Bipartite covering graphs

Archdeacon, Dan; Kwak, Jin Ho; Lee, Jaeun; Sohn, Moo Young

doi:10.1016/s0012-365x(99)00196-x

Cited by 11 publications

(16 citation statements)

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“…Theorem 2.1: Let W be a walk in a voltage graph X with initial vertex v. Then for each vertex (v, g) in X α for g ∈ G, there is a unique walk W g in X α that starts at (v, g) and projects down to W . Assume W = e Voltage graphs have been successfully used to obtain many instances of graphs with extremal properties; see [9], [4], [5], [3].…”

Section: Preliminariesmentioning

confidence: 99%

LDPC codes from voltage graphs

Kelley

Walker

2008

2008 IEEE International Symposium on Information Theory

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Abstract-Several well-known structure-based constructions of LDPC codes, for example codes based on permutation and circulant matrices and in particular, quasi-cyclic LDPC codes, can be interpreted via algebraic voltage assignments. We explain this connection and show how this idea from topological graph theory can be used to give simple proofs of many known properties of these codes. In addition, the notion of abelianinevitable cycle is introduced and the subgraphs giving rise to these cycles are classified. We also indicate how, by using more sophisticated voltage assignments, new classes of good LDPC codes may be obtained.

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Section: Preliminariesmentioning

confidence: 99%

LDPC codes from voltage graphs

Kelley

Walker

2008

2008 IEEE International Symposium on Information Theory

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“…Assume G is connected. In [24] it is explained that the ordinary derived graph G α ′ is connected if and only if G ′ = G. In [4], a similar result is explained for a permutation derived graph, which involves the structure of the orbits when the local group acts on the set {1, 2, . .…”

Section: Connectivitymentioning

confidence: 89%

“…For any assignment α of edges in G to voltages in a group G, it is possible to find for any spanning tree T of G, a voltage assignment α ′ of edges in G to G, where the edges of T are assigned the identity permutation under α ′ and the resulting graphs G α and G α ′ are isomorphic [4], [24]. We will therefore focus on how to assign voltages to the edges that lie outside of a chosen spanning tree, also called the co-tree.…”

Section: Connectivitymentioning

confidence: 99%

“…Similarly we let d act on the group G to give the other generator of the permutation group (a permutation we will also call d). Our labeling gives that the permutations c and d are, in cycle notation, c : (1,2,3,4,5,6,7,8,9,10,11)· (12,16,17,18,19,20,21,22,23,24,25) Using c and d as generators and the relation c 3 d = dc, we obtain the nonabelian permutation group P of order 55.…”

Section: B Examplesmentioning

confidence: 99%

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On codes designed via algebraic lifts of graphs

Kelley

2008

2008 46th Annual Allerton Conference on Communication, Control, and Computing

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Abstract-Over the past few years several constructions of protograph codes have been proposed that are based on random lifts of suitably chosen base graphs. More recently, an algebraic analog of this approach was introduced using the theory of voltage graphs. The strength of the voltage graph framework is the ability to analyze the resulting derived graph algebraically, even when the voltages themselves are assigned randomly. Moreover, the theory of voltage graphs provides insight to designing lifts of graphs with particular properties. In this paper we illustrate how the properties of the derived graphs and the corresponding codes relate to the voltage assignments. In particular, we present a construction of LDPC codes by giving an algebraic method of choosing the permutation voltages and illustrate the usefulness of the proposed technique via simulation results.

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“…[2]) Let G be a non-bipartite graph with a generating voltage assignment ν in A which derives the covering graphG. ThenG is bipartite if and only if there exists a subgroup A e of index two in A such that for every cycle C, ν(C) ∈ A e if and only if the length of C is even.It is obvious that χ G (H) = 1 if and only if G is a null graph.…”

mentioning

confidence: 99%