An isolated singularity x of a complex surface (X,\({\mathcal O}_ x)\) is \textit{perfect}, or \textit{homological trivial}, if the local fundamental group \(\pi_ 1(\partial U_ x)\) is a perfect group, where \(U_ x\) is a contractible neighborhood of x in X. A graph \(\Gamma\) is called \textit{perfect} if there exist integer weights \(n_ i\) on the vertices of \(\Gamma\) for which \(\Gamma (n_ 1,...,n_ k)\) is the weighted dual intersection graph of the minimal resolution of a perfect surface singularity whose minimal resolution is normal. In this paper we use techniques for graphical evaluation of determinants to characterize most kinds of perfect graphs and to relate this problem to Diophantine questions involving partial fraction representations of integers. These questions, in turn, have independent interest in number theory, involving techniques of continued fractions and Egyptian fractions.