Graph-theoretical approach for quantifying galaxy distributions in a cold dark matter universe

We use a graph theory for quantifying galaxy distributions in the cold dark matter (CDM) universe. Cosmological N-body simulations with CDM spectra are performed, and a constellation graph is constructed from these simulations. We apply graph theory to these constellation graphs and calculate the distribution functions of the eigenvalues of the adjacency matrices. In addition to a three-dimensional analysis, a two-dimensional analysis, an analysis in a slicelike geometry, and a redshift space analysis are also carried out. From our analyses, we find that the kurtosis and the average deviation of the distribution function of the eigenvalues are useful statistical measures for quantifying the galaxy distributions. We also find that the graph-theoretical approach possesses a discriminative ability with regard to the two-dimensional galaxy distributions. The slicelike geometry, which covers a rather narrow region of the sky, is not sufficient for the graph theoretical analysis. However, we find that the discriminative ability of the graph theory is recovered in redshift space.