co-identity join graph of lattices

Let $pounds$ be a lattice with $1$ and $0$. the co-identity join graph of $pounds$, denoted by $mathbb{cg} (pounds)$, is an undirected simple graph whose vertices are all nontrivial elements (i.e. different from $1$ and $0$) of $pounds$ and two distinct elements $x$ and $y$ are adjacent if and only if $x vee y neq 1$. the basic properties and possible structures of this graph are studied and the interplay between the algebraic properties of $pounds$ and the graph-theoretic structure of $mathbb{cg} (pounds)$ is investigated.