dimensional) lattices with impurities, the corresponding effective field theory and its upper critical dimension are not known to us.

In section \[sec:applis\], we give two applications of our results. We show in particular in section \[sec:loc\] that for any $\alpha$ the core of a random graph only carries a small number of 0 eigenvalues of the adjacency matrix and that the emergence of the core has a direct impact on the localized and delocalized eigenvectors with eigenvalue 0. In section \[sec:guard\], we show that for $\alpha < e$, the problem of finding minimal vertex covers or maximal edge disjoint subsets (matchings) can be handled very simply in polynomial time (in fact, in linear time once the graph is encoded in a suitable form). While the matching problem can always be solved in polynomial time, the minimal vertex cover problem is believed to be NP-hard for general graphs, and the same should be true on the core of a random graph for $\alpha > e$.

The formal proof that the core is a well-defined object is given in appendix.

General definitions {#sec:defs}
===================

We start with a few standard definitions. This section should be used only for reference.

**Graph**: A graph 