Perfect non-commuting graphs of matrices over chains

Abstract

In this paper, we study the non-commuting graph [Formula: see text] of strictly upper triangular [Formula: see text] matrices over an [Formula: see text]-element chain [Formula: see text]. We prove that [Formula: see text] is a compact graph. From [Formula: see text], we construct a poset [Formula: see text]. We further prove that [Formula: see text] is a dismantlable lattice and its zero-divisor graph is isomorphic to [Formula: see text]. Lastly, we prove that [Formula: see text] is a perfect graph.