Minimum number of distinct eigenvalues of distance-regular and signed Johnson graphs
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915029803073536 |
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| author | Fallat, Shaun Gupta, Himanshu Herman, Allen Parenteau, Johnna |
| author_facet | Fallat, Shaun Gupta, Himanshu Herman, Allen Parenteau, Johnna |
| contents | We study the minimum number of distinct eigenvalues over a collection of matrices associated with a graph. Lower bounds are derived based on the existence or non-existence of certain cycle(s) in a graph. A key result proves that every Johnson graph has a signed variant with exactly two distinct eigenvalues. We also explore applications to weighing matrices, linear ternary codes, tight frames, and compute the minimum rank of Johnson graphs. Further results involve the minimum number of distinct eigenvalues for graphs in association schemes, distance-regular graphs, and Hamming graphs. We also draw some connections with simplicial complexes and higher-order Laplacians. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_00250 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimum number of distinct eigenvalues of distance-regular and signed Johnson graphs Fallat, Shaun Gupta, Himanshu Herman, Allen Parenteau, Johnna Combinatorics Primary 05C50, 05E30, Secondary 05C22, 15A18 We study the minimum number of distinct eigenvalues over a collection of matrices associated with a graph. Lower bounds are derived based on the existence or non-existence of certain cycle(s) in a graph. A key result proves that every Johnson graph has a signed variant with exactly two distinct eigenvalues. We also explore applications to weighing matrices, linear ternary codes, tight frames, and compute the minimum rank of Johnson graphs. Further results involve the minimum number of distinct eigenvalues for graphs in association schemes, distance-regular graphs, and Hamming graphs. We also draw some connections with simplicial complexes and higher-order Laplacians. |
| title | Minimum number of distinct eigenvalues of distance-regular and signed Johnson graphs |
| topic | Combinatorics Primary 05C50, 05E30, Secondary 05C22, 15A18 |
| url | https://arxiv.org/abs/2411.00250 |