Narrow Results

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.

In this paper, we study the non-commuting graph $\mathrm{\Gamma }(M_3^{\mathrm{\text{su}}}(C_n))$ of strictly upper triangular $3\times 3$ matrices over an $n$-element chain $C_n$. We prove that $\mathrm{\Gamma }(M_3^{\mathrm{\text{su}}}(C_n))$ is a compact graph. From $\mathrm{\Gamma }(M_3^{\mathrm{\text{su}}}(C_n))$, we construct a poset $P$. We further prove that $P\oplus 1$ is a dismantlable lattice and its zero-divisor graph is isomorphic to $\mathrm{\Gamma }(M_3^{\mathrm{\text{su}}}(C_n))$. Lastly, we prove that $\mathrm{\Gamma }(M_3^{\mathrm{\text{su}}}(C_n))$ is a perfect graph.