On the structure of the character degree graphs having diameter three

The structure of the character degree graphs $\mathrm{\Delta }(G)$, i.e. the prime graphs on the set $\mathrm{c}\mathrm{d}(G)$ of the irreducible character degrees of a finite group $G$, such that $G$ is solvable and $\mathrm{\Delta }(G)$ has diameter three, remains an intriguing area of study. However, a comprehensive understanding of these structures remains elusive. In this paper, we prove some properties and provide an infinite series of examples of this class of graphs, building on the ideas of Mark Lewis.