Families of small regular graphs of girth 5

Abreu, M.; Araujo‐Pardo, Gabriela; Balbuena, C.; Labbate, Domenico

doi:10.1016/j.disc.2012.05.020

Cited by 21 publications

(61 citation statements)

References 27 publications

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“…As an example, the variable nodes are in the sequence {v 1 Figure 6A, and the names of v 2 and v 4 are swapped in Figure 6B. In Figure 7, H 1 and H 2 are the parity-check matrices for the BGs in Figure 6A,B, respectively.…”

Section: Symmetry In the Mdd Solution Spacementioning

confidence: 99%

“…DeleteEdge picks the codes with the girth smaller than g, and deletes edges until we end up with a BG with girth g. Deleting an edge {c i , v j } from a BG increases the slacks t i and s j , and the objective function value. Hence, we choose an edge {c i , v j }, which is in a small cycle, that is, small (i, j) < g, has small degree deviations t i and s j , and small objective function coefficients 1 (c i ) and 1 (v j ) . This means we choose the entry (i*, j*) to delete, that is, set x i * j * = 0, using Equation (12).…”

Section: Solution Polishing Algorithmsmentioning

confidence: 99%

“…The message passing between the variable and check nodes continues until all check nodes are satisfied (a codeword is found), or a termination criterion (such as iteration limit or no candidate bit to flip) is met. In Figure 4B, notice that all of the adjacent check nodes of erroneously received bits v 1 and v 2 are in the length-4 cycle (v 1 , c 1 , v 2 , c 2 ). Gallager A fails to recover the bits v 1 and v 2 , since the number of satisfied check nodes equals the number of unsatisfied nodes.…”

Section: Introduction and Literature Reviewmentioning

confidence: 99%

“…In Figure 4B, notice that all of the adjacent check nodes of erroneously received bits v 1 and v 2 are in the length-4 cycle (v 1 , c 1 , v 2 , c 2 ). Gallager A fails to recover the bits v 1 and v 2 , since the number of satisfied check nodes equals the number of unsatisfied nodes. Hence, Gallager A terminates with the vector v = (0 0 1 0 1) due to the nonpresence of a candidate bit to flip.…”

Section: Introduction and Literature Reviewmentioning

confidence: 99%

“…This algorithm is used to approximate the girth of a graph. Some families of small regular graphs of girth 5 are described in [1]. Small regular BGs of girth 6 can be constructed with the method given in [3].…”

Section: Introduction and Literature Reviewmentioning

confidence: 99%

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A branch‐and‐cut algorithm for a bipartite graph construction problem in digital communication systems

2019

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We study a bipartite graph (BG) construction problem that arises in digital communication systems. In a digital communication system, information is sent from one place to another over a noisy communication channel using binary symbols (bits). The original information is encoded by adding redundant bits, which are then used to detect and correct errors that may have been introduced during transmission. Harmful structures, such as small cycles, severely deteriorate the error correction capability of a BG. We introduce an integer programming formulation to generate a BG for a given smallest cycle length. We propose a branch-and-cut algorithm for its solution and investigate the structural properties of the problem to derive valid inequalities and variable fixing rules. We also introduce heuristics to obtain feasible solutions for the problem. The computational experiments show that our algorithm can generate BGs without small cycles in an acceptable amount of time for practically relevant dimensions.

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Section: Symmetry In the Mdd Solution Spacementioning

confidence: 99%

Section: Solution Polishing Algorithmsmentioning

confidence: 99%

Section: Introduction and Literature Reviewmentioning

confidence: 99%

Section: Introduction and Literature Reviewmentioning

confidence: 99%

Section: Introduction and Literature Reviewmentioning

confidence: 99%

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A branch‐and‐cut algorithm for a bipartite graph construction problem in digital communication systems

2019

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Generalized pentagonal geometries

Forbes

Rutherford

2021

J of Combinatorial Designs

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A stability result for girth‐regular graphs with even girth

Kiss

Miklavič

Szőnyi

2021

Journal of Graph Theory

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Let Γ denote a finite, connected, simple graph. For an edge e of Γ let n(e) denote the number of girth-cycles containing e. For a vertex v of Γ let {e 1 , e 2 , . . . , e k } be the set of edges incident to v orderd such that n(e 1 ) ≤ n(e 2 ) ≤ • • • ≤ n(e k ). Then (n(e 1 ), n(e 2 ), . . . , n(e k )) is called the signature of v. The graph Γ is said to be girth-regular, if all of its vertices have the same signature.Let Γ be a girth-regular graph with girth g = 2d and signature (a 1 , a 2 , . . . , a k ) . It is known that in this case we have a k ≤ (k − 1) d . In this paper we show that if a k = (k − 1) d − for some non-negative integer < k − 1, then = 0.

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Families of small regular graphs of girth 5

Cited by 21 publications

References 27 publications

A branch‐and‐cut algorithm for a bipartite graph construction problem in digital communication systems

A branch‐and‐cut algorithm for a bipartite graph construction problem in digital communication systems

Generalized pentagonal geometries

A stability result for girth‐regular graphs with even girth

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