# Orthogonality Graphs of Matrices Over Skew Fields

## Abstract

The paper is devoted to studying the orthogonality graph of the matrix ring over a skew field. It is shown that for n ≥ 3 and an arbitrary skew field 𝔻, the orthogonality graph of the ring M{sub n}(𝔻) of n × n matrices over a skew field 𝔻 is connected and has diameter 4. If n = 2, then the graph of the ring M{sub n}(𝔻) is a disjoint union of connected components of diameters 1 and 2. As a corollary, the corresponding results on the orthogonality graphs of simple Artinian rings are obtained.

- Authors:

- Lomonosov Moscow State University (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22773927

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Sciences

- Additional Journal Information:
- Journal Volume: 232; Journal Issue: 6; Other Information: Copyright (c) 2018 Springer Science+Business Media, LLC, part of Springer Nature; http://www.springer-ny.com; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1072-3374

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; FIELD THEORIES; GRAPH THEORY; MATRICES; RINGS

### Citation Formats

```
Guterman, A. E., E-mail: guterman@list.ru, and Markova, O. V., E-mail: ov-markova@mail.ru.
```*Orthogonality Graphs of Matrices Over Skew Fields*. United States: N. p., 2018.
Web. doi:10.1007/S10958-018-3909-7.

```
Guterman, A. E., E-mail: guterman@list.ru, & Markova, O. V., E-mail: ov-markova@mail.ru.
```*Orthogonality Graphs of Matrices Over Skew Fields*. United States. doi:10.1007/S10958-018-3909-7.

```
Guterman, A. E., E-mail: guterman@list.ru, and Markova, O. V., E-mail: ov-markova@mail.ru. Wed .
"Orthogonality Graphs of Matrices Over Skew Fields". United States. doi:10.1007/S10958-018-3909-7.
```

```
@article{osti_22773927,
```

title = {Orthogonality Graphs of Matrices Over Skew Fields},

author = {Guterman, A. E., E-mail: guterman@list.ru and Markova, O. V., E-mail: ov-markova@mail.ru},

abstractNote = {The paper is devoted to studying the orthogonality graph of the matrix ring over a skew field. It is shown that for n ≥ 3 and an arbitrary skew field 𝔻, the orthogonality graph of the ring M{sub n}(𝔻) of n × n matrices over a skew field 𝔻 is connected and has diameter 4. If n = 2, then the graph of the ring M{sub n}(𝔻) is a disjoint union of connected components of diameters 1 and 2. As a corollary, the corresponding results on the orthogonality graphs of simple Artinian rings are obtained.},

doi = {10.1007/S10958-018-3909-7},

journal = {Journal of Mathematical Sciences},

issn = {1072-3374},

number = 6,

volume = 232,

place = {United States},

year = {2018},

month = {8}

}

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