The notion of graph generated by the mutual orthogonality relation for the elements of an associative ring is introduced. The main attention is paid to the commutative rings and to the matrix ring over a field and its various subrings and subsets. In particular, the diameters of the orthogonality graphs of the full matrix algebra over an arbitrary field and its subsets consisting of diagonal, diagonalizable, triangularizable, and nilpotent matrices are computed. Bibliography: 36 titles.

Abstract. We study the connectedness and the diameter of orthogonality graphs of upper triangular matrix algebras over arbitrary fields.Mathematics subject classification (2010): 15A24, 05C50.

Abstract. We study the connectedness and the diameter of orthogonality graphs of upper triangular matrix algebras over arbitrary fields.Mathematics subject classification (2010): 15A24, 05C50.

We study pairs of mutually orthogonal normal matrices with respect to tropical multiplication. Minimal orthogonal pairs are characterized. The diameter and girth of three graphs arising from the orthogonality equivalence relation are computed.

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