A note on automorphisms of the zero-divisor graph of upper triangular matrices.
The concept of zero-divisor graph was first defined and studied for commutative rings by \textit{I. Beck} [J. Algebra 116, No. 1, 208-226 (1988; Zbl 0654.13001)], and further studied by many authors. Now let \(F_q\) be a finite field with \(q\) elements, \(n\) (\(\geq 3\)) a positive integer, \(T(n,q)\) the set of all \(n\times n\) upper triangular matrices over \(F_q\). In [\textit{D. Wong. X. Ma}, and \textit{J. Zhou}, Linear Algebra Appl. 460, 242-258 (2014; Zbl 1300.05187)], the zero-divisor graph of \(T(n,q)\), written as \(\mathcal T\), is defined to be a graph with all nonzero zero-divisors in \(T(n,q)\) as vertices, and there is a directed edge from a vertex \(X\) to a vertex \(Y\) if and only if \(XY=0\). The subgraph of \(\mathcal T\) induced by all rank one matrices in \(T(n,q)\) is denoted by \(\mathcal R\). Wong et al. [loc. cit.] determined the automorphisms of \(\mathcal R\) and left the automorphisms of \(\mathcal T\) unsolved. In this note, the author solves this problem.
scientific article; zbMATH DE number 6370307
scientific article