1 Introduction

Clusters of galaxies are the largest (partially) virialized structures in the Universe and the cosmological parameters may have a strong influence on their profile. The determination of the cluster density profile shape is a crucial question. A high value of $\Omega$ gives for example steeper asymptotic profiles in the simulations (i.e. Crone et al. 1994; Jing et al. 1995). It is then possible to recover the value of $\Omega$ with the shape of the clusters. We note however that other studies (see Navarro et al. 1995, 1996) argue that the dark matter profiles in clusters deduced from the CDM model are identical whatever the details of the model.

Similarly, it is important to know if the cluster profiles exhibit a cusp (e.g. Adami et al. 1998). After an analysis of a subsample of the ENACS clusters, Adami et al. (1998) conclude that clusters have a core if we consider the galaxies brighter than b_j = 20. The ENACS clusters (e.g. Katgert et al. 1996 or Mazure et al. 1996) obey the model of a relaxed system (e.g. King 1962). However, the bright galaxies ($B_j\leq -18.5$) are equally fitted by a profile with core or with cusp. It is then crucial to know how the shape of the galaxy distribution vary with the magnitude of the tested galaxies. Adami et al. (1998) have fitted different profiles with different shapes for the considered magnitudes. This method is very efficient and quantitative but also time consuming and complex according to the large number of parameters.

We develop here a new way to characterize the variation with magnitude of the aggregation of the galaxies in clusters, without any profile fitting. We use the Minimal Spanning Tree (or MST hereafter) which is common in astronomy to the study of the very large scale structures (e.g. Barrow et al. 1985; Graham et al. 1995; Bhavsar & Splinter 1996 or Krzewina & Saslaw 1996). It is also used in physic to study order and disorder of a given set of points (e.g. Dussert et al. 1986). We use here this last aspect to study the density profiles of clusters of galaxies by using only a bidimensionnal analysis. The first part of the article is about the MST theory and the calibration of the method. We apply the method in the second part to a subsample of 15 very rich and very regular clusters in order to calibrate the method. The last part is our conclusions.

We use H₀=100 km Mpc^-1 s^-1 and q₀=0.