Spectral gap of $G(n, αn, α^2 n)$ graphs and the giant component theorem
M. Koshelev, A. Raigorodskii
TL;DR
The paper analyzes the spectrum of Johnson graphs in the perturbed regime $G(n,\alpha n+f_r(n), \alpha^2 n+f_s(n))$ and derives a sharp upper bound on the nontrivial eigenvalues in terms of $f_r(n)$ and $f_s(n)$: $\lambda\left(G\left(n,\alpha n+f_r(n), \alpha^2 n+f_s(n)\right)\right) = O\left( \frac{\max\{1, |f_s(n)-2\alpha f_r(n)|, f_r^2(n)/n\}}{n} d\left(G\left(n,\alpha n+f_r(n), \alpha^2 n+f_s(n)\right)\right)\right)$. The proof builds on a Kos–Eig lemma to bound $E_{r-s}(i)$ via combinatorial coefficients, followed by a trace argument on $A^2$ to relate eigenvalues to these bounds. This spectral control enables applying a general $(n,d,\lambda)$-graph giant-component framework, yielding a giant-component theorem for $G(n,\alpha n+f_r(n), \alpha^2 n+f_s(n))$: for $0<\alpha<1$ all components are small, and for $\alpha>1$ a single giant component of size $(1+o(1))(1-\overline{\alpha}/\alpha)N_n$ appears, with $N_n=\binom{n}{\alpha n+f_r(n)}$. Overall, the work extends spectral-gap techniques to a broad class of perturbed Johnson graphs and provides a concrete threshold for giant-component emergence in this family.
Abstract
The spectrum of a graph $G$ is the set of the eigenvalues of its adjacency matrix. It turns out that one can say a lot about a graph with the only knowledge being the spectrum of this graph. In this paper we obtain new results about the spectrum of $G(n, αn, α^2 n)$ graphs. We then apply these results to get a giant component theorem for them.