Let $G$ be a connected graph with $n$ vertices, and the degree of its vertices are $d_1,…,d_n$. I want to find an upper bound on the diameter of $G$, using only $n,d_1,…,d_n$.
One result I know is $$D(G)\leq1+\frac{\cosh^{-1}(n-1)}{\cosh^{-1}(\frac{\lambda_n+\lambda_2}{\lambda_n-\lambda_2})}$$ where $\lambda_n\geq…\geq\lambda_2\geq0$ are eigenvalues of the graph Laplacian, from "An upper bound on the diameter of a graph from eigenvalues associated with its Laplacian." But this bound involves information on the "adjacency" of vertices.
Best Answer
Have a look at this paper, there they claim to have a lot of bounds, based on the degree sequence.
https://www.sciencedirect.com/science/article/pii/S0893965911003739
For example they have the bound $diam(G)\leq n-t+1 $, where $n$ is the number of vertices, and $t$ is the size of degree sequence as a set, i.e. the number of different degrees.
Overall, there are bounds in terms of minimal, maximal degree, and it seems that the problem of determining sharp bounds is hard.