The biological motivated problem that we want to solve in this paper is to predict the new members of a partially known protein complex (i.e. complex membership determination). In this problem, we are given a core set of proteins (i.e. the queries) making up a protein complex. However, the biologist experts do not know whether this core set is complete or not. Our objective is to find more potential members of the protein complex by ranking proteins in protein-protein interaction network. One of the solutions to this problem is a network reliability based method. Due to high time complexity of this method, the random walk on graphs method has been proposed to solve this complex membership determination problem. However, the random walk on graphs method is not the current state of the art network-based method solving bioinformatics problem. In this paper, the novel un-normalized graph (p-) Laplacian based ranking method will be developed based on the un-normalized graph p-Laplacian operator definitions such as the curvature operator of graph (i.e. the un-normalized graph 1-Laplacian operator) and will be used to solve the complex membership determination problem. The results from experiments shows that the un-normalized graph p-Laplacian ranking methods are at least as good as the current state of the art network-based ranking method (p=2) but often lead to better ranking accuracy performance measures.