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Linkability in iterated line graphs

Thomas Böhme, Martin Knor, L'udovít Niepel
2006 Discrete Mathematics  
We prove that for every graph H with the minimum degree 5, the third iterated line graph L 3 (H ) of H contains K √ −1 as a minor.  ...  Using this fact we prove that if G is a connected graph distinct from a path, then there is a number k G such that for every i k G the i-iterated line graph of G is 1 2 (L i (G))-linked.  ...  In [3] and [2] , Hartke and Higgins study the growth of the minimum and the maximum degree of iterated line graphs, respectively.  ... 
doi:10.1016/j.disc.2005.11.018 fatcat:psh67ltgpfhvnjx3spfcoqi3ti

Free Pseudodistance Growth Rates for Spatially Coupled LDPC Codes over the BEC [article]

Cunlu Zhou, David G. M. Mitchell, Roxana Smarandache
2018 arXiv   pre-print
The minimum pseudoweight is an important parameter related to the decoding performance of LDPC codes with iterative message-passing decoding.  ...  In this paper, we consider ensembles of periodically time-varying spatially coupled LDPC (SC-LDPC) codes and the pseudocodewords arising from their finite graph covers of a fixed degree.  ...  Fig. 1 . 1 Minimum pseudodistance growth rates of degree-2 covers (solid lines) and degree-3 covers (dashed lines) of terminated and tail-biting SC-LDPC code ensembles with calculated upper and lower bounds  ... 
arXiv:1809.04253v1 fatcat:lexailglpvgivirkb74e7jpmra

Maximum Degree Growth of the Iterated Line Graph

Stephen G. Hartke, Aparna W. Higgins
1999 Electronic Journal of Combinatorics  
Let $\Delta_k$ denote the maximum degree of the $k^{\rm th}$ iterated line graph $L^k(G)$. For any connected graph $G$ that is not a path, the inequality $\Delta_{k+1}\leq 2\Delta_k-2$ holds.  ...  Niepel, Knor, and Šoltés have conjectured that there exists an integer $K$ such that, for all $k\geq K$, equality holds; that is, the maximum degree $\Delta_k$ attains the greatest possible growth.  ...  Acknowledgments The first author wishes to thank the University of Dayton Honors Program for support of his Honors Thesis, of which this work is a part.  ... 
doi:10.37236/1460 fatcat:csm7ytkvpzezbd5rmmhgb2ryqm

Progressive edge growth LDPC Encoder with spiral search algorithm

Anand Anbalgan, Senthil Kumar.P
2017 International Journal of Engineering & Technology  
The lowest odd degree (minimal) TS is increasing the formation of more unsaturated nodes in iterative decoding.  ...  Progressive edge growth (PEG) Low-density parity check code (LDPC) [2] avoidance of trapping sets are mainly based on the distance and degree calculation of successive CN.  ...  This graph is modified as a line graph.  ... 
doi:10.14419/ijet.v7i1.3.10673 fatcat:k4n6f245gzhxldmy3fvroluuh4

Spatially Coupled LDPC Codes Constructed From Protographs

David G. M. Mitchell, Michael Lentmaier, Daniel J. Costello
2015 IEEE Transactions on Information Theory  
linear growth of minimum distance with block length.  ...  When all variable nodes in the coupled chain have degree greater than two, asymptotically the error probability converges at least doubly exponentially with decoding iterations and we obtain sequences  ...  iterative BP decoding thresholds and linear growth of minimum distance with block length.  ... 
doi:10.1109/tit.2015.2453267 fatcat:fpmpzpoy5rcwnlohs6us4ccfbe

Page 62 of Mathematical Reviews Vol. , Issue 2003A [page]

2003 Mathematical Reviews  
The graph G is said to be clique divergent if the sequence of the orders o(k”G) of the iterated clique graphs of G tends to infinity with n, and G is said to have linear growth if this divergent sequence  ...  Summary: “We study the dynamical behaviour of simple graphs under the iterated application of the clique graph operator &, which transforms each finite graph G into the intersection graph kG of its (maximal  ... 

Page 1679 of Mathematical Reviews Vol. , Issue 2000c [page]

2000 Mathematical Reviews  
1679 Graph theory 2000c:05087 05C35 Hartke, Stephen G. (1-DYTN; Dayton, OH); Higgins, Aparna W. (1-DYTN; Dayton, OH) Maximum degree growth of the iterated line graph. (English summary) Electron. J.  ...  graph L”(G) has the maximum degree growth property.  ... 

On a Low-Rate TLDPC Code Ensemble and the Necessary Condition on the Linear Minimum Distance for Sparse-Graph Codes [article]

Iryna Andriyanova, Jean-Pierre Tillich
2010 arXiv   pre-print
This paper addresses the issue of design of low-rate sparse-graph codes with linear minimum distance in the blocklength.  ...  The asymptotic analysis of the ensemble shows that its iterative threshold is situated close to the Shannon limit.  ...  variable nodes of degrees 1 and 2 in the code structure But the presence of a large number of variable nodes of low degrees is not favorable for the minimum distance growth.  ... 
arXiv:1010.1911v1 fatcat:iykphlyv7jai7nfpfbjchroljm

Designing a Good Low-Rate Sparse-Graph Code

Iryna Andriyanova, Jean-Pierre Tillich
2012 IEEE Transactions on Communications  
The asymptotic analysis of the ensemble shows that its iterative threshold is close to the Shannon limit.  ...  The condition is formulated in terms of degree-1 and degree-2 variable nodes and of low-weight codewords of the underlying code, and it generalizes results known for turbo codes [8] and LDPC codes.  ...  In this way, a non-zero fraction of degree-2 variable nodes may be allowed for a linear minimum distance growth. b 0 b 1 b 2 b 3 b r−3 b r−2 b r−1 2) Structure of the bipartite graph: A structure on the  ... 
doi:10.1109/tcomm.2012.082712.100205 fatcat:4kykctugdfgaxhfhfjzqtm3tzi

Page 5974 of Mathematical Reviews Vol. , Issue 2004h [page]

2004 Mathematical Reviews  
Loic Foissy (F-REIMS-LM; Reims) 2004h:05036 05C07 Hartke, Stephen G. (1-RTG; Piscataway, NJ); Higgins, Aparna W. (1-DYTN; Dayton, OH) Minimum degree growth of the iterated line graph.  ...  Summary: “Let 0, denote the minimum degree of the kth iterated line graph L*(G). For any connected graph G that is not a path, the inequality 6,.; > 26, —2 holds. L. Niepel, M. Knor and L.  ... 

MASTIFF

Mohsen Koohi Esfahani, Peter Kilpatrick, Hans Vandierendonck
2022 Proceedings of the 36th ACM International Conference on Supercomputing  
In this paper, we study the MSF algorithm from the perspective of graph structure and investigate the implications of the power-law degree distribution of real-world graphs on this algorithm.  ...  The Minimum Spanning Forest (MSF) problem finds usage in many different applications.  ...  First author is supported by a scholarship of the Queen's University Belfast and the Department for the Economy, Northern Ireland.  ... 
doi:10.1145/3524059.3532365 fatcat:lgzf2nmy2ne4vb4eb7dmeds7pm

Regular and irregular progressive edge-growth tanner graphs

Xiao-Yu Hu, E. Eleftheriou, D.M. Arnold
2005 IEEE Transactions on Information Theory  
Lower bounds on the girth of PEG Tanner graphs and on the minimum distance of the resulting low-density parity-check (LDPC) codes are derived in terms of parameters of the graphs.  ...  Index Terms-Girth, low-density parity-check (LDPC) codes, LDPC codes over GF( ), progressive edge growth (PEG), PEG Tanner graphs.  ...  They thank the Associate Editor and two anonymous reviewers for constructive comments, which have greatly helped improve the exposition of the correspondence.  ... 
doi:10.1109/tit.2004.839541 fatcat:xngdgthvdracnpxxccls4d7c3e

An Optimized Algorithm for Constructing LDPC Code with Good Performance
고성능 LDPC 코드를 생성하기 위한 최적화된 알고리듬

Hee-Jong Suh
2013 The Journal of the Korea institute of electronic communication sciences  
In this paper, an algorithm having new edge growth with depth constraints for constructing Tanner graph of LDPC(Low density parity check) codes is proposed.  ...  This algorithm reduces effectively the number of small stoping set in the graph and has lower complexity than other algorithm.  ...  decoding of LDPC codes under the gauss channel Table 1 . 1 The variable node degree and minimum depth standard The number of minimum depth The degree of the variable node ∞     5   ... 
doi:10.13067/jkiecs.2013.8.8.1149 fatcat:xiuf6646wvcqbcrf54o6tpe2ee

Connectivity of iterated line graphs

Martin Knor, L'udovı́t Niepel
2003 Discrete Applied Mathematics  
Moreover, if a hypothesis on the growth of the minimum degree of the i-iterated line graph is true, then an analogous result is true for an arbitrary graph G if i is su ciently large. ?  ...  In this paper we present lower bounds for the connectivity of the i-iterated line graph L i (G) of a graph G.  ...  In [5] , Hartke and Higgins study the growth of the maximum degree of iterated line graphs, and very recently, in [8] Xiong and Liu characterize the graphs whose i-iterated line graphs are Hamiltonian  ... 
doi:10.1016/s0166-218x(02)00197-x fatcat:eulexxafjvesjk3icuwzjwdgce

Spatially Coupled Generalized LDPC Codes: Asymptotic Analysis and Finite Length Scaling [article]

David G. M. Mitchell, Pablo M. Olmos, Michael Lentmaier, Daniel J. Costello
2021 arXiv   pre-print
of SC-GLDPC code ensembles demonstrating capacity approaching thresholds via the threshold saturation effect; (2) an asymptotic analysis of the minimum distance and free distance properties of SC-GLDPC  ...  In this paper, we study spatially coupled generalized low-density parity-check (SC-GLDPC) codes and present a comprehensive analysis of these codes, including: (1) an iterative decoding threshold analysis  ...  However, Gallager showed that regular constructions, where the variable and check node degrees of the Tanner graph representation of H are fixed, maintain linear minimum distance growth with block length  ... 
arXiv:1910.14110v2 fatcat:bxpac3kconfsndrucjspaobpty
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