Graphs defined by orthogonality

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, 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 and its subsets consisting of diagonal, diagonalizable, triangularizable, and nilpotent matrices over an arbitrary field are computed.