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.