  For a finite group $G$, let $\Delta(G)$ denote the character graph built on
the set of degrees of the irreducible complex characters of $G$. In this paper,
we show that if the diameter of $\Delta(G)$ is equal to three, then the
complement of $\Delta(G)$ is bipartite. Also in this case, we determine the
structure of the character graph $\Delta(G)$.
