Strongly Regular Graphs

Brouwer, AE Andries

doi:10.1007/978-1-4614-1939-6_9

Cited by 72 publications

(171 citation statements)

References 18 publications

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“…But this weight is not trivial. The eigenvalues and multiplicities are: 4].) The folded n-cube has antipodal pairs of vertices in the n-cube as its vertices, with two such pairs adjacent if the possible distances between them are 1 and n − 1.…”

Section: Observe That For Certain Triplesmentioning

confidence: 99%

“…Thus we have a procedure for building non-trivial regular weights from trivial ones on known schemes. For example, a new regular weight on SRG(126, 25,8,4) is obtained from a trivial weight on the Odd graph on 9 points, a distance regular graph of diameter 4, by merging the distance 1 and 2 graphs of the scheme.…”

Section: Introductionmentioning

confidence: 99%

“…Brouwer's parameter tables for SRGs [4] and distance-regular graphs [3] were accessed, as were the association schemes of small order provided by Hanaki and Miyamoto [10].…”

Section: Introductionmentioning

confidence: 99%

“…We obtain (26,10,3,4); the Chang and the T (8) graphs (28,12,6,4); and the complete bipartite graphs. See [4,15,20,21] for more.…”

mentioning

confidence: 99%

“…Non-Higmanian (28,12,6,4) Higmanian, closure (36, 14,4,6) Non-Higmanian (36, 15,6,6) Amorphic, closure L 1,3 (6); also in L 2,2 (6); neither exists (50, 21,8,9) Non-Higmanian (64, 27,10,12) Amorphic, occurs in another primitive scheme (64, 28,12,12) Amorphic, closure L Amorphic, occurs also in primitive scheme (100, 45,20,20) Amorphic, closure…”

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confidence: 99%

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Weighted association schemes, fusions, and minimal coherent closures

Sankey

2014

J Algebr Comb

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A weighted association scheme is a scheme with an edge weight function, which for our purposes will take values ±1. When the scheme has a coherent fusiona merging of classes resulting in another association scheme-the edge weights on the fusion scheme are inherited. The reverse process involves the coherent closure of a weighted scheme: the smallest coherent algebra containing the weighted adjacency matrices. The weight function applied to this closure is necessarily trivial, meaning constant on classes of the associated configuration. In this work there are two main objects of study: minimal rank coherent closures of strongly regular graphs with regular weights; and strongly regular graphs with regular weights which are obtained as fusions of association schemes with trivial regular weights. Both of these extend work of D. Taylor on regular two-graphs and their interactions with strongly regular graphs. We obtain regular weights on strongly regular graphs with 125 and 256 vertices as fusions of rank 5 weighted schemes, and a family of rank 6, primitive, non-metric schemes which are minimal-rank closures of weighted lattice graphs. Our main result is the classification of these coherent closures of rank 4. They are imprimitive rank 4 association schemes, which have been studied generally by E. R. van Dam, and Y. Chang and T. Huang.

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Section: Observe That For Certain Triplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Brouwer's parameter tables for SRGs [4] and distance-regular graphs [3] were accessed, as were the association schemes of small order provided by Hanaki and Miyamoto [10].…”

Section: Introductionmentioning

confidence: 99%

“…We obtain (26,10,3,4); the Chang and the T (8) graphs (28,12,6,4); and the complete bipartite graphs. See [4,15,20,21] for more.…”

mentioning

confidence: 99%

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confidence: 99%

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Weighted association schemes, fusions, and minimal coherent closures

Sankey

2014

J Algebr Comb

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A Family of Partial Geometric Designs from Three‐Class Association Schemes

Nowak

Ölmez

Song

2016

J of Combinatorial Designs

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In this paper, we show that partial geometric designs can be constructed from certain three‐class association schemes and ternary linear codes with dual distance three. In particular, we obtain a family of partial geometric designs from the three‐class association schemes introduced by Kageyama, Saha, and Das in their article [“Reduction of the number of associate classes of hypercubic association schemes,” Ann Inst Statist Math 30 (1978)]. We also give a list of directed strongly regular graphs arising from the partial geometric designs obtained in this paper.

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Self‐Dual Codes and the Nonexistence of a Quasi‐Symmetric 2‐(37,9,8) Design with Intersection Numbers 1 and 3

Harada

Munemasa

Tonchev

2017

J of Combinatorial Designs

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We prove that a certain binary linear code associated with the incidence matrix of a quasi-symmetric 2-(37, 9, 8) design with intersection numbers 1 and 3 must be contained in an extremal doubly even self-dual code of length 40. Using the classification of extremal doubly even self-dual codes of length 40, we show that a quasi-symmetric 2-(37, 9, 8) design with intersection numbers 1 and 3 does not exist.

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Strongly Regular Graphs

Cited by 72 publications

References 18 publications

Weighted association schemes, fusions, and minimal coherent closures

Weighted association schemes, fusions, and minimal coherent closures

A Family of Partial Geometric Designs from Three‐Class Association Schemes

Self‐Dual Codes and the Nonexistence of a Quasi‐Symmetric 2‐(37,9,8) Design with Intersection Numbers 1 and 3

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