New strongly regular decompositions of the complete graphs with prime power vertices

Momihara, Koji; Okumura, Yuki

doi:10.1016/j.ffa.2015.07.001

“…10.3]. Further cyclotomic constructions have been proposed, based on sophisticated number-theoretic techniques [62,63,65,67,68,69,71].…”

Section: Proposition 22 (Fourier Inversion Formula) Let G Be An Abelian Group and Letmentioning

confidence: 99%

“…The construction of partial difference sets is therefore of great interest. We refer to [13,60] for excellent surveys of partial difference sets and equivalent structures, and to [1,2,4,5,8,9,11,12,14,15,17,18,19,20,21,22,24,25,26,27,28,29,30,31,32,37,38,42,43,44,45,46,47,48,49,50,51,52,54,58,59,61,62,63,64,65,67,68,69,…”

Section: Introductionmentioning

confidence: 99%

Packings of partial difference sets

Jedwab¹,

Li²

2021

Combinatorial Theory

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A packing of partial difference sets is a collection of disjoint partial difference sets in a finite group G. This configuration has received considerable attention in design theory, finite geometry, coding theory, and graph theory over many years, although often only implicitly. We consider packings of certain Latin square type partial difference sets in abelian groups having identical parameters, the size of the collection being either the maximum possible or one smaller. We unify and extend numerous previous results in a common framework, recognizing that a particular subgroup reveals important structural information about the packing. Identifying this subgroup allows us to formulate a recursive lifting construction of packings in abelian groups of increasing exponent, as well as a product construction yielding packings in the direct product of the starting groups. We also study packings of certain negative Latin square type partial difference sets of maximum possible size in abelian groups, all but one of which have identical parameters, and show how to produce such collections using packings of Latin square type partial difference sets.

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“…Strongly regular decompositions of the complete graph have been investigated by Kharaghani et al [14], Momihara and Okumura [18], van Dam [22], among others. In the setting of Section 2, we now consider the case in which

Γ_{0}

is complete with srg parameters

(n = n_{1} + n_{2}, n - 1, n - 2, 0)

.…”

Section: Strongly Regular Decompositions Of the Complete Graphmentioning

confidence: 99%

On strongly regular designs admitting fusion to strongly regular decomposition

Sankey

¹

2021

J of Combinatorial Designs

0

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A strongly regular decomposition of a strongly regular graph is a partition of the vertex set into two parts on which the induced subgraphs are strongly regular, or cliques or cocliques. Strongly regular designs (srd's) as defined by D. G. Higman are coherent configurations of rank 10 with two fibers in which the homogeneous configuration on each fiber is a strongly regular graph. Haemers and Higman proved the equivalence between strongly regular decompositions, excluding special cases, and srd's with certain parameter conditions. Here we obtain this result by examining the srd's that admit a fusion to a strongly regular graph on the full vertex set. We derive equivalent conditions to Theorem 2.8 of Haemers and Higman by elementary methods. Incorporating recent works of Hanaki and Klin and Reichard, a table of feasible parameter sets for this class of srd's is presented along with a discussion of known constructions. In two cases, nonexistence is observed due to nonexistence of the strongly regular graph obtained through fusion. One of these is also ruled out by Hobart's generalized Krein conditions, applied to srd's. As strongly regular decompositions of the complete graph have sparked interest with recent papers we observe that in our situation this occurs only when the constituent graphs are also complete and the design is trivial.

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“…10.3]. Further cyclotomic constructions have been proposed based on sophisticated numbertheoretic techniques [62,63,65,67,68,69,71].…”

Section: Historical Overviewmentioning

confidence: 99%

“…The construction of partial difference sets is therefore of great interest. We refer to [13,60] for excellent surveys of partial difference sets and equivalent structures, and to [1,2,4,5,8,9,11,12,14,15,17,18,19,20,21,22,24,25,26,27,28,29,30,31,32,37,38,42,43,44,45,46,47,48,49,50,51,52,54,58,59,61,62,63,64,65,67,68,69,…”

Section: Introductionmentioning

confidence: 99%

Packings of partial difference sets

Jedwab¹,

Li²

2020

Preprint

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A packing of partial difference sets is a collection of disjoint partial difference sets in a finite group G. This configuration has received considerable attention in design theory, finite geometry, coding theory, and graph theory over many years, although often only implicitly. We consider packings of certain Latin square type partial difference sets in abelian groups having identical parameters, the size of the collection being either the maximum possible or one smaller. We unify and extend numerous previous results in a common framework, recognizing that a particular subgroup reveals important structural information about the packing. Identifying this subgroup allows us to formulate a recursive lifting construction of packings in abelian groups of increasing exponent, as well as a product construction yielding packings in the direct product of the starting groups. We also study packings of certain negative Latin square type partial difference sets of maximum possible size in abelian groups, all but one of which have identical parameters, and show how to produce such collections using packings of Latin square type partial difference sets.

show abstract

New strongly regular decompositions of the complete graphs with prime power vertices

Abstract: We construct two infinite families of strongly regular decompositions of complete graphs on finite fields, which consist of four negative Latin square type Cayley graphs. Our construction is based on the existence of cyclotomic strongly regular graphs.

Cited by 5 publications

References 18 publications

Packings of partial difference sets

Packings of partial difference sets

On strongly regular designs admitting fusion to strongly regular decomposition

Packings of partial difference sets

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