Spectral gap and embedded trees for the Laplacian of the Erd\H{o}s-R\'enyi graph

TL;DR

This paper analyzes the spectral properties of Erdős-Rényi graphs, revealing that the smallest non-zero Laplacian eigenvalue is determined by embedded line subgraphs of size t, with high probability.

Contribution

It provides a precise asymptotic description of the smallest non-zero Laplacian eigenvalue in Erdős-Rényi graphs based on embedded line structures.

Findings

Smallest non-zero eigenvalue converges to a specific value related to line subgraphs.

Eigenvalue linked to a line of size t connected by a single edge.

Results hold with high probability for specified degree ranges.

Abstract

For the Erd\H{o}s-R\'enyi graph of size $N$ with mean degree $(1+o(1))\frac{\log N}{t+1}\leq d\leq(1-o(1))\frac{\log N}{t}$ where $t\in\mathbb{N}^{*}$, with high probability the smallest non zero eigenvalue of the Laplacian is equal to $2-2\cos(\pi(2t+1)^{-1})+o(1)$. This eigenvalue arises from a small subgraph isomorphic to a line of size $t$ linked to the giant connected component by only one edge.