Incomparability graphs of dismantable lattices

We prove some characterizations of the incomparability graphs of some dismantlable lattices. We discuss the realizability of some standard graphs as a graph of a dismantlable lattice. We also find the minimum covering energy of the incomparability graph of some dismantlable lattices.

Perfect non-commuting graphs of matrices over chains

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.