve the original version of the paper.  ... 
doi:10.1002/(sici)1097-0118(199711)26:3<147::aid-jgt5>3.3.co;2-t fatcat:x7zbhupj5bc6niulekqq2bx22q

On the structure of extremal graphs of high girth

Felix Lazebnik, Ping Wang
1997 Journal of Graph Theory  
Let n ≥ 3 be a positive integer, and let G be a simple graph of order v containing no cycles of length smaller than n + 1 and having the greatest possible number of edges (an extremal graph).  ...  On the other hand we prove that the answer to the question is negative for v = 2n + 2 ≥ 26.  ...  Hartnell for his suggestions on the proof of Theorem 3, and to referees, whose comments helped them to improve the original version of the paper.  ... 
doi:10.1002/(sici)1097-0118(199711)26:3<147::aid-jgt5>3.0.co;2-r fatcat:rjirxo3sifhbtk6lav5fagwtoq

The Norms of Graph Spanners

Eden Chlamtác, Michael Dinitz, Thomas Robinson, Michael Wagner
2019 International Colloquium on Automata, Languages and Programming  
To interpolate between these two extremes, in this paper we initiate the study of graph spanners with respect to the p-norm of their degree vector, thus simultaneously modeling the number of edges (the  ...  girth conjecture).  ...  While the structure of these extremal graphs is simple, the dependence of the parameters of these graphs on t and p is quite complex.  ... 
doi:10.4230/lipics.icalp.2019.40 dblp:conf/icalp/ChlamtacDR19 fatcat:shp5tewygjdzvbts6h3uhknjuq

On the homogeneous algebraic graphs of large girth and their applications

T. Shaska, V. Ustimenko
2009 Linear Algebra and its Applications  
Families of finite graphs of large girth were introduced in classical extremal graph theory.  ...  We consider some results on such algebraic graphs over any field. The upper bound on the dimension of variety of edges for algebraic graphs of girth 2d is established.  ...  On the applications of graphs with high girth to coding theory and cryptography The following directions of applied data security are motivations of studies of extremal properties of balanced graphs of  ... 
doi:10.1016/j.laa.2008.08.023 fatcat:dlt53j7ywzcq5enffof4age6lu

Extremal absorbing sets in low-density parity-check codes

Emily McMillon, Allison Beemer, Christine A. Kelley
2021 Advances in Mathematics of Communications  
Absorbing sets are combinatorial structures in the Tanner graphs of low-density parity-check (LDPC) codes that have been shown to inhibit the high signal-to-noise ratio performance of iterative decoders  ...  In this paper, we determine the sizes of absorbing sets that can occur in general and leftregular LDPC code graphs, with emphasis on the range of b for a given a for which an (a, b)-absorbing set may exist  ...  Acknowledgments We thank the anonymous reviewers for their helpful comments that improved the quality of the paper, including a nicer proof of Theorem 3.2.  ... 
doi:10.3934/amc.2021003 fatcat:5bqyowmvgjalnbc3aprwm4fg4a

Using simulated annealing to construct extremal graphs

P. Wang, G.W. Dueck, S. MacMillan
2001 Discrete Mathematics  
The girth of graph G is the length of the shortest cycle in G; if G has no cycle, we deÿne the girth of G as inÿnite.  ...  We shall discuss an important property, i.e. the degree distribution of extremal graphs.  ...  In [14] , Lazebnik and Wang discuss some structural properties of extremal graphs with girth ¿n + 1.  ... 
doi:10.1016/s0012-365x(00)00265-x fatcat:42745t45ebgzjgn3szjxqmydim

On the implementation of cryptoalgorithms based on algebraic graphs over some commutative rings

Kotorowicz, Ustimenko
2008 Condensed Matter Physics  
The paper is devoted to computer implementation of some graph based stream ciphers. We compare the time performance of this new algorithm with fast, but no very secure RC4, and with DES.  ...  It turns out that some of new algorithms are faster than RC4. They satisfy the Madryga requirements, which is unusual for stream ciphers (like RC4).  ...  Erdös' and his school and some new results on extremal directed graphs, observe the results on cryptographical properties of graphs of high girth and implementations of algorithms based on automata related  ... 
doi:10.5488/cmp.11.2.347 fatcat:afdr65zbyzalxbz4vrmwutleim

On Extremal Graph Theory, Explicit Algebraic Constructions of Extremal Graphs and Corresponding Turing Encryption Machines [chapter]

Vasyl Ustimenko, Urszula Romańczuk
2013 Studies in Computational Intelligence  
We observe recent results on the applications of extremal graph theory to cryptography.  ...  Classical Extremal Graph Theory contains Erdős Even Circuite Theorem and other remarkable results on the maximal size of graphs without certain cycles.  ...  Correlation with Expansion Properties Our applications of the Graph theory to Cryptography is based on the use of the graphs of high girth.  ... 
doi:10.1007/978-3-642-29694-9_11 fatcat:t22qvy6blrdgzes7o74r752wce

A new series of dense graphs of high girth [article]

Felix Lazebnik, Vasiliy A. Ustimenko, Andrew J. Woldar
1995 arXiv   pre-print
These graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of order v and girth at least g, g> 5, g = 11,12.  ...  We construct a bipartite, q-regular, edge-transitive graph CD(k,q) of order v < 2q^k-t+1 and girth g > k+5. If e is the the number of edges of CD(k,q), then e =Ω(v^1+ 1k-t+1).  ...  Acknowledgment We are indebted to Alexander Schliep, whose computer programs gave us an insight into the structure of the graphs D(k , q) .  ... 
arXiv:math/9501231v1 fatcat:zsya653vjjfiza33ivlok7s4im

A New Series of Dense Graphs of High Girth

F. Lazebnik, V. A. Ustimenko, A. J. Woldar
1995 Bulletin of the American Mathematical Society  
These graphs provide the best known asymptotic lower bound for the greatest number of edges in graphs of order v and girth at least g, g ≥ 5, g = 11, 12.  ...  We construct a bipartite, q-regular, edge-transitive graph CD(k, q) of order v ≤ 2q k−t+1 and girth g ≥ k + 5. If e is the the number of edges of CD(k, q), then e = Ω(v 1+ 1 k−t+1 ).  ...  Acknowledgment We are indebted to Alexander Schliep, whose computer programs gave us an insight into the structure of the graphs D(k, q).  ... 
doi:10.1090/s0273-0979-1995-00569-0 fatcat:x4qkolpt25dw5h6tyxjcalwz2m

A Combinatorial Classic — Sparse Graphs with High Chromatic Number [chapter]

Jaroslav Nešetřil
2013 Bolyai Society Mathematical Studies  
Acknowledgment: I thank to Martin Bálek and Andrew Goodall for the help when preparing this article.  ...  The existence of high girth non cover graph of posets (proved in [72] ) led to the proof that the following recognition problem is NP-complete [78] , [14] : Input: A graph G.  ...  It is not even known whether high girth subcubic graphs have circular chromatic number < 3.  ... 
doi:10.1007/978-3-642-39286-3_15 fatcat:plrnkchibbdv5jm3jfap4dmk5u

Shortest-path metric approximation for random subgraphs

Jan Vondrák
2006 Random structures & algorithms (Print)  
We consider graph optimization problems where the cost of a solution depends only on the shortest-path metric in the graph, such as Steiner Tree or Traveling Salesman.  ...  subgraph of G, with high probability.  ...  Thus the extremal number of edges for graphs of given girth provides an upper bound on the size of spanners as well.  ... 
doi:10.1002/rsa.20150 fatcat:hu4fvapi3vfddbp7pfcwsersgu

Design of cages with a randomized progressive edge-growth algorithm

A. Venkiah, D. Declercq, C. Poulliat
2008 IEEE Communications Letters  
, this new instance of the PEG algorithm allows to construct cages, i.e. graphs with the minimal size such that a graph of girth g exists, which is the best result one might hope for.  ...  In this letter, we propose some improvements in the PEG algorithm which greatly improve the girth properties of the resulting graphs: given a graph size, they increase the girth g achievable by the algorithm  ...  crucial to focus on the girth properties of the underlying Tanner graph.  ... 
doi:10.1109/lcomm.2008.071843 fatcat:zkgjh6oqjzgync7hylxrfg2geu

Some problems of Graph Based Codes for Belief Propagation decoding [article]

Vasiliy Usatyuk
2015 arXiv   pre-print
In paper we show fast methods to measure first components of TS enumerator using EMD=ACE constrains for high girth codes on the graph using graph spectral classification method.  ...  Short survey about code on the graph by example of hardware friendly quasi-cycle LDPC code.  ...  on the middle or high dense paritycheck matrix.  ... 
arXiv:1511.00133v3 fatcat:fuopsxk33zbx3h7ekneskry72y

Low-density parity check codes and iterative decoding for long-haul optical communication systems

B. Vasic, I.B. Djordjevic, R.K. Kostuk
2003 Journal of Lightwave Technology  
We present a class of structured codes based on mutually orthogonal Latin rectangles. Such codes have high rates and can lend themselves to very low-complexity encoder/decoder implementations.  ...  We show that LDPC codes provide a significant system performance improvement with respect to the state-of-the-art FEC schemes employed in optical communications systems.  ...  The structure of our codes is of crucial importance for high-speed implementations, because these codes can lend themselves to extremely simple encoders, and because we can be assured of large minimum  ... 
doi:10.1109/jlt.2003.808769 fatcat:re3uhfa2e5b53lwifuh57o7j4i
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