Critical slope and leading-eigenfunction conjecture for inhomogeneous random graphs

Let

be an irreducible kernel on a ground space $(,)$, and let

be its associated integral operator with norm

. Let $c_0$ be the critical parameter, $\rho$ the giant-component fraction, and let equation (t5) denote the expansion referenced in the source. **Critical slope conjecture.** Equation (t5) holds with the larger error term $o$ whenever

has an eigenfunction

$$of eigenvalue$$

satisfying $<$; conversely, $=0$ if $>0$ but no such eigenfunction exists or an eigenfunction exists with $=$. This is presented as a further expected extension, with the rank-one case cited for comparison; the supplied text gives no resolution status.