Total graph of a lattice

In this paper, we prove that the study of the subgraph \(T(Z^*(L))\) of the total graph T(L) of a lattice L is essentially the study of the zero-divisor graph of a poset. Also, we prove that the graph \(T^c(Z^*(L))\) is weakly perfect whereas \(T(Z^*(L))\) is not weakly perfect. The graph \(T(Z^*(L))\) and its complement \(T^c(Z^*(L))\) are shown to be a perfect graph if and only if L has at most four atoms. In the concluding section, we establish that, in the context of a commutative reduced ring R, the total graph, the annihilating ideal graph, the complement of the co-annihilating ideal graph, and the complement of the comaximal ideal graph coincide.