Friedman spectral expansion in the uniform simple model

Fix $d\ge 3$ and $\epsilon >0$. For a uniformly sampled labelled simple $d$-regular graph on $n$ vertices, $\mathbb{P}(\max \{{\lambda }_2(A),\mathrm{\mid }{\lambda }_n(A)\mathrm{\mid }\}\ge 2\sqrt{d-1}+\epsilon )\to 0$ as $n\to \mathrm{\infty }$ along $nd$ even. This includes all nonconstant eigenvalues with multiplicity. It gives a fixed-degree, asymptotic bound, not an exact Ramanujan conclusion.

Here $d$ is an integer, $\epsilon$ is fixed before the limit, and $n\ge d+1$. The eigenvalues of the adjacency matrix $A$ are ordered $d={\lambda }_1\ge \cdots \ge {\lambda }_n$ with multiplicity. The finite graph space exists by Feasible labelled simple regular graph orders, and its relation to conditional pairings is the exact event identity in Simple-model spectral events as conditional pairing counts. The spectral maximum is ${\rho }_0(A)=\mathrm{\parallel }A-(d\mathrm{/}n)J{\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}$ by Ordered eigenvalues and the centered regular norm. In particular extra eigenvalues $d$ from disconnection and $-d$ from a bipartite component are retained, as made explicit in Disconnected and bipartite spectral witnesses. The equivalent normalized threshold is $2\sqrt{d-1}\mathrm{/}d+\epsilon \mathrm{/}d$ by Exact normalized spectral threshold transfer, with all asymptotic quantifiers as in Asymptotic quantifiers and uniform parameter ranges.

Bordenave, Section 1, Theorem 1 (PDF/printed p. 2), explicitly imports the outstanding probabilistic estimate for a uniform member of ${\mathcal{G}}_d(n)$ stated above. The preceding definition on that page specifies labelled simple graphs. Its ordered eigenvalues are denoted ${\mu }_i$ there; replacing that letter by ${\lambda }_i$ changes no hypothesis or conclusion. The five local intermediate items establish feasibility, ordered deflation, obstruction witnesses, normalization, and exact event/model encoding before this import. They do not prove the probability bound.

Application is direct: the graph law is precisely the uniform simple law by Simple-model spectral events as conditional pairing counts, the spectral event is exactly the source's ordered event by Ordered eigenvalues and the centered regular norm, and the limits keep $d$ and $\epsilon$ fixed by Asymptotic quantifiers and uniform parameter ranges. Thus Theorem 1 applies without an additional connectedness condition, a permutation-model identification, or any claimed local reconstruction of its proof. The assertion about exceptional eigenvalues follows from Disconnected and bipartite spectral witnesses, and the normalized version follows from Exact normalized spectral threshold transfer. The exact source contract and actual reading limits are recorded in the reference metadata.