Orthogonality graphs of matrices over commutative rings
O. G. Styrt
TL;DR
This paper investigates the structure of orthogonality graphs of matrix rings over commutative rings, revealing their connectivity, diameter, and proximity of vertices to scalar matrices.
Contribution
It provides new criteria for the diameter of these graphs and characterizes the distance of vertices from scalar matrices over rings with zero-divisors.
Findings
Orthogonality graph is connected for matrices over rings with zero-divisors.
Diameter of the graph is either 3 or 4, with criteria distinguishing the cases.
Vertices are at most distance 2 from some scalar matrix.
Abstract
The paper is devoted to studying the orthogonality graph of the matrix ring over a commutative ring. It is proved that the orthogonality graph of the ring of matrices with size greater than 1 over a commutative ring with zero-divisors is connected and has diameter 3 or 4; a criterion for each value is obtained. It is also shown that each of its vertices has distance at most 2 from some scalar matrix.
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Taxonomy
Topicsgraph theory and CDMA systems · Enterprise Management and Information Systems · Intuitionistic Fuzzy Systems Applications