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On the structure of extremal graphs of high girth

1997
*
Journal of Graph Theory
*

Let n ≥ 3 be a positive integer, and let G be a simple

doi:10.1002/(sici)1097-0118(199711)26:3<147::aid-jgt5>3.3.co;2-t
fatcat:x7zbhupj5bc6niulekqq2bx22q
*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. ...##
###
On the structure of extremal graphs of high girth

1997
*
Journal of Graph Theory
*

Let n ≥ 3 be a positive integer, and let G be a simple

doi:10.1002/(sici)1097-0118(199711)26:3<147::aid-jgt5>3.0.co;2-r
fatcat:rjirxo3sifhbtk6lav5fagwtoq
*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. ...##
###
The Norms of Graph Spanners

2019
*
International Colloquium on Automata, Languages and Programming
*

To interpolate between these two

doi:10.4230/lipics.icalp.2019.40
dblp:conf/icalp/ChlamtacDR19
fatcat:shp5tewygjdzvbts6h3uhknjuq
*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. ...##
###
On the homogeneous algebraic graphs of large girth and their applications

2009
*
Linear Algebra and its Applications
*

Families

doi:10.1016/j.laa.2008.08.023
fatcat:dlt53j7ywzcq5enffof4age6lu
*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*...##
###
Extremal absorbing sets in low-density parity-check codes

2021
*
Advances in Mathematics of Communications
*

Absorbing sets are combinatorial

doi:10.3934/amc.2021003
fatcat:5bqyowmvgjalnbc3aprwm4fg4a
*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. ...##
###
Using simulated annealing to construct extremal graphs

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. ...

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

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 ...

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

2013
*
Studies in Computational Intelligence
*

We observe recent results

doi:10.1007/978-3-642-29694-9_11
fatcat:t22qvy6blrdgzes7o74r752wce
*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*. ...##
###
A new series of dense graphs of high girth
[article]

1995
*
arXiv
*
pre-print

These

arXiv:math/9501231v1
fatcat:zsya653vjjfiza33ivlok7s4im
*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) . ...##
###
A New Series of Dense Graphs of High Girth

1995
*
Bulletin of the American Mathematical Society
*

These

doi:10.1090/s0273-0979-1995-00569-0
fatcat:x4qkolpt25dw5h6tyxjcalwz2m
*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). ...##
###
A Combinatorial Classic — Sparse Graphs with High Chromatic Number
[chapter]

2013
*
Bolyai Society Mathematical Studies
*

Acknowledgment: I thank to Martin Bálek and Andrew Goodall for

doi:10.1007/978-3-642-39286-3_15
fatcat:plrnkchibbdv5jm3jfap4dmk5u
*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. ...##
###
Shortest-path metric approximation for random subgraphs

2006
*
Random structures & algorithms (Print)
*

We consider

doi:10.1002/rsa.20150
fatcat:hu4fvapi3vfddbp7pfcwsersgu
*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. ...##
###
Design of cages with a randomized progressive edge-growth algorithm

2008
*
IEEE Communications Letters
*

, this new instance

doi:10.1109/lcomm.2008.071843
fatcat:zkgjh6oqjzgync7hylxrfg2geu
*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*. ...##
###
Some problems of Graph Based Codes for Belief Propagation decoding
[article]

2015
*
arXiv
*
pre-print

In paper we show fast methods to measure first components

arXiv:1511.00133v3
fatcat:fuopsxk33zbx3h7ekneskry72y
*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. ...##
###
Low-density parity check codes and iterative decoding for long-haul optical communication systems

2003
*
Journal of Lightwave Technology
*

We present a class

doi:10.1109/jlt.2003.808769
fatcat:re3uhfa2e5b53lwifuh57o7j4i
*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 ...
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