Spectra of large diluted but bushy random graphs

Nathanael Enriquez (1, 2) , Laurent Menard (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 MODAL'X - Modélisation aléatoire de Paris X

We compute an asymptotic expansion in $1/c$ of the limit in $n$ of the empirical spectral measure of the adjacency matrix of an Erd\H{o}s-Rényi random graph with $n$ vertices and parameter $c/n$. We present two different methods, one of which is valid for the more general setting of locally tree-like graphs. The second order in the expansion gives some information about the edge of the spectrum and leads to a conjecture about the second largest eigenvalue.

Mots clés: random matrices

Mots clés: random graphs; random trees; adjacency matrix; random matrices.

Domaines: Probabilités [math.PR]

Nathanael Enriquez, Laurent Menard. Spectra of large diluted but bushy random graphs. 2013. ⟨hal-00878739v1⟩