Conditions for Singular Incidence Matrices

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Abstract

Suppose one looks for a square integral matrixN, for which NN has a prescribed form.Then the Hasse-Minkowski invariants and the determinant of NN lead to necessary conditions for existence.The Bruck-Ryser-Chowla theorem gives a famous example of such conditions in case N is the incidence matrix of a square block design.This approach fails when N is singular.In this paper it is shown that in some cases conditions can still be obtained if the kernels of N and N are known, or known to be rationally equivalent.This leads for example to non-existence conditions for selfdual generalised polygons, semi-regular square divisible designs and distance-regular graphs.
Original languageEnglish
Place of PublicationTilburg
PublisherOperations research
Number of pages6
Volume2003-66
Publication statusPublished - 2003

Publication series

NameCentER Discussion Paper
Volume2003-66

Fingerprint

Singular matrix
Incidence Matrix
Generalized Polygon
Distance-regular Graph
Block Design
Divisible
Nonexistence
Determinant
kernel
Necessary Conditions
Invariant
Theorem

Keywords

  • singularities
  • matrices
  • graphs

Cite this

Haemers, W. H. (2003). Conditions for Singular Incidence Matrices. (CentER Discussion Paper; Vol. 2003-66). Tilburg: Operations research.
Haemers, W.H. / Conditions for Singular Incidence Matrices. Tilburg : Operations research, 2003. (CentER Discussion Paper).
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Haemers, WH 2003 'Conditions for Singular Incidence Matrices' CentER Discussion Paper, vol. 2003-66, Operations research, Tilburg.

Conditions for Singular Incidence Matrices. / Haemers, W.H.

Tilburg : Operations research, 2003. (CentER Discussion Paper; Vol. 2003-66).

Research output: Working paperDiscussion paperOther research output

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AU - Haemers, W.H.

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N2 - Suppose one looks for a square integral matrixN, for which NN has a prescribed form.Then the Hasse-Minkowski invariants and the determinant of NN lead to necessary conditions for existence.The Bruck-Ryser-Chowla theorem gives a famous example of such conditions in case N is the incidence matrix of a square block design.This approach fails when N is singular.In this paper it is shown that in some cases conditions can still be obtained if the kernels of N and N are known, or known to be rationally equivalent.This leads for example to non-existence conditions for selfdual generalised polygons, semi-regular square divisible designs and distance-regular graphs.

AB - Suppose one looks for a square integral matrixN, for which NN has a prescribed form.Then the Hasse-Minkowski invariants and the determinant of NN lead to necessary conditions for existence.The Bruck-Ryser-Chowla theorem gives a famous example of such conditions in case N is the incidence matrix of a square block design.This approach fails when N is singular.In this paper it is shown that in some cases conditions can still be obtained if the kernels of N and N are known, or known to be rationally equivalent.This leads for example to non-existence conditions for selfdual generalised polygons, semi-regular square divisible designs and distance-regular graphs.

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KW - graphs

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Haemers WH. Conditions for Singular Incidence Matrices. Tilburg: Operations research. 2003. (CentER Discussion Paper).