Deza graphs: A generalization of strongly regular graph

Erickson, Martin; Fernando, S.; Haemers, Willem H.; Hardy, Darel W.; Hemmeter, J.

doi:10.1002/(sici)1520-6610(1999)7:6<395::aid-jcd1>3.0.co;2-u

Cited by 54 publications

(15 citation statements)

References 5 publications

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“…Deza graphs (see [8]) are k-regular graphs which are not strongly regular, and where the number of common neighbors of two distinct vertices takes just two values. So proper DDGs, which are not isomorphic to mK n or the complement, are Deza graphs.…”

Section: Definition 11mentioning

confidence: 99%

“…The DDGs of Construction 4.8 are improper whenever λ 1 = λ 2 , which is the case if and only if n = 4. For the two Hadamard matrices presented above, this leads to DDGs with parameters (24, 10,6,3,3,8) and (24, 6, 2, 1, 3, 8), respectively.…”

Section: Hadamard Matricesmentioning

confidence: 99%

“…Thus we can get two DDGs from one strongly regular graph with μ − λ = 1, unless the strongly regular graph is isomorphic to the complement (which is the case for the Paley graphs). For example the Petersen graph and its complement lead to DDGs with parameters (v, k, λ 1 , λ 2 , m, n) = (20, 7, 6, 2, 10, 2) and (20, 13,12,8,10,2), respectively. The pentagon, which is a strongly regular graph with parameters (5, 2, 0, 1), leads once more to the example of Fig.…”

Section: Construction 410mentioning

confidence: 99%

“…In particular the partial complement of the Klein graph is a DDG with parameters (24, 14,7,8,8,3). Taking partial complements often gives improper DDGs.…”

Section: Partial Complementsmentioning

confidence: 99%

“…By taking the union of five classes in this Hoffman coloring, we obtain an equitable partition into two parts of size 20. The partial complement with respect to this partition gives a DDG with parameters (40, 17,8,6,2,20).…”

Section: Partial Complementsmentioning

confidence: 99%

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Divisible design graphs

Haemers

Kharaghani

Meulenberg

2011

Journal of Combinatorial Theory, Series A

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A divisible design graph is a graph whose adjacency matrix is the incidence matrix of a divisible design. Divisible design graphs are a natural generalization of (v, k, λ)-graphs, and like (v, k, λ)-graphs they make a link between combinatorial design theory and algebraic graph theory. The study of divisible design graphs benefits from, and contributes to, both parts. Using information of the eigenvalues of the adjacency matrix, we obtain necessary conditions for existence. Old results of Bose and Connor on symmetric divisible designs give other conditions and information on the structure. Many constructions are given using various combinatorial structures, such as (v, k, λ)-graphs, distance-regular graphs, symmetric divisible designs, Hadamard matrices, and symmetric balanced generalized weighing matrices. Several divisible design graphs are characterized in terms of the parameters.

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Section: Definition 11mentioning

confidence: 99%

Section: Hadamard Matricesmentioning

confidence: 99%

Section: Construction 410mentioning

confidence: 99%

“…In particular the partial complement of the Klein graph is a DDG with parameters (24, 14,7,8,8,3). Taking partial complements often gives improper DDGs.…”

Section: Partial Complementsmentioning

confidence: 99%

Section: Partial Complementsmentioning

confidence: 99%

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Divisible design graphs

Haemers

Kharaghani

Meulenberg

2011

Journal of Combinatorial Theory, Series A

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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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