Partial geometric designs having circulant concurrence matrices

We classify small partial geometric designs (PGDs) by spectral characteristics of their concurrence matrices. It is well known that the concurrence matrix of a PGD can have at most three distinct eigenvalues, all of which are nonnegative integers. The matrix contains useful information on the incide...

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Bibliographic Details
Published in:Journal of combinatorial designs Vol. 30; no. 6; pp. 420 - 460
Main Authors: Song, Sung‐Yell, Tranel, Theodore
Format: Journal Article
Language:English
Published: Hoboken Wiley Subscription Services, Inc 01.06.2022
Subjects:
1 1 2 $1\frac{1}{2}$‐design
association scheme
Combinatorial analysis
Eigenvalues
Mathematical analysis
Matrices (mathematics)
partial geometric difference set
partial geometry
special partially balanced incomplete block design
strongly regular graph
t $t$‐design
ISSN:1063-8539, 1520-6610
Online Access:Get full text
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Summary:We classify small partial geometric designs (PGDs) by spectral characteristics of their concurrence matrices. It is well known that the concurrence matrix of a PGD can have at most three distinct eigenvalues, all of which are nonnegative integers. The matrix contains useful information on the incidence structure of the design. An ordinary 2‐ ( v , k , λ ) $(v,k,\lambda )$ design has a single concurrence λ $\lambda $, and its concurrence matrix is circulant, a partial geometry has two concurrences 1 and 0, and a transversal design TD λ ( k , u ) ${\text{TD}}_{\lambda }(k,u)$ has two concurrences λ $\lambda $ and 0, and its concurrence matrix is circulant. In this paper, we survey the known PGDs by highlighting their concurrences and constructions. Then we investigate which symmetric circulant matrices are realized as the concurrence matrices of PGDs. In particular, we try to give a list of all PGDs of order up to 12 each of which has a circulant concurrence matrix. We then describe these designs along with their combinatorial properties and constructions. This work is part of the second author's Ph.D. dissertation [46].
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ISSN:1063-8539
1520-6610
DOI:10.1002/jcd.21834