Geodesic
Clique numbers of the total graphs of $2\times n$ and $3\times 3$ matrices
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXVII, Tome 534 (2024), pp. 128-146 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The total graph of the space of $m\times n$ matrices over a field $\mathbb F$ is the graph with vertex set $M_{m\times n}(\mathbb F)$ in which distinct matrices $A$ and $B$ are connected by an edge if and obly if rank$(A+B)< \min(m,n)$. It is proved that over a field of order $q$, where $q$ is a power of an odd prime, the clique number of the total graph of $2\times n$ matrices equals $q^n$, whereas that of $3\times 3$ matrices is $O(q^6)$. Up to now, this issue has only been examined for $2\times 2$ matrices. 
@article{ZNSL_2024_534_a5,
     author = {A. M. Maksaev and V. V. Promyslov and D. S. Sheshenya},
     title = {Clique numbers of the total graphs of $2\times n$ and $3\times 3$ matrices},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {128--146},
     year = {2024},
     volume = {534},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2024_534_a5/}
}
TY  - JOUR
AU  - A. M. Maksaev
AU  - V. V. Promyslov
AU  - D. S. Sheshenya
TI  - Clique numbers of the total graphs of $2\times n$ and $3\times 3$ matrices
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2024
SP  - 128
EP  - 146
VL  - 534
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2024_534_a5/
LA  - ru
ID  - ZNSL_2024_534_a5
ER  - 
%0 Journal Article
%A A. M. Maksaev
%A V. V. Promyslov
%A D. S. Sheshenya
%T Clique numbers of the total graphs of $2\times n$ and $3\times 3$ matrices
%J Zapiski Nauchnykh Seminarov POMI
%D 2024
%P 128-146
%V 534
%U http://geodesic.mathdoc.fr/item/ZNSL_2024_534_a5/
%G ru
%F ZNSL_2024_534_a5
A. M. Maksaev; V. V. Promyslov; D. S. Sheshenya. Clique numbers of the total graphs of $2\times n$ and $3\times 3$ matrices. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXXVII, Tome 534 (2024), pp. 128-146. http://geodesic.mathdoc.fr/item/ZNSL_2024_534_a5/

[1] B. R. Bakhadly, A. E. Guterman, O. V. Markova, “Grafy, opredelennye ortogonalnostyu”, Zap. nauchn. semin. POMI, 428, 2014, 49–80

[2] S. Akbari, H. Bidkhori, A. Mohammadian, “Commuting graphs of matrix algebras”, Commun. Algebra, 36:11 (2008), 4020–4031 | DOI | MR | Zbl

[3] S. Akbari, M. Jamaali, S.A. Seyed Fakhari, “The clique numbers of regular graphs of matrix algebras are finite”, Linear Algebra Appl., 431 (2009), 1715–1718 | DOI | MR | Zbl

[4] D. F. Anderson, A. Badawi, “The total graph of a commutative ring”, J. Algebra, 320 (2008), 2706–2719 | DOI | MR | Zbl

[5] D. F. Anderson, P. S. Livingston, “The zero-divisor graph of a commutative ring”, J. Algebra, 217:2 (1999), 434–447 | DOI | MR | Zbl

[6] I. Beck, “Coloring of commuatitive rings”, J. Algebra, 116:1 (1988), 208–226 | DOI | MR | Zbl

[7] K. Morrison, “Integer sequences and matrices over finite fields”, J. Integer Seq., 9 (2006), 06.2.1, 28 pp. | MR | Zbl

[8] J. Zhou, D. Wong, X. Ma, “Automorphism group of the total graph over a matrix ring”, Linear Multilinear Algebra, 65 (2017), 572–581 | DOI | MR | Zbl