Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2605.14190 |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917493854961664 |
|---|---|
| author | Atkins, Eli |
| author_facet | Atkins, Eli |
| contents | We describe a general method for constructing representations of finite integral symmetric relation algebras from distance-regular graphs. Given a distance-regular graph of diameter $d$, the distances between vertices induces a coloring of the complete graph with $d$ colors, and we show that this coloring yields a representation of finite integral symmetric relation algebra on $d+1$ atoms. We then introduce a necessary and sufficient condition for when such a representation is algebraic, proving that this occurs if and only if the distance-regular graph is also distance-transitive.
We study the diameter-3 case of this method in detail, and we express a condition for the representation's mandatory cycles in terms of the distance-regular graph's intersection array. We apply this result to give a positive answer to an open question of Roger Maddux; namely, whether the relation algebra $30_{65}$ has a representation on a finite set. The representation is given on 42 points, and arises from the second subconstituent of the Hoffman-Singleton graph. We further use this method to describe an infinite class of finite representations of $26_{65}$ and the smallest possible representation of $31_{65}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14190 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Relation Algebra Representations from Distance-Regular Graphs Atkins, Eli Combinatorics Logic 05C12, 03G15, 05E30 We describe a general method for constructing representations of finite integral symmetric relation algebras from distance-regular graphs. Given a distance-regular graph of diameter $d$, the distances between vertices induces a coloring of the complete graph with $d$ colors, and we show that this coloring yields a representation of finite integral symmetric relation algebra on $d+1$ atoms. We then introduce a necessary and sufficient condition for when such a representation is algebraic, proving that this occurs if and only if the distance-regular graph is also distance-transitive. We study the diameter-3 case of this method in detail, and we express a condition for the representation's mandatory cycles in terms of the distance-regular graph's intersection array. We apply this result to give a positive answer to an open question of Roger Maddux; namely, whether the relation algebra $30_{65}$ has a representation on a finite set. The representation is given on 42 points, and arises from the second subconstituent of the Hoffman-Singleton graph. We further use this method to describe an infinite class of finite representations of $26_{65}$ and the smallest possible representation of $31_{65}$. |
| title | Relation Algebra Representations from Distance-Regular Graphs |
| topic | Combinatorics Logic 05C12, 03G15, 05E30 |
| url | https://arxiv.org/abs/2605.14190 |