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Perfect state transfer on distance regular graphs
| Content Provider | Semantic Scholar |
|---|---|
| Author | Babai, László Cameron, Peter J. Conder, Marston D. E. Coutinho, Gabriel Guo, Krystal Godsil, Chris D. |
| Copyright Year | 2014 |
| Abstract | László Babai Symmetry versus regularity Abstract: Symmetry is usually defined in terms of automorphisms; regularity in terms of numerical parameters. Symmetry conditions imply various regularity constraints, but not conversely. We shall discuss two areas of this relationship. First we review some old results by Peter Frankl on how symmetry and regularity constraints affect extremal set systems. Our second topic involves recent results on how regularity constraints restrict the order and the structure of the automorphism group, especially for Steiner designs, strongly regular graphs, and coherent configurations. These results were found by various subsets of Xi Chen (Columbia U), Xiaorui Sun (Columbia U), Shang-Hua Teng (U Southern California), John Wilmes (U Chicago), and the speaker. Symmetry is usually defined in terms of automorphisms; regularity in terms of numerical parameters. Symmetry conditions imply various regularity constraints, but not conversely. We shall discuss two areas of this relationship. First we review some old results by Peter Frankl on how symmetry and regularity constraints affect extremal set systems. Our second topic involves recent results on how regularity constraints restrict the order and the structure of the automorphism group, especially for Steiner designs, strongly regular graphs, and coherent configurations. These results were found by various subsets of Xi Chen (Columbia U), Xiaorui Sun (Columbia U), Shang-Hua Teng (U Southern California), John Wilmes (U Chicago), and the speaker. Eiichi Bannai On tight relative t-designs in association schemes Abstract: The concept of relative t-designs in association schemes was introduced by Delsarte in his paper: Pairs of vectors in the space of an association scheme (1977). This concept, in a sense, predicted the concept of Euclidean t-designs, introduced later by Neumaier and Seidel (1988). However, it seems that the study of Euclidean t-designs has preceded the study of relative t-designs in association schemes, and that the study of the latter has just started recently in a way modeling the study of Euclidean t-designs. In this talk we first review the similarities between the studies on ”spherical tdesigns and Euclidean t-designs” and on ”t-designs and relative t-designs in association schemes”, putting the emphasis on the study of tight t-designs in each situation. The purpose of this talk is to try to convince the reader that we should study the classification problems of tight relative t-designs in association schemes in a systematic way. More details will be available in the following papers initiating the study in this direction. [1] E. Bannai, E. Bannai, S. Suda, and H. Tanaka: On relative t-designs in polynomial association schemes, preprint, arXiv:1303.7163. [2] E. Bannai, E. Bannai, and H. Bannai: On the existence of tight relative 2-designs on binary Hamming association schemes, Discrete Mathematics 314 (2014), 17–37. [3] Z. Xiang: A Fisher type inequality for weighted regular t-wise balanced designs, J. Combin. Theory Ser. A 119 (2012), 1523–1527. [4] Y. Zhu, E. Bannai, and E. Bannai: On tight relative 2-designs on the Johnson association schemes (a tentative title, in preparation). The concept of relative t-designs in association schemes was introduced by Delsarte in his paper: Pairs of vectors in the space of an association scheme (1977). This concept, in a sense, predicted the concept of Euclidean t-designs, introduced later by Neumaier and Seidel (1988). However, it seems that the study of Euclidean t-designs has preceded the study of relative t-designs in association schemes, and that the study of the latter has just started recently in a way modeling the study of Euclidean t-designs. In this talk we first review the similarities between the studies on ”spherical tdesigns and Euclidean t-designs” and on ”t-designs and relative t-designs in association schemes”, putting the emphasis on the study of tight t-designs in each situation. The purpose of this talk is to try to convince the reader that we should study the classification problems of tight relative t-designs in association schemes in a systematic way. More details will be available in the following papers initiating the study in this direction. [1] E. Bannai, E. Bannai, S. Suda, and H. Tanaka: On relative t-designs in polynomial association schemes, preprint, arXiv:1303.7163. [2] E. Bannai, E. Bannai, and H. Bannai: On the existence of tight relative 2-designs on binary Hamming association schemes, Discrete Mathematics 314 (2014), 17–37. [3] Z. Xiang: A Fisher type inequality for weighted regular t-wise balanced designs, J. Combin. Theory Ser. A 119 (2012), 1523–1527. [4] Y. Zhu, E. Bannai, and E. Bannai: On tight relative 2-designs on the Johnson association schemes (a tentative title, in preparation). Peter Cameron Endomorphisms and synchronization Abstract: There are two natural correspondences between graph endomorphisms and transformation semigroups. One way round, every graph has a semigroup of endomorphisms, which acts on the set of vertices of the graph. The other way, from a transformation semigroup S we can build a graph Gr(S), having various nice properties: it is complete if and only if S is a permutation group, and null if and only if S is synchronizing; and it has clique number equal to chromatic number. These two correspondences There are two natural correspondences between graph endomorphisms and transformation semigroups. One way round, every graph has a semigroup of endomorphisms, which acts on the set of vertices of the graph. The other way, from a transformation semigroup S we can build a graph Gr(S), having various nice properties: it is complete if and only if S is a permutation group, and null if and only if S is synchronizing; and it has clique number equal to chromatic number. These two correspondences |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.fields.utoronto.ca/programs/scientific/13-14/godsil65/abstracts.pdf |
| Alternate Webpage(s) | https://www.fields.utoronto.ca/programs/scientific/13-14/godsil65/abstracts.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
