Divisible design graphs

Haemers, Willem H.; Kharaghani, Hadi; Meulenberg, Maaike A.

doi:10.1016/j.jcta.2010.10.003

Cited by 41 publications

(71 citation statements)

References 15 publications

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“…The neighborhood design of a (v, k, λ)-graph is a symmetric (v, k, λ)-design. Divisible design graphs have been introduced in [4] as a generalization of (v, k, λ)-graphs. Definition 1.1 A k-regular graph on v vertices is a divisible design graph (DDG for short) with parameters (v, k, λ 1 , λ 2 , m, n) whenever the vertex set can be partitioned into m classes of size n, such that two distinct vertices from the same class have exactly λ 1 common neighbors, and two vertices from different classes have exactly λ 2 common neighbors.…”

Section: Divisible Design Graphsmentioning

confidence: 99%

“…For necessary proofs and more details we refer to [4]. As usual, we denote by I v and J v the v × v identity and all-ones matrix, respectively.…”

Section: The Adjacency and The Quotient Matrixmentioning

confidence: 99%

“…Bose [1] proved that the canonical partition of a symmetric divisible design is a tactical decomposition (also called: equitable partition) of its incidence matrix A, which means that each block A i,j has constant row and column sum (see [4] for a short proof). This enables us to define the matrix R = [r i,j ], where r i,j is the row (and column) sum of A i,j .…”

Section: The Adjacency and The Quotient Matrixmentioning

confidence: 99%

“…The first construction generalizes Construction 4.8 from [4]. Theorem 3.1 Let H be a regular graphical Hadamard matrix of order 4u 2 with diagonal entries −1 and row sum 2u (u can be negative), and let A be the adjacency matrix of an (n, k , λ)-graph.…”

Section: Constructionsmentioning

confidence: 99%

“…In a disconnected DDG λ 2 = 0 and each component is an (n, k, λ)-graph, or the incidence graph of a symmetric (n, k, λ)-design (see [4], Proposition 4.7). Such a graph is not walk-regular if both types of components are present, and walk-regular otherwise.…”

Section: Walk-regular Ddgsmentioning

confidence: 99%

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More About Divisible Design Graphs

Crnković

Haemers

2011

SSRN Journal

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Divisible design graphs (DDG for short) have been recently defined by Kharaghani, Meulenberg and the second author as a generalization of (v, k, λ)-graphs. In this paper we give some new constructions of DDGs, most of them using Hadamard matrices and (v, k, λ)-graphs. For three parameter sets we give a nonexistence proof. Furthermore, we find conditions for a DDG to be walk-regular. It follows that most of the known examples are walk-regular, but some are not. In case walk-regularity of a DDG is forced by the parameters, necessary conditions for walk-regularity lead to new nonexistence results for DDGs. We examine all feasible parameter sets for DDGs on at most 27 vertices, establish existence in all but one cases, and decide on existence of a walk-regular DDG in all cases.

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Section: Divisible Design Graphsmentioning

confidence: 99%

“…For necessary proofs and more details we refer to [4]. As usual, we denote by I v and J v the v × v identity and all-ones matrix, respectively.…”

Section: The Adjacency and The Quotient Matrixmentioning

confidence: 99%

Section: The Adjacency and The Quotient Matrixmentioning

confidence: 99%

Section: Constructionsmentioning

confidence: 99%

Section: Walk-regular Ddgsmentioning

confidence: 99%

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More About Divisible Design Graphs

Crnković

Haemers

2011

SSRN Journal

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Deza graphs with parameters and

Goryaĭnov

Haemers

Kabanov

et al. 2018

J of Combinatorial Designs

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A Deza graph with parameters ( n , k , b , a ) is a k‐regular graph with n vertices, in which any two vertices have a or b ( a ≤ b) common neighbours. A Deza graph is strictly Deza if it has diameter 2, and is not strongly regular. In an earlier paper, the two last authors et al characterised the strictly Deza graphs with b = k − 1 and β > 1, where β is the number of vertices with b common neighbours with a given vertex. Here, we start with a characterisation of Deza graphs (not necessarily strictly Deza graphs) with parameters ( n , k , k − 1 , 0 ). Then, we deal with the case β = 1 and a > 0, and thus complete the characterisation of Deza graphs with b = k − 1. It follows that all Deza graphs with b = k − 1, β = 1 and a > 0 can be made from special strongly regular graphs, and in fact are strictly Deza except for K 2. We present several examples of such strongly regular graphs. A divisible design graph (DDG) is a special Deza graph, and a Deza graph with β = 1 is a DDG. The present characterisation reveals an error in a paper on DDGs by the second author et al. We discuss the cause and the consequences of this mistake and give the required errata.

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