  McKay proved that the limiting spectral measures of the ensembles of
$d$-regular graphs with $N$ vertices converge to Kesten's measure as
$N\to\infty$. In this paper we explore the case of weighted graphs. More
precisely, given a large $d$-regular graph we assign random weights, drawn from
some distribution $\mathcal{W}$, to its edges. We study the relationship
between $\mathcal{W}$ and the associated limiting spectral distribution
obtained by averaging over the weighted graphs. Among other results, we
establish the existence of a unique `eigendistribution', i.e., a weight
distribution $\mathcal{W}$ such that the associated limiting spectral
distribution is a rescaling of $\mathcal{W}$. Initial investigations suggested
that the eigendistribution was the semi-circle distribution, which by Wigner's
Law is the limiting spectral measure for real symmetric matrices. We prove this
is not the case, though the deviation between the eigendistribution and the
semi-circular density is small (the first seven moments agree, and the
difference in each higher moment is $O(1/d^2)$). Our analysis uses
combinatorial results about closed acyclic walks in large trees, which may be
of independent interest.
