# Divisible design graphs

W.H. Haemers, H. Kharaghani, M.A. Meulenberg

Research output: Contribution to journalArticleScientificpeer-review

### Abstract

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.
Original language English 978-992 Journal of Combinatorial Theory, Series A, Structures designs and application combinatorics 118 3 https://doi.org/10.1016/j.jcta.2010.10.003 Published - 2011

### Fingerprint

Graph Design
Divisible
Balanced Generalized Weighing Matrix
Graph in graph theory
Graph Distance
Combinatorial Design
Distance-regular Graph
Incidence Matrix
Graph theory
Weighing
Eigenvalue
Necessary Conditions

### Cite this

@article{64fc67a03ae8426eae68ff747d803885,
title = "Divisible design graphs",
abstract = "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.",
author = "W.H. Haemers and H. Kharaghani and M.A. Meulenberg",
note = "Appeared earlier as CentER DP 2010-19",
year = "2011",
doi = "10.1016/j.jcta.2010.10.003",
language = "English",
volume = "118",
pages = "978--992",
journal = "Journal of Combinatorial Theory, Series A, Structures designs and application combinatorics",
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Divisible design graphs. / Haemers, W.H.; Kharaghani, H.; Meulenberg, M.A.

In: Journal of Combinatorial Theory, Series A, Structures designs and application combinatorics, Vol. 118, No. 3, 2011, p. 978-992.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Divisible design graphs

AU - Haemers, W.H.

AU - Kharaghani, H.

AU - Meulenberg, M.A.

N1 - Appeared earlier as CentER DP 2010-19

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

U2 - 10.1016/j.jcta.2010.10.003

DO - 10.1016/j.jcta.2010.10.003

M3 - Article

VL - 118

SP - 978

EP - 992

JO - Journal of Combinatorial Theory, Series A, Structures designs and application combinatorics

JF - Journal of Combinatorial Theory, Series A, Structures designs and application combinatorics

SN - 0097-3165

IS - 3

ER -