Necessary and sufficient conditions are provided for the existence of a simple graph, or a simple connected graph with given numbers $m_{ij}$ of edges with end-degrees $i,j$ for $i\leq j\in \left\{ 1,2,\ldots,\Delta \right\}$, where $\Delta$ is the maximum degree. Moreover this allows to determine the $k^{th}$ minimum or maximum value of all Adriatic indices together with the corresponding graphs.