Geodesic
Orthogonality graphs of matrices over skew fields
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXX, Tome 463 (2017), pp. 81-93

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The paper is devoted to studying the orthogonality graph of the matrix ring over a skew field. It is shown that for $n\geq3$ the orthogonality graph of the $n\times n$ matrix ring $M_n(\mathbb D)$ over a skew field $\mathbb D$ is connected and has diameter $4$ for an arbitrary skew field $\mathbb D$. If $n=2$, then the graph of the ring $M_n(\mathbb D)$ is a disjoint union of connected components of diameters $1$ and $2$. As a corollary, we obtain related results on the orthogonality graphs of simple Artinian rings. 
@article{ZNSL_2017_463_a6,
     author = {A. E. Guterman and O. V. Markova},
     title = {Orthogonality graphs of matrices over skew fields},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {81--93},
     publisher = {mathdoc},
     volume = {463},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_463_a6/}
}
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A. E. Guterman; O. V. Markova. Orthogonality graphs of matrices over skew fields. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part XXX, Tome 463 (2017), pp. 81-93. http://geodesic.mathdoc.fr/item/ZNSL_2017_463_a6/