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Let $G$ be a finite solvable group and let $\Delta(G)$ denote the prime degree graph constructed on the set of primes associated to the set of irreducible character degree of $G$. It is well known that, whenever $\Delta(G)$ is connected, its diameter is at most $3$ and this is the best possible bound. In the present talk we will discuss some structural properties of such a group by outlining some main results that have been proved so far and by discussing some new results on such groups with special type of degree graphs.