In this talk, a new distance to compare two large-scale graphs based on the theory of quantum probability is introduced. Our proposed distance between two graphs is defined as the distance between the corresponding moment matri- ces of their spectral distributions. An explicit form for the spectral distribution of the corresponding adjacency matrix of a graph is established. It is shown that the spectral distributions of their adjacency matrices in a vector state includes information not only about their eigenvalues, but also about the corresponding eigenvectors. Also, such distance includes enough information of the structure of a graph. Moreover, we prove that such distance is graph invariant and sub- structure invariant. Various examples are given, and distances between graphs with few vertices are checked. Computational results for real large-scale net- works show that its accuracy is better than any existing methods and time cost is extensively cheap.