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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| Vydáno v: | Journal of combinatorial designs Ročník 30; číslo 6; s. 420 - 460 |
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01.06.2022
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| ISSN: | 1063-8539, 1520-6610 |
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| Abstract | 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]. |
|---|---|
| AbstractList | 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 TDλ(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]. 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]. 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‐ design has a single concurrence , and its concurrence matrix is circulant, a partial geometry has two concurrences 1 and 0, and a transversal design has two concurrences 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]. |
| Author | Tranel, Theodore Song, Sung‐Yell |
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| Cites_doi | 10.1090/S0002-9947-1938-1501951-4 10.1007/978-3-642-74341-2 10.1006/jcta.1997.2796 10.4153/CJM-1970-067-9 10.4153/CJM-1954-005-4 10.1007/s10114-016-6151-6 10.1214/aoms/1177729027 10.1006/ffta.1997.0184 10.1002/jcd.21416 10.1007/s10623-015-0111-5 10.1016/0012-365X(76)90088-1 10.1080/01621459.1952.10501161 10.1007/s10623-017-0400-2 10.1016/0012-365X(76)90030-3 10.1007/s10623-019-00644-7 10.1016/0097-3165(88)90043-X 10.1002/jcd.21354 10.1016/j.laa.2004.09.015 10.1002/jcd.21518 10.1016/0021-8693(78)90220-X 10.1214/aoms/1177729382 10.1016/0012-365X(74)90068-5 10.1017/CBO9780511987045 10.1016/j.ejc.2012.01.011 10.1007/BF01111042 10.1017/CBO9780511623714 10.1017/CBO9780511549533 10.1016/0097-3165(80)90067-9 10.1016/j.disc.2018.06.008 10.1007/s00373-013-1364-2 10.1214/aoms/1177705806 10.1142/S1005386717000232 10.2140/pjm.1963.13.389 10.1007/BF01835977 10.1002/jcd.21767 |
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| Snippet | We classify small partial geometric designs (PGDs) by spectral characteristics of their concurrence matrices. It is well known that the concurrence matrix of a... |
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| SubjectTerms | 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 |
| Title | Partial geometric designs having circulant concurrence matrices |
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