Distributed computing network-systems are modeled as directed/undirected graphs with vertices representing compute elements and adjacency-edges capturing their uni- or bi-directional communication. To quantify an intuitive tradeoff between two graph-parameters: minimum vertex-degree and diameter of the underlying graph, we formulate an extremal problem with the two parameters: for all positive integers n and d, the extremal value \(\nabla (n, d)\) denotes the least minimum vertex-degree among all connected order-n graphs with diameters of at most d. We prove matching upper and lower bounds on the extremal values of \(\nabla (n, d)\) for various combinations of n- and d-values.